Managerial Decision Modeling with Spreadsheets 3rd Edition

Answer: C Page Ref: 2 Topic: Types of Decision Models Difficulty: Moderate Answer: C Page Ref: 7 Topic: Steps Involved in Decision Modeling Difficulty: Easy Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall 1 Answer: E Page Ref: 2 Topic:

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall 12) Consider the following PERT network, along with the completion times for each activity, in days. Use Crystal Ball to enumerate the completion times of all paths in the network. Typically, the longest path in the network dictates

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 10 Simulation Modeling 10.1 Chapter Questions 1) John Smith is planning to refinance his home mortgage to take advantage of the lower current interest rates.

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 11 Forecasting Models 11.1 Chapter Questions 1) Consider the following data that was fitted using a Linear Trend. Period Actual value (or) Y Period

33 22) Refer to the table above. Forecast next quarterly sales using a 3-period weighted moving average. Use Solver to determine the optimal weights that minimize MSE. Answer: Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Copyright (c) 2013 Pearson

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall 11) The following data were collected on the height (inches) and weight (pounds) of 5 students. Height 72 70 62 65 67 Weight 180 172 125 132 145 a. Develop a regression model to predict weight based on

Answer: C Page Ref: 12-6 Topic: Economic Order Quantity: Determining How Much to Order Difficulty: Moderate AACSB: Analytic Skills Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall 1 Answer: A Page Ref: 12-6 Topic: Economic Order Quantity: Determining How Much to Order

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 2 Linear Programming Models: Graphical and Computer Methods 2.1 Chapter Questions 1) Consider the following linear programming model: Max X12 + X2 +

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall 5) A computer retail store sells two types of flat screen monitors: 17 inches and 19 inches, with a profit contribution of $300 and $250, respectively. The monitors are ordered each week from an outside supplier. As an

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall 9) Consider the following data that illustrates demand, production costs, and expected capacity for a given product. Month 1 2 3 4 Demand (units) 100 125 200

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 3 Linear Programming Modeling Applications with Computer Analyses in Excel 3.1 Chapter Questions 1) In a multi-period production scheduling application, the decision variables typically designate:

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Answer: E Page Ref: 163 Topic: Introduction Difficulty: Easy 1 Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 5 Transportation, Assignment, and Network Models 5.1 Chapter Questions 1) Which of

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall 7) A company produces four products each of which must undergo assembly, inspection, and packaging. The information below summarizes the hours required for each operation by each product, the total hourly capacity for each operation, and the fixed cost

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 6 Integer, Goal, and Nonlinear Programming Models 6.1 Chapter Questions 1) These are problems in which some, but not all, the decision variables must have

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 7 Project Management 7.1 Chapter Questions Use this information to answer the following questions. Consider the following Activity-On-Node (AON) project (activity completion times are

14 6) Refer to the information above. Assume that that time required to complete each activity is as follows. Activity Time (Days) A 3 B 4 C 5 D 3 E 2

1 Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 8 Decision Analysis 8.1 Chapter Questions 1) Determining the best payoff for each alternative and choosing the alternative with the "best of the best" is the approach called: A) maximax B) maximin C) Laplace

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall 9) Refer to the information above. Assume that the bakery (see problem 7) has obtained the following probability information regarding demand for the pies: P(50) = 0.3, P(100) = 0.5, and P(150) = 0.2. a. Develop a decision

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Use this information to answer the following questions. A small suburban post office is frequented by customers at the rate of 15 per hour. The post office has a single employee who typically processes each customer's request at

Copyright (c) 2013 Pearson Education, Inc. publishing as Prentice Hall Managerial Decision Modeling w/ Spreadsheets, 3e (Balakrishnan/Render/Stair) Chapter 9 Queuing Models 9.1 Chapter Questions 1) Reneging refers to customers who: A) do not join a queue B) switch queues C) join a queue

exclusive events can be seen in flipping a coin. The outcome of any one trial can either be a head or a tail. Thus, the events of getting a head and a tail are mutually exclusive because only one of these events can occur on any one trial. This assumes,

(b) The probability of getting a 5-inch nail is = 0.19. (c) The probability of getting a nail 3 inches or shorter is the probability of getting a nail 1 inch, 2 inches, or 3 inches in length (the events are mutually exclusive). The probability is thus P(1 or 2

manufacturing and service sectors. For example, deciding how many sections of a course to offer during a semester can be modeled as a deterministic model since the costs and benefits of offering each section are known. Probabilistic (also called stochastic) models assume that

Problem 1-21 Refurbish old equipment Make major modifications Purchase new equipment Known parameters: Known parameters: Known parameters: Selling price per unit $10.00 Selling price per unit $10.00 Selling price per unit $10.00 Fixed cost $70,000.00 Fixed cost $135,000.00 Fixed cost $230,000.00 Variable cost per unit $7.20 Variable cost per unit $5.25 Variable cost per unit

Problem 1-23 Students Tuition Total Members 960 $8,000 $7,680,000 Non-members 240 $10,500 $2,520,000 Tuition Dollars $10,200,000 Budget $15,000,000 Church appropriation $4,800,000 Tuition funds needed $10,200,000 Tuition for non-member students would be $10,500.

Problem 1-25 Known parameters: Selling price per unit $6.00 Fixed cost $23,000.00 Variable cost per unit $3.82 Input Data Number of units, X 10550 Results Total revenue $63,302.75 Fixed cost $23,000.00 Total variable cost $40,302.75 Total cost $63,302.75 Profit $0.00

Problem 1-27 With current equipment With new equipment Known parameters: Known parameters: Selling price per unit $1.10 Selling price per unit $1.20 Fixed cost $15,500.00 Fixed cost $23,500.00 Variable cost per unit $0.60 Variable cost per unit $0.70 Input Data Input Data Number of units, X 32000 Number of units, X 50000 Results Results Total revenue $35,200.00 Total revenue

Problem 1-29 Machine A Machine B Known parameters: Known parameters: Selling price per unit $28.00 Selling price per unit $28.00 Fixed cost $75,000.00 Fixed cost $62,000.00 Variable cost per unit $16.00 Variable cost per unit $20.00 Input Data Input Data Number of units, X 3250 Number of units, X 3250 Results Results Total revenue $91,000.00 Total revenue $91,000.00 Fixed cost

Problem 1-19 Known parameters: Selling price per unit $0.06 Fixed cost $15,000.00 Variable cost per unit $0.02 Input Data Number of units, X 375000 Results Total revenue $22,500.00 Fixed cost $15,000.00 Total variable cost $7,500.00 Total cost $22,500.00 Profit $0.00

Problem 1-20 With current equipment With new equipment Known parameters: Known parameters: Selling price per unit $1.10 Selling price per unit $1.10 Fixed cost $15,500.00 Fixed cost $23,500.00 Variable cost per unit $0.60 Variable cost per unit $0.70 Input Data Input Data Number of units, X 32000 Number of units, X 50000 Results Results Total revenue $35,200.00 Total revenue

Problem 1-22 Students Tuition 1,200 $8,500 Tuition Dollars $10,200,000 Budget $15,000,000 Church appropriation $4,800,000 Tuition funds needed $10,200,000 The new tuition will have to be $8,500. Hence, tuition will have to be increased by $500.

Problem 1-24 Students Tuition Total Members 900.00 $8,000 $7,200,000 Non-members 333.33 $9,000 $3,000,000 Tuition Dollars $10,200,000 Budget $15,000,000 Church appropriation $4,800,000 Tuition funds needed $10,200,000 Recruit 34 more non-member students, for a total of 334.

Problem 1-26 Without salary With salary Known parameters: Known parameters: Selling price per unit $34.00 Selling price per unit $34.00 Fixed cost $190,000.00 Fixed cost $230,000.00 Variable cost per unit $18.00 Variable cost per unit $18.00 Input Data Input Data Number of units, X 11875 Number of units, X 14375 Results Results Total revenue $403,750.00 Total revenue $488,750.00 Fixed cost

Problem 1-28 Manual process Mechanized process Known parameters: Known parameters: Selling price per unit $2.50 Selling price per unit $2.50 Fixed cost $43,400.00 Fixed cost $84,600.00 Variable cost per unit $1.80 Variable cost per unit $1.30 Input Data Input Data Number of units, X 62000 Number of units, X 70500 Results Results Total revenue $155,000.00 Total revenue $176,250.00 Fixed cost

Problem 1-30 Proposal A Proposal B Known parameters: Known parameters: Selling price per unit $18.00 Selling price per unit $18.00 Fixed cost $65,000.00 Fixed cost $34,000.00 Variable cost per unit $10.00 Variable cost per unit $14.00 Input Data Input Data Number of units, X 8125 Number of units, X 8500 Results Results Total revenue $146,250.00 Total revenue $153,000.00 Fixed cost

Abjar Transport Company Data Table Calculations set to Manual. Press F9 to recalculate simulation values. 29 820 30 820 31 514 32 822 33 10 26 34 923 35 821 36 719 37 821 38 11 29 39 821 40 924 41 718 42 923 Monthly freight volume to be handled: Run C Rigs NC Rigs 21 820 22 10 26 23 822 24 925 25 10 27 26 924 27 822 28

Data Table Calculations set to Manual. Press F9 to recalculate. Custo- mer # Time between calls Call arrival time Call start service Service time Call end service Wait time Agent idle time RN interval lower limit Time Proba- bility Run Avg wait time Max wait time Idle time % 0 0 0 0 0 0 0 0 18.8 28 6.7% 12 342 45 449 3 0 0.39 30.18 13

Biales Data Table Calculations set to Manual. Press F9 to recalculate. Trip No. of barrels Rev per barrel Run Profit 143 60 RN intvl LL Barrels Probability 1 -6220 243 60 0.00 28 0.12 2 -3970 748 60 7 -9330 843 60 8 -9100 938 60 RN intvl LL Revenue Probability 9 -2900 11 33 60 0.20 60 0.44 11 -9220 12 43 70 0.64

Data Table Calculations set to Manual. Press F9 to recalculate. Production per day (tons) 100 Number of failures 1 Run Cost Days lost Profit per ton $12 RN intvl LL Failures Probability 1 $28,800 24 Average production days lost 26.5 0.970 6 0.010 6 N/A 8 $75,600 63 Average cost per month $31,842 0.980 7

CHAPTER 10 SIMULATION MODELING Internet Case Study: Biales Waste Disposal, GmbH Each trip out of Italy costs DM 900 for shipment and another DM 120 for loading and unloading. If there are 25 trips, the annual cost to operate the Italy office is (DM 1,020 x 25

CHAPTER 10 SIMULATION MODELING Internet Case Study: Buffalo Alkali and Plastics To analyze this problem, we develop two simulation models. The first model, which is shown in file Ch10 Internet Case Buffalo Alkali.XLS, sheet Current, assumes that repair

techniques. Problems that require quick answers that cannot wait for a simulation model to be built are a second category. probabilistic) elements through random sampling. Monte Carlo steps: (1) set up probability distribution(s), (2) set up cumulative probabilities, (3)

Data Table Calculations set to Manual. Press F9 to recalculate. Week Demand Profit Run Avg profit Prob >= 200 20 42 $2,100.00 20 $1,700.96 15.38% 21 30 $1,500.00 21 $1,704.81 3.85% 22 38 $1,900.00 22 $1,768.27 17.31% 23 31 $1,550.00 23 $1,730.77 13.46% 24 39 $1,950.00 24 $1,806.73 21.15% 25 32 $1,600.00 25 $1,786.54 23.08% 26 32 $1,600.00 26

(Calculations set to Manual. Press F9 to recalculate). Month Demand 125 Demand Probability RN intvl LL 230 10 0.02 0.00 325 15 0.07 0.02 33 35 34 25 35 35 36 35 37 25 38 15 39 25 40 40 41 35 42 25 43 35 44 30 45 35 46 40 47 35 48 35 17 30 18 45 19 35 20 50 21 30 22 35 23 20 24 40 25 30 26 35 27 45 28 25 29

P10-17 Data Table Calculations set to Manual. Press F9 to recalculate. Month Begin Bal Income Expense End Bal Run End bal P(neg bal) 1 $1,500 $950 $1,000 $1,450 Income Probability RN intvl LL 1$400 0.0000 2 $1,450 $900 $800 $1,550 $850 0.35 0 2 $400 0.0000 3 $1,550 $900 $900 $1,550 $900 0.25 0.35 3 $450

29 100 $300.0 $200.0 $100.0 29 $4,356 30 175 $300.0 $200.0 $100.0 30 $4,975 31 125 $300.0 $200.0 $100.0 31 $4,638 32 75 $243.8 $200.0 $43.8 32 $4,525 33 150 $300.0 $200.0 $100.0 33 $4,919 34 125 $300.0 $200.0 $100.0 34 $4,581 35 75 $243.8 $200.0 $43.8 35 $4,469 36 125 $300.0 $200.0 $100.0 36 $4,356 37 100

Data Table Calculations set to Manual. Press F9 to recalculate. Day Demand Today Revenue Cost Next-Day Revenue Profit Demand Probability RN intvl LL Run Profit 42 $32,625.00 43 $33,750.00 44 $27,375.00 45 $30,000.00 46 $31,875.00 47 $31,500.00 48 $27,375.00 49 $32,625.00 50 $33,750.00 51 $29,250.00 52 $32,250.00 53 $33,375.00 54 $30,000.00 55 $32,250.00 56 $29,625.00 57 $31,500.00 58 $28,500.00 59 $32,250.00 130 $3,000.00 $1,500.00 $0.00 $1,500.00 15 0.08 0.00 1 $33,375.00 28 $31,125.00 29 $31,500.00 30 $30,000.00 31

Data Table (Calculations set to Manual. Press F9 to recalculate). Day Demand Revenue Cost Profit Run Profit 129 $435.00 $120.00 $315.00 Mean = 30 1 $7,515.00 43 $7,560.00 44 $7,290.00 45 $7,395.00 46 $7,185.00 47 $7,275.00 48 $7,320.00 49 $7,725.00 50 $6,825.00 34 $7,545.00 35 $7,635.00 36 $7,080.00 37 $7,350.00 38 $7,455.00 39 $7,080.00 40 $7,380.00 41 $7,185.00 42 $7,275.00 Demand Distribution Problem 10-20 225 $375.00 $120.00 $255.00 Std dev = 6 2

P10-22 Data Table Calculations set to Manual. Press F9 to recalculate. Stude nt Sched appt time Actual arrive Sched time needed Act time needed Start service Finish service Run Finish service 1 0 -10 20 20 020 Minutes Probability RN intvl LL 1215 Based on 200 replications: 18 205 Finish service at: 221.68 19 230 20 215 21 200 22 225 23 220 24 210 25 220 26 220 27 210 33 205 34 215 35

Data Table (Calculations set to Manual. Press F9 to recalculate). Day Begin Inv Units Rec Avail Inv Dem and Demand Filled End Inv Stock out End Inv + Order Place order? Lead time Arrive on day Inv cost Stockout cost Order cost Demand Probability RN intvl LL Run Total cost % stoc- kout 1150 0150 60 60 90 090 1 4 6 $67.50 $0 $75 20 0.08 0.00 1 $81,630.00 10.43% 49 40

Data Table (Calculations set to Manual. Press F9 to recalculate). Day Begin Inv Units Rec Avail Inv Dem and Demand Filled End Inv Stock out End Inv + Order Place order? Lead time Arrive on day Inv cost Stockout cost Order cost Demand Probability RN intvl LL Run Total cost % stoc- kout 1150 0150 50 50 100 0100 1 5 7 $75.00 $0 $75 20 0.08 0.00 1 $45,585.00 4.85% 46 140

Data Table Calculations set to Manual. Press F9 to recalculate. Day Begin Inv Units Rec Avail Inv Dem- and Dem- and filled End Inv Stock out End Inv + Order Place order? Lead time Arrive on day Inv cost Stockout cost Order cost Run Total cost % stoc- kout 1300 0300 49 49 251 0251 0 0 0 $125.50 $0 $0 Minimum demand 30 1 $17,537.00 14.38% 38 23 023 36 23 013 300

Data Table Calculations set to Manual. Press F9 to recalculate. Day Begin Inv Units Rec Avail Inv Dem- and Dem- and filled End Inv Stock out End Inv + Order Place order? Lead time Arrive on day Inv cost Stockout cost Order cost Run Total cost % stoc- kout 1300 0300 38 38 262 0262 0 0 0 $131.00 $0 $0 Minimum demand 30 1 $19,818.50 16.62% 41 50 050 41 41 9 0

Data Table Calculations set to Manual. Press F9 to recalculate. Day Begin Inv Units Rec Avail Inv Dem- and Demand Filled End Inv Stockout End Inv + Order Place order? Lead time Arrive on day Inv cost Stock out cost Order cost Lead time Proba- bility RN intvl LL Run Total cost % stoc- kout 1100 0100 18 18 82 082 0 0 0 $16.40 $0 $0 1 0.10 0.00 1 $7,131.40 8.23% 45 5 0

Problem 10-30 Data Table (Calculations set to Manual. Press F9 to recalculate). Maximum res accepted = 9 RN intvl LL Value Probability Run Profit Demand = 7 1 $174 Actual res accepted = 7 2$24 Number no shows = 0 0.00 5 0.05 3 $582 18 $582 19 $324 Fixed Cost $450 22 $840 23 $195 Aircraft capacity 6 24

Data Table Calculations set to Manual. Press F9 to recalculate. Custo- mer # Time between arrivals Arrival time Start service Service time End service Wait time Arrive within 1 hour? RN intvl LL Time Proba- bility Run Wait time 0 0 0 0 0 0 0 1 1 294.0 1 1 1 1 5 6 0 12 78.0 2 2 3 6 5 11 3 10.00 1 0.20 3 133.0 18

Data Table Calculations set to Manual. Press F9 to recalculate. 25 Minor $74 27 $23,841 26 Minor $106 28 $15,863 27 Minor $102 29 $16,369 28 Minor $115 30 $15,241 29 Minor $110 31 $19,963 30 Minor $137 32 $20,454 Number of projects: 40 to 55 No. of jobs: 48 Run Profit Probability of minor 0.65 1 $15,955 Minor

Data Table Calculations set to Manual. Press F9 to recalculate. 25 Dry 276 276 $270.00 25 $7,711.25 26 Dry 315 300 $300.00 26 $7,395.00 27 Dry 284 284 $280.00 27 $7,726.25 28 Dry 292 292 $290.00 28 $7,688.75 29 Dry 326 300 $300.00 29 $7,567.50 30 Dry 289 289 $286.25 30 $7,220.00 31 $7,446.25 32 $7,811.25 33 $7,430.00 34 $7,707.50 35

Data Table Calculations set to Manual. Press F9 to recalculate. Probability of residential 0.70 No. of projects: 14 Run Profit Resid revenue (Normal) 1 $10,323 Mean $500 Project Type Revenue 2 $6,312 Standard deviation $50 1 Resid $508 3 $6,712 10 0.15 0.00 8 Resid $440 10 $12,665 12 0.20 0.35 10 Resid

Data Table (Calculations set to Manual. Press F9 to recalculate). Time Between Arrival Begin Wait for Service End Time in Run Wait System Utilization Customer 37 4108 109 1 3 112 439 3.46 6.03 0.915 38 3111 112 1 4 116 540 1.49 4.00 0.839 39 1112 116 4 3 119 741 6.16

Problem 10-39 Data Table Calculations set to Manual. Press F9 to recalculate. Demand 153 12 $110.00 23.53% Sales 153 13 $170.00 0.00% 14 $170.00 0.00% Revenue $229.50 15 $144.50 10.00% Cost $85.00 16 $102.50 26.47% Number of pretzels available 170 Run Profit %unsold Cost per pretzel $0.50 1 $144.50 10.00% 10 $111.50 22.94% Blowout?

Data Table (Calculations set to Manual. Press F9 to recalculate). Time Between Arrival Begin Wait for Service End Time in Run Wait P(T>=3) Customer 29 486 86 0 3 89 331 2.67 0.335 30 187 89 2 4 93 632 1.73 0.295 31 592 93 1 2 95 333 3.25 0.445 32 193 95

Data Table Calculations set to Manual. Press F9 to recalculate. Fail # Time between failures Actual failure time Begin maint Wait for maint Maint time End maint Time in system Begin maint Wait for maint Maint time End maint Time in system Run Annual downtime savings 4 23.63 178.27 180.99 2.72 32.20 213.19 34.91 178.27 0.00 16.46 194.73 16.46 Time between failures (hrs) 30 4 $19,998.24 5

Data Table (Calculations set to Manual. Press F9 to recalculate). Unit cost $12 Run Profit 31 $287,100 32 $177,820 33 $201,240 34 $232,100 35 $219,980 36 $208,980 37 $226,350 38 $102,460 39 $219,990 40 $194,610 41 $247,370 42 $265,430 23 $196,680 24 $249,550 25 $299,560 26 $239,600 27 $239,430 28 $222,090 29 $228,500 30 $272,430 Problem 10-42 Original price $30 1 $180,650 Sale price $20 2 $228,250 Order quantity 17,000 3 $234,990 4 $207,710 5 $229,080 Mean Std

Data Table (Calculations set to Manual. Press F9 to recalculate). Custo- mer Time between arrivals Arrival time Start service Service time End service Wait time Queue length RN intvl LL Arrival Proba- bility Run Queue Length Prob (Wait>=3) 0 0 0 0 0 0 0 0 1 2.73 0.80 35 5116 128 1129 12 436 1.45 0.61 36 3119 129 4133 10 437 6.68 0.97 37 4123 133 3136 10

B I E C D H F A G Data Table Calculations set to Manual. Activity Immed. pred. Optimistic time (a) Most likely time (m) Pessimistic time (b) Expected time Std Dev Activity time Start time Finish time Run Compl Time A—3 6 8 5.83 0.83 6.15 0.00 6.15 1 35.11 10 33.53 Based on 1 replication: 11 34.25 Project completion time = 35.11 12 33.91 13 35.51 Based on 3000 replications: 14 36.21 Project completion time =

F I A D K C B H G E Start J L End Data Table Calculations set to Manual. Press F9 to recalculate simulation values. Run Time Activity Immed. pred. Mean Std dev Activity time Start time Finish time F B, D, E 10.00 0.33 10.03 33.03 43.06 Prob (>= 75 days) = 0.0350 7 64.04 G B, D, E 7.33 0.67 6.06 33.03 39.09 8 72.39 H F 15.00 0.33 15.21 43.06

C I G E B H F A D Data Table Calculations set to Manual. Activity Immed. pred. Optimistic time (a) Most likely time (m) Pessimistic time (b) Expected time Std Dev Activity time Start time Finish time Run Compl Time Based on 3000 replications: 14 51.34 Project completion time = 52.57 15 49.69 Prob (>= 50 days) = 0.7750 16 53.59 17 53.30 19 48.36 21 50.98 23 49.68 25 54.59 27 60.79 29 51.49 31 50.87 33 48.37 35 45.22 37 48.62 39 50.67 41 55.32 43

3 2.0% $61,200.00 $6,120.00 11.33% $13,630.85 $19,750.85 Max raise % 8 2 $1,309,763 5 1.0% $64,902.60 $6,490.26 -2.93% $26,422.58 $32,912.84 Min return % -5 4 $880,822 7 8.0% $74,300.50 $7,430.05 10.03% $43,625.27 $51,055.32 6 $1,341,662 9 5.0% $81,916.30 $8,191.63 -1.64% $67,351.19 $75,542.82 8 $876,945 11 4.0% $86,896.81 $8,689.68 8.75% $94,657.19 $103,346.87 10 $1,075,667 13 8.0%

P10-14 Month Failures 1 4 Lwr Lmt Failures Probability 2 3 0 0 0.10 13 5 Expected failures/month 2.48 18 3 19 3 20 3 21 2 22 2 23 1 24 1 25 1 26 3 27 4 33 3 34 3 35 3 36 2 Page 1 14 3 15 3 16 1 17 4 28 1 29 4 30 4 31 1 32 1 Failure Distribution Problem 10-15 3 0 0.10 10.17 4 4 0.27 20.21 5 2 0.48 30.28 6

P10-21 Data Table Calculations set to Manual. Press F9 to recalculate. Run Total time Cost/hr Total time Cost/hr # Time (Hrs) # Time (Hrs) Hours Probability RN intvl LL 1800 $14.50 2,470 $10.69 130 1130 10 0.05 0 2 750 $15.47 2,440 $10.82 36 910 $12.75 2,400 $11.00 37 700 $16.57 2,280 $11.58 38 770 $15.06 2,340

P10-23 Data Table Calculations set to Manual. Press F9 to recalculate. Patient Sched appt time Actual arrive Sched time needed Act time needed Start service Finish service Run Finish service Ada 0 0 20 16 016 Minutes Probability RN intvl LL 1177 Bro 15 25 15 12 25 37 -20 0.20 0.00 2 169 Factor Probability RN intvl LL 9175 33 162 34 175 35 171 36 171 37

P10-31 Calculations set to Manual. Press F9 to recalculate. Pati- ent Room 1 Room 2 Room 3 Room 4 Room 5 Room 6 Roo m 7 Roo m 8 Room 9 Room 10 Room 11 Room 12 Room 13 Room 14 Room 15 Num. x-ray To Prob RN intvl LL Run P(multi x-ray) 4 Initial Out Out Out Out Out Out Out Out Out Out Out Out Out Out 0 Op 0.15 0.45 4

P10-33 Data Table Calculations set to Manual. Press F9 to recalculate. Month Begin Inv Units Rec Avail Inv Demand Demand Filled End Inv Stock out End Inv + Order Place order? Lead time Arrive on day Inv cost Stockout cost Order cost Value Proba- bility RN intvl LL Run Avg stockout Cost per month 35 4.04 $31,371 36 3.08 $24,417 37 4.13 $32,004 38 3.54 $27,704 39 4.58 $35,192 40 3.88 $30,204 41 2.83 $22,600 42 4.13 $31,921 43 4.33 $33,375 44 2.33

Data Table Calculations set to Manual. Press F9 to recalculate. House Home? Donate? Type? $15? $25? $35? Other $ Run Total 27 Yes Donate $25 $0 $25 $0 $0 27 $60 28 Yes Decline N/A $0 $0 $0 $0 28 $250 29 No N/A N/A $0 $0 $0 $0 29 $240 30 Yes Decline N/A

Data Table Calculations set to Manual. Press F9 to recalculate. Custo- mer # Time between arrivals Arrival time Start service Service time End service Wait time Queue length Run Total cost per day 0 0.00 0.00 0.00 0.00 0.00 0.00 0 Arrival Distribution (Exponential) 1 $1,307.20 4 23.18 54.86 54.86 7.30 62.16 0.00 0 Mean (minutes) 9 5 $1,131.20 6 12.25 67.46 69.02 7.09

A H F K C B I D Start E G J End Data Table Calculations set to Manual. Press F9 to recalculate simulation values. Run Profit Activity Immed. pred. Mean Std dev Activity time Start time Finish time H F 6.00 1.00 5.31 25.36 30.67 9 33.78 K H, I 7.33 1.33 4.96 30.67 35.63 12 35.58 End J, K 0.00 0.00 0.00 35.63 35.63 13 34.93 14 39.63 16 33.69 18 37.97 20 34.97 22 34.88 24 38.92 26

H A D C F B I G E J Start Data Table Calculations set to Manual. Press F9 to recalculate simulation values. Run Time Activity Immed. pred. Minimu m time Maximum time Activity time Start time Finish time 16 54.00 18 58.00 20 44.00 22 62.00 24 56.00 26 51.00 28 51.00 30 47.00 32 46.00 34 41.00 36 59.00 38 56.00 40 63.00 42 49.00 44 49.00 46 50.00 48 49.00 50 58.00 52 47.00 C A, B 3 8 5.0 17.0 22.0 4 55.00 D C 5

Prob(>300K) 62.00% Average $347,249 Initial employees = 1000 1 $333,555 Monthly premium = $100 2 $538,318 Prob of major claim = 0.10 to 0.15 D Uniform 3 -$9,367 12 $824,977 13 $412,693 14 $330,392 15 $890,055 16 $116,103 17 $754,309 18 $590,850 19 $393,652 20 $192,881 21 $484,136 22 $252,467 23 $115,872 24 $219,318 25 $315,590 26 -$90,338 27 $250,424 28 -$59,195 29 -$60,173 30 $441,419 31 $124,603 32 $471,183 33 $492,338 34 $401,380 35 $178,737 36 $415,293 37 $97,389 38

Data Table Calculations set to Manual. Press F9 to recalculate. Week Begin Inv Units Rec Avail Inv Dem- and Demand Filled End Inv Stockout End Inv + Order Place order? Lead time Arrive on week Inv cost Stockout cost Order cost Lead time Proba- bility RN intvl LL Run Total cost % stoc- kout 1140 0140 13 13 127 0127 0 0 0 $2,540 $0 $0 10.15 0.00 1$145,560 4.72% 35 180 0180 10 10

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error Year 1 51.8 6512 53.020 -1.220 1.220 1.489 2.36% North-South Airline Year 2 54.92 8404 57.937 -3.017 3.017 9.103 5.49% Year 3 69.7 11077 64.884 4.816 4.816 23.197 6.91% Year 4 68.9 11717 66.547 2.353 2.353

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error 1996 68 22 67.122 0.878 0.878 0.772 1.29% 1997 75 25 76.081 -1.081 1.081 1.169 1.44% Internet Case: Kwik Lube (#1) 1998 75 24 73.095 1.905 1.905 3.631 2.54% 1999 78 26 79.068 -1.068 1.068 1.140

where 1996 = year 1, 1997 = year 2, and so on. R2 = 77.6%. MAPE = 9.94%. The forecasts for 2006 (year 11) and 2007 (year 12) are 145,523 and 154,720, respectively. Using a weighted average of $2.875 to represent gate receipts per person, revenues for 2006 and

an indicator variable for three of the four seasons. We do not include a variable for the fourth season since this would make the columns linearly dependent. The resulting equation (see file Ch11 Internet Case Human Resources.XLS, sheet Regression) is: Average participants = 54.028

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error 1996 53353 1 53549.709 -196.709 196.709 38694.466 0.37% Internet Case: Akron Zoological Park (#1) 1997 61417 2 62747.018 -1330.018 1330.018 1768948.364 2.17% 1998 63853 3 71944.327 -8091.327 8091.327 65469577.034 12.67% 1999 63034 4 81141.636 -18107.636 18107.636

Forecasting 2 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Day 1 622 Day 2 418 Problem 11-15 Day 3 608 520.000 88.000 88.000 7744.000 14.47% Day 4 752 513.000 239.000 239.000 57121.000 31.78% Day 5 588 680.000 -92.000 92.000 8464.000 15.65% Day 6 656 670.000 -14.000 14.000 196.000

Forecasting 2 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Week 1 22 Week 2 27 Week 3 30 24.500 5.500 5.500 30.250 18.33% Week 4 21 28.500 -7.500 7.500 56.250 35.71% Problem 11-17 Week 5 33 25.500 7.500 7.500 56.250 22.73% Week 6 28 27.000 1.000 1.000 1.000

Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Year 1 480 440.000 Problem 11-19 (a) Year 2 525 454.000 71.000 71.000 5041.000 13.52% Year 3 548 478.850 69.150 69.150 4781.723 12.62% Year 4 593 503.053 89.948 89.948 8090.553 15.17% Year 5 614 534.534 79.466 79.466 6314.825 12.94% Average 77.391 6057.025

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error Month 1 3 4 4.238 -1.238 1.238 1.533 41.27% Month 2 6 5 5.210 0.790 0.790 0.625 13.17% Problem 11-21 Month 3 7 8 8.124 -1.124 1.124 1.263 16.05% Month 4 5 7 7.152 -2.152 2.152

Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Month 17 55 32.640 7.360 7.360 54.175 18.40% Month 18 40 33.744 21.256 21.256 451.830 38.65% Month 19 35 36.932 -1.932 1.932 3.733 5.52% Month 20 60 36.642 -11.642 11.642 135.544 46.57% Month 21 75 34.896 20.104 20.104 404.172 36.55% Month

Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Week 1 21 22.000 Week 2 25 21.700 3.300 3.300 10.890 13.20% Week 3 23 22.690 0.310 0.310 0.096 1.35% Problem 11-24 Week 4 27 22.783 4.217 4.217 17.783 15.62% Week 5 22 24.048 -2.048 2.048 4.195 9.31% Week 6 20 23.434 -3.434 3.434

Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Month 1 83.0 78.000 Month 2 81.5 79.000 2.500 2.500 6.250 3.07% Problem 11-25(a) Month 3 77.8 79.500 -1.700 1.700 2.890 2.19% Month 4 84.7 79.160 5.540 5.540 30.692 6.54% Month 5 84.3 80.268 4.032 4.032 16.257 4.78% Month 6 85.8 81.074

Forecasting Linear trend analysis Input Data Forecast Error Analysis Period Actual value (or) Y Period number (or) X Forecast Error Absolute error Squared error Absolute % error Year 1 30 1 27.018 2.982 2.982 8.891 9.94% Year 2 27 2 30.303 -3.303 3.303 10.910 12.23% Problem 11-27 Year 3 34 3 33.588 0.412 0.412 0.170 1.21% Year 4 35 4 36.873 -1.873 1.873

Season Demand Average demand Ratio Season index Year 3 demand Fall Y1 125 175.0 0.71 0.86 192.9 Winter Y1 225 175.0 1.29 1.43 321.4 Problem 11-29 Spring Y1 75 175.0 0.43 0.47 106.1 Summer Y1 210 175.0 1.20 1.24 279.6 Fall Y2 175 175.0 1.00 Winter Y2 275 175.0 1.57 Spring Y3 90 175.0 0.51 Summer Y4 225 175.0 1.29

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error Year 1 0.29 2.34 0.357 -0.067 0.067 0.004 23.11% Problem 11-31 Year 2 0.14 1.70 0.133 0.007 0.007 0.000 5.29% Year 3 0.19 1.89 0.199 -0.009 0.009 0.000 4.85% Year 4 0.16 1.94 0.217 -0.057 0.057

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error SE 1532.393 Forecast 10861.754 46700 Correlation -0.850 r-squared 0.722 Lower 95% interval $8,341.19 Upper 95% interval $13,382.32 Problem 11-33 Car 1 14200 11600 14214.718 -14.718 14.718 216.606 0.10% Car 2 14000 22800 13144.826 855.174 855.174 731322.727 6.11% Car 3 10500

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period GPA CBAT Forecast Error Absolute error Squared error Absolute % error Student 1 3.70 690 3.737 -0.037 0.037 0.001 0.99% Student 2 3.00 610 3.537 -0.537 0.537 0.289 17.91% Student 3 3.25 480 3.214 0.036 0.036 0.001 1.11% Slope 0.002 SE 0.290 Predicted GPA 3.587 630 Correlation 0.663 r-squared 0.439 Lower 90% intvl 3.22 Upper

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error Year 1 41 4000 39.610 1.390 1.390 1.933 3.39% Year 2 38 3500 38.327 -0.327 0.327 0.107 0.86% Problem 11-37 Year 3 36 1800 33.967 2.033 2.033 4.131 5.65% Year 4 43 2800 36.532 6.468 6.468

Forecasting Multiplicative decomposition 12 seasons Input Data Seasonal Index Computation Forecast Error analysis Period Actual value (Y) Time period (X) Centered average Seasonal ratio Seasonal index Unseasona lized value Unseasona lized Forecast Seasonaliz ed Forecast Error Absolute error Squared error Absolute % error Month 1 59042 1 1.306 45211.783 40731.268 53190.902 5851.098 5851.098 34235344.895 9.91% Month 2 62659 2 1.505 41627.096 40847.757 61485.904 1173.096 1173.096 1376155.205 1.87% Month 10 58435 10

Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Quarter 1 9027 9027.000 Quarter 2 9092 9027.000 65.000 65.000 4225.000 0.71% Quarter 3 9194 9092.000 102.000 102.000 10404.000 1.11% Quarter 4 9367 9194.000 173.000 173.000 29929.000 1.85% Problem 11-41 (a) Quarter 5 9505 9367.000 138.000 138.000 19044.000 1.45% Quarter 6 9588

Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Month 2 2.955 2.972 -0.017 0.017 0.000 0.58% Month 3 2.991 2.964 0.027 0.027 0.001 0.89% Month 4 3.117 2.976 0.141 0.141 0.020 4.51% Month 5 3.178 3.040 0.138 0.138 0.019 4.35% Month 6 3.148 3.102 0.046 0.046 0.002 1.46% Month

Forecasting Multiplicative decomposition 12 seasons Input Data Seasonal Index Computation Forecast Error analysis Unseason Unseason Seasonali Month 9 3.280 9 3.245 1.011 1.013 3.239 3.246 3.288 -0.008 0.008 0.000 0.24% Month 10 3.274 10 3.284 0.997 1.003 3.265 3.258 3.267 0.007 0.007 0.000 0.21% Month 11 3.264 11 3.314 0.985 0.995 3.282 3.270 3.252 0.012 0.012 0.000

Forecasting 4 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Quarter 1 263 Quarter 2 300 Quarter 3 245 Quarter 4 381 Quarter 5 312 297.250 14.750 14.750 217.563 4.73% Problem 11-45 Quarter 6 323 309.500 13.500 13.500 182.250 4.18% Quarter 7 298 315.250 -17.250 17.250 297.563 5.79% Quarter 8 390

Forecasting 4 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Quarter 1 294 Quarter 2 499 Quarter 3 437 Quarter 4 344 Quarter 5 335 393.500 -58.500 58.500 3422.250 17.46% Problem 11-47 Quarter 6 507 403.750 103.250 103.250 10660.563 20.36% Quarter 7 529 405.750 123.250 123.250 15190.563 23.30% Quarter 8 285

2. Select the items or quantities that are to be forecasted. 4. Select the forecasting model. 6. Validate the forecasting model. 8. Implement the results. account any factors that may influence the quantity being forecasted. Causal models use historical data as well. Time-series models use only historical data. useful when

1999 78 26 2000 99 33 2001 104 35 2002 120 39 2003 133 44 Average Kwik Lube sales = 1.419 + 2.986 × (Average industry sales) Forecasted average Kwik Lube sales for 2004 and 2005 can be

Forecasting Multiplicative decomposition 5 seasons Input Data Seasonal Index Computation Forecast Error analysis Game 9 25300 9 35340.000 0.716 0.747 33860.962 36526.391 27291.537 -1991.537 1991.537 3966218.761 7.87% Game 10 36200 10 35340.000 1.024 1.015 35652.064 36998.787 37567.421 -1367.421 1367.421 1869840.031 3.78% Game 11 35900 11 37400.000 0.960 1.025 35019.788 37471.184 38413.011 -2513.011 2513.011 6315226.481

Forecasting Multiplicative decomposition 4 seasons Input Data Seasonal Index Computation Forecast Error analysis Period Actual value (Y) Time period (X) Centered average Seasonal ratio Seasonal index Unseason alized value Unseason alized Forecast Seasonali zed Forecast Error Absolute error Squared error Absolute % error Quarter 28 94 28 90.000 1.044 1.020 92.188 86.636 88.339 5.661 5.661 32.043 6.02% Quarter 29 93 29 90.000 1.033 1.052 88.438 87.868 92.400 0.600 0.600

Forecasting 2 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Period 1 4 Period 2 8 Problem 11-13(a) Period 3 6 6.000 0.000 0.000 0.000 0.00% Period 4 6 7.000 -1.000 1.000 1.000 16.67% Period 5 9 6.000 3.000 3.000 9.000 33.33% Period 6 8 7.500 0.500 0.500 0.250

Forecasting 3 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Semester 1 2.4 Semester 2 2.9 Semester 3 3.1 Problem 11-14(a) Semester 4 3.2 2.800 0.400 0.400 0.160 12.50% Semester 5 3 3.067 -0.067 0.067 0.004 2.22% Semester 6 2.9 3.100 -0.200 0.200 0.040 6.90% Semester 7 2.8 3.033 -0.233

Forecasting Exponential smoothing Input Data Forecast Error Analysis Period Actual value Forecast Error Absolute error Squared error Absolute % error Day 1 622 622.000 Day 2 418 622.000 -204.000 204.000 41616.000 48.80% Problem 11-16(a) Day 3 608 564.477 43.523 43.523 1894.278 7.16% Day 4 752 576.749 175.251 175.251 30712.820 23.30% Day 5 588 626.166 -38.166 38.166 1456.639 6.49% Day 6 656 615.404

Forecasting 3 period weighted moving average Input Data Forecast Error Analysis Period Actual value Weights Forecast Error Absolute error Squared error Absolute % error Week 1 22 0.000 Week 2 27 0.417 Problem 11-18(a) Week 3 30 0.583 Week 4 21 28.750 -7.750 7.750 60.063 36.90% Week 5 33 24.750 8.250 8.250 68.063 25.00% Week 6 28 28.000 0.000 0.000 0.000

Forecasting 3 period weighted moving average Input Data Forecast Error Analysis Period Actual value Weights Forecast Error Absolute error Squared error Absolute % error Week 1 22 1 Week 2 26 3 Problem 11-20(a) Week 3 23 5 Week 4 27 23.889 3.111 3.111 9.679 11.52% Week 5 21 25.556 -4.556 4.556 20.753 21.69% Week 6 25 23.222 1.778 1.778 3.160

Forecasting 3 period moving average Input Data Forecast Error Analysis Period Actual Value Forecast Error Absolute error Squared error Absolute % error Month 1 9 Month 2 12 Month 3 14 Month 4 811.667 -3.667 3.667 13.444 45.83% Month 5 13 11.333 1.667 1.667 2.778 12.82% Problem 11-22(a) Month 6 15 11.667 3.333 3.333 11.111 22.22% Month 7 912.000 -3.000 3.000 9.000

Forecasting Linear trend analysis Input Data Forecast Error Analysis Period Actual value (or) Y Period number (or) X Forecast Error Absolute error Squared error Absolute % error Year 1 14 1 12.026 1.974 1.974 3.898 14.10% Year 2 9 2 13.763 -4.763 4.763 22.690 52.93% Problem 11-26 (a) Year 3 12 3 15.501 -3.501 3.501 12.258 29.18% Year 4 14 4 17.239 -3.239

Forecasting Causal regression analysis Input Data Forecast Error Analysis Period Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error Year 1 30 67.3 27.003 2.997 2.997 8.982 9.99% Year 2 27 70.6 28.802 -1.802 1.802 3.246 6.67% Problem 11-28 Year 3 34 82.4 35.233 -1.233 1.233 1.519 3.63% Year 4 35 84.7 36.486 -1.486 1.486

Sales Seasonal index Seasonalized sales $130,000 1.25 $162,500 Problem 11-30 $150,000 0.95 $142,500 $170,000 0.75 $127,500 $190,000 1.05 $199,500

Forecasting Causal regression analysis Input Data Forecast Error Analysis Student Dep Variable (or) (Y) Indep Variable (or) (X) Forecast Error Absolute error Squared error Absolute % error D 3.55 730 3.469 0.081 0.081 0.007 2.27% F 3.98 430 2.614 1.366 1.366 1.865 34.31% H 2.63 520 2.871 -0.241 0.241 0.058 9.16% J 2.09 540 2.928 -0.838 0.838 0.702 40.09% L 4.00 750 3.526

Forecasting Multiple regression Input Data Forecasts and Error Analysis Y x 1 x 2 Forecast Error Absolute error Squared error Absolute % error Car 8 $7,900.00 73,100 68822.383 -922.383 922.383 850790.303 11.68% Car 9 $6,200.00 77,500 87634.869 -1434.869 1434.869 2058849.257 23.14% Car 10 $7,500.00 85,700 96805.330 694.670 694.670 482566.403 09.26% Average 1187.537 1765683.169 12.50% Regression Line MAD MSE MAPE Car 6 $10,900.00

Forecasting Multiple regression Input Data Forecasts and Error Analysis Y x 1 x 2 Forecast Error Absolute error Squared error Absolute % error Student 1 3.700 690.000 34.000 3.798 -0.098 0.098 0.010 2.64% Student 2 3.000 610.000 29.000 3.474 -0.474 0.474 0.225 15.81% Problem 11-36 Student 3 3.250 480.000 24.000 3.056 0.194 0.194 0.038 5.97% Student 4 4.000 740.000 39.000 4.065

Forecasting Multiple regression Input Data Forecasts and Error Analysis Y x 1 x 2 Forecast Error Absolute error Squared error Absolute % error Year 1 41.000 4000.000 82.000 42.449 -1.449 1.449 2.101 03.54% Year 2 38.000 3500.000 64.000 38.553 -0.553 0.553 0.306 01.46% Problem 11-38 Year 3 26.000 1800.000 52.000 30.447 -4.447 4.447 19.772 17.10% Year 4 43.000 2800.000 60.000 35.313

Forecasting Multiplicative decomposition 4 seasons Input Data Seasonal Index Computation Forecast Error analysis Period Actual value (Y) Time period (X) Centered average Seasonal ratio Seasonal index Unseason alized value Unseason alized Forecast Seasonali zed Forecast Error Absolute error Squared error Absolute % error Quarter 1 42.9 1 0.800 53.597 53.180 42.566 0.334 0.334 0.112 0.78% Quarter 2 48.5 2 0.890 54.519 54.043 48.077 0.423 0.423 0.179 0.87% Quarter 3 54.1

Forecasting Multiplicative decomposition 4 seasons Input Data Seasonal Index Computation Forecast Error analysis Period Actual value (Y) Time period (X) Centered average Seasonal ratio Seasonal index Unseason alized value Unseason alized Forecast Seasonali zed Forecast Error Absolute error Squared error Absolute % error Quarter 1 9027 1 1.001 9014.982 9051.176 9063.242 -36.242 36.242 1313.488 0.40% Quarter 2 9092 2 1.001 9085.075 9163.686 9170.671 -78.671 78.671 6189.106 0.87% Quarter 3 9194

Forecasting Multiplicative decomposition 4 seasons Input Data Seasonal Index Computation Forecast Error analysis Period Actual value (Y) Time period (X) Centered average Seasonal ratio Seasonal index Unseason alized value Unseason alized Forecast Seasonali zed Forecast Error Absolute error Squared error Absolute % error Quarter 1 263 1 0.918 286.491 304.928 279.925 -16.925 16.925 286.471 6.44% Quarter 2 300 2 0.940 318.981 308.002 289.674 10.326 10.326 106.618 3.44% Seasonal Ratios Season 1

Forecasting Multiplicative decomposition 4 seasons Input Data Seasonal Index Computation Forecast Error analysis Period Actual value (Y) Time period (X) Centered average Seasonal ratio Seasonal index Unseason alized value Unseason alized Forecast Seasonali zed Forecast Error Absolute error Squared error Absolute % error Quarter 1 294 1 0.806 364.846 381.687 307.571 -13.571 13.571 184.163 4.62% Quarter 2 499 2 1.159 430.492 389.786 451.816 47.184 47.184 2226.342 9.46% Quarter

Inventory EOQ Model Input Data Demand rate, D 499.5 (in thousands of feet) Ordering cost, Cs$50.00 Total ordering cost $719.01 Total purchase cost $206,793.00 Total cost $208,231.03 Reorder point, ROP 115.27 Internet Case: Laplace Power & Light Co. (Optimal) Carrying cost %, I 10.00% (percentage) Unit purchase cost, P $414.00 Daily demand rate, d 1.92 Lead time (in days), L 60 Results Economic order quantity,

Inventory Quantity Discount Model Input Data Range 1 Range 2 Range 3 Minimum quantity 0 101 501 Unit purchase cost, P $25.00 $24.00 $22.00 Results Range 1 Range 2 Range 3 Adjusted order quantity 391.92 400.00 501.00 Total holding cost 1,224.74 1,200.00 1,377.75 Total ordering cost 1,224.74 1,200.00 958.08 Total purchase cost 240,000.00 230,400.00 211,200.00 Total cost 242,449.49 232,800.00 213,535.83 Economic order

complete system, additional tape units could be demanded. An increase in demand could change the outcome of the quantity discount model. The second strategy would also have an impact on the results of the analysis. At this time, two

Inventory Quantity Discount Model Input Data Demand rate, D 192000 Results Range 1 Range 2 Range 3 Range 4 Adjusted order quantity 678.82 7001.00 8001.00 20001.00 Total holding cost 25,455.84 241,534.50 264,033.00 630,031.50 Total ordering cost 25,455.84 2,468.22 2,159.73 863.96 Total purchase cost 48,000,000.00 44,160,000.00 42,240,000.00 40,320,000.00 Total cost 48,050,911.69 44,404,002.72 42,506,192.73 40,950,895.46 Economic order quantity, Q* 678.82 707.72 723.63

Inventory EOQ Model Input Data Demand rate, D 5000 Ordering cost, Cs$20.00 Results Maximum inventory 182.57 Average inventory 91.29 Number of orders 27.39 Total holding cost $547.72 Total ordering cost $547.72 Total purchase cost $300,000.00 Total cost $301,095.45 Reorder point, ROP 200 Economic order quantity, Q* 182.57 Sturdivant Sound Systems (Optimal) Carrying cost, Ch$6.00 (fixed amount) Unit purchase cost, P $60.00 Daily demand rate, d 20 Lead time

Inventory EOQ Model Input Data Demand rate, D 2000 Ordering cost, Co$10.00 Results Average inventory 91.06 Number of orders 10.98 Total holding cost $109.82 Total ordering cost $109.82 Total purchase cost $20,100.00 Total cost $20,319.64 Economic order quantity, Q* 182.12 Maximum inventory 182.12 Internet Case: Western Ranchman Outfitters (Optimal) Carrying cost %, I 12.00% (percentage) Unit purchase cost, P $10.05 Inventory EOQ Model Input Data Demand rate, D

Inventory EOQ Model Input Data Demand rate, D 439 Ordering cost, Co$65.00 Martin-Pullin Bicycle Corp. (EOQ) Carrying cost, Ch$12.24 (fixed amount) Unit purchase cost, P Results Economic order quantity, Q* 68.28 Maximum inventory 68.28 Average inventory 34.14 Number of orders 6.43 Total holding cost $417.89 Total ordering cost $417.89 Total purchase cost $0.00 Total cost $835.79 Inventory Safety Stock (Normal DDLT) Input Data Mean DDLT, m 36.58 Std

ordering extra quantities judiciously, Mr. Randell has managed to keep WRO almost totally supplied with the requisite number of Levi 501s. Further, since the actual solution is so close to the model solution, and since we have seen that the EOQ is a robust model, Mr. Veta can feel that

Problem 12-17 Inventory EOQ Model Input Data Demand rate, D 45,000 D = (EOQ^2)*(I x P) / (2 x Co) Ordering cost, Co$150.00 Carrying cost %, I 25.00% (percentage) Unit purchase cost, P $600.00 Results Economic order quantity, Q* 300.00 Maximum inventory 300.00 Average inventory 150.00 Number of orders 150.00 Total holding cost $22,500.00 Total ordering cost $22,500.00 Total purchase cost $27,000,000.00 Total cost $27,045,000.00

Problem 12-19 (a) Inventory EOQ Model Input Data Demand rate, D 80000 Results Average inventory 478.09 Number of orders 83.67 Total holding cost $3,346.64 Total ordering cost $3,346.64 Total purchase cost $0.00 Total cost $6,693.28 Reorder point, ROP 11250 Economic order quantity, Q* 956.18 Maximum inventory 956.18 Ordering cost, Cs$40.00 Carrying cost, Ch$7.00 (fixed amount) Unit purchase cost, P Daily demand rate, d 450 Lead time (in days),

Problem 12-21 Inventory EOQ Model Inventory EOQ Model Input Data Input Data Results Results Economic order quantity, Q* 50.00 = 5,000 / 100 Economic order quantity, Q* 200.00 200 * 100 = 20,000 cu ft Maximum inventory 50.00 Maximum inventory 200.00 Average inventory 25.00 Average inventory 100.00 Number of orders 100.00 Number of orders 25.00 Total holding cost

Problem 12-23 Freq Ordering cost EOQ Inventory EOQ Model 85 $45 1341.64 95 $46 1356.47 90 $47 1371.13 Input Data 80 $48 1385.64 Results Economic order quantity, Q* 1373.07 Note down the EOQ value each time. Maximum inventory 1373.07 Average inventory 686.54 Number of orders 21.85 Total holding cost $1,029.81 Total ordering cost $1,029.81 Total purchase cost $0.00 Total cost $2,059.61 Demand rate, D

Problem 12-25 Inventory Economic Production Quantity Model Input Data Demand rate, D 100000 Setup cost, Cs$400.00 Carrying cost, Ch$6.00 (fixed amount) Daily production rate, p 5000 Daily demand rate, d 500 Unit purchase cost, P Results Economic production quantity, Q* 3849.00 Maximum inventory 3464.10 Average inventory 1732.05 Number of setups 25.98 Total holding cost $10,392.30 Total setup cost $10,392.30 Total production cost $0.00 Total cost, TC $20,784.61

Stockout cost per unit = $50 Alternative (ROP) 40 50 60 70 80 90 EMV 50 $50 $0 $3,500 $7,000 $10,500 $14,000 $5,605 60 $100 $50 $0 $3,500 $7,000 $10,500 $3,170 70 $150 $100 $50 $0 $3,500 $7,000 $1,445 80 $200 $150 $100 $50 $0 $3,500 $430 90 $250 $200 $150 $100 $50 $0 $125 Best Probability

Inventory ABC Analysis Input Data Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % U1 1110 $2.00 $2,220.00 2.21% 2.21% V2 1110 $5.40 $5,994.00 5.96% 8.17% ABC Analysis: Items sorted in descending order by dollar volume Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % Y5 1200 $5.84 $7,008.00 6.97% 88.84% V2 1110 $5.40 $5,994.00 5.96% 12.94% U1 1110

Problem 12-31 Inventory EOQ Model Input Data Demand rate, D 5000 Results Number of orders 9.13 Total holding cost $136.93 Total ordering cost $136.93 Total purchase cost $0.00 Total cost $273.86 Economic order quantity, Q* 547.72 Maximum inventory 547.72 Average inventory 273.86 3-week option Let EOQ = Q* Demand during lead time (DDLT) = 3 x 100 = 300 Order cost = (5000/Q*) x Co =

Problem 12-33 Inventory Quantity Discount Model Input Data Demand rate, D 1000 Results Range 1 Range 2 Range 3 Adjusted order quantity 156.35 157.67 501.00 Total holding cost 351.78 348.84 1,092.81 Total ordering cost 351.78 348.84 109.78 Total purchase cost 18,000.00 17,700.00 17,450.00 Total cost 18,703.56 18,397.67 18,652.59 Economic order quantity, Q* 156.35 157.67 158.79 Ordering cost, Co55 Carrying cost %, I 25.00%

Problem 12-35 Inventory Quantity Discount Model Input Data Demand rate, D 3000 Range 1 Range 2 Range 3 Range 4 Minimum quantity 1 601 1101 1551 Unit purchase cost, P $10.00 $9.92 $9.86 $9.80 Results Range 1 Range 2 Range 3 Range 4 Adjusted order quantity 1449.14 1449.14 1449.14 1501.00 Total holding cost 724.57 724.57 724.57 750.50 Total ordering cost 724.57

Problem 12-37 Inventory Safety Stock (Normal DDLT) Input Data Mean DDLT, m 250 Std deviation of DDLT, s25 Service level desired 98.00% (percentage) Carrying cost, Ch Z-value to use 2.05 Results Safety stock, SS 51.00 ROUND function used in cell B12 Reorder point, ROP 301.00 SS carrying cost $0.00

Inventory ABC Analysis Input Data Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % Item 1 410 $0.75 $307.50 0.63% 0.63% Item 2 330 $17.00 $5,610.00 11.49% 12.12% ABC Analysis: Items sorted in descending order by dollar volume Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % Item Item 5 240 $110.00 $26,400.00 54.07% 54.07% A Item 9 100 $125.00

Inventory ABC Analysis Input Data Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % A1500 $4.00 $6,000.00 16.76% 16.76% B1500 $1.00 $1,500.00 4.19% 20.95% C700 $3.00 $2,100.00 5.87% 26.82% ABC Analysis: Items sorted in descending order by dollar volume Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % Item I1200 $8.00 $9,600.00 26.82% 26.82% A A1500 $4.00 $6,000.00 16.76% 43.58% A H800 $7.00

locations within the organization. This can be done by determining how much inventory is to be ordered and when the inventory should be ordered. Therefore, a delicate balance must be sought between increased carrying costs and shortages

units/month × 12 months/yr.). For Collins (see file Ch12 Internet Case Drake Radio.XLS, sheet Collins), the lowest total cost is $213,535.83 with an associated EOQ of 501

CHAPTER 12 INVENTORY CONTROL MODELS Internet Case Study: Laplace Power and Light Co. The optimal order quantity is 34,740 feet (see file Ch12 Internet Case Laplace Power.XLS, sheet Optimal). The reorder point is

Problem 12-15 (a) Inventory EOQ Model Input Data Demand rate, D 150000 Results Average inventory 19364.92 Number of orders 3.87 Total holding cost $116.19 Total ordering cost $116.19 Total purchase cost $0.00 Total cost $232.38 Economic order quantity, Q* 38729.83 Maximum inventory 38729.83 Ordering cost, Co$30.00 Carrying cost, Ch$0.006 (fixed amount) Unit purchase cost, P Problem 12-15 (b) Inventory EOQ Model Input Data Demand rate, D 150000 Results Average inventory 19364.92 Number

Problem 12-16 Inventory EOQ Model Input Data Demand rate, D 6000 Ordering cost, Cs$45.00 Results Average inventory 118.59 Number of orders 25.30 Total holding cost $1,138.42 Total ordering cost $1,138.42 Total purchase cost $720,000.00 Total cost $722,276.84 Reorder point, ROP 240 Economic order quantity, Q* 237.17 Maximum inventory 237.17 Carrying cost %, I 8.00% (percentage) Unit purchase cost, P $120.00 Daily demand rate, d 24 (d and L

Problem 12-18 Inventory EOQ Model Input Data Demand rate, D 1000 Results Average inventory 70.71 Number of orders 7.07 Total holding cost $141.42 Total ordering cost $141.42 Total purchase cost $10,000.00 Total cost $10,282.84 Economic order quantity, Q* 141.42 Maximum inventory 141.42 Ordering cost, Co$20.00 Carrying cost %, I 20.00% (percentage) Unit purchase cost, P $10.00

Inventory EOQ Model Input Data Demand rate, D 24000 Results Average inventory 685.99 Number of orders 17.49 Total holding cost $2,099.14 Total ordering cost $2,099.14 Total purchase cost $408,000.00 Total cost $412,198.29 Economic order quantity, Q* 1371.99 Maximum inventory 1371.99 Problem 12-20 (Current supplier) Ordering cost, Co$120.00 Carrying cost %, I 18.00% (percentage) Unit purchase cost, P $17.00 Inventory EOQ Model Input Data Demand rate, D 24000 Results Economic order

Problem 12-22 Inventory EOQ Model Inventory EOQ Model Input Data Input Data Demand rate, D 48000 Demand rate, D 48000 Ordering cost, Co$75.00 Ordering cost, Co$75.00 Carrying cost, Ch$45.00 (fixed amount) Carrying cost, Ch$45.00 (fixed amount) Unit purchase cost, P Unit purchase cost, P Results Results Economic order quantity, Q* 360.00 (=0.9*16000/40) Economic order quantity, Q* 400.00 Maximum inventory 360.00

Problem 12-24 Inventory Economic Production Quantity Model Input Data Demand rate, D 75000 Setup cost, Cs$800.00 Carrying cost, Ch$0.90 (fixed amount) Daily production rate, p 1000 Daily demand rate, d 250 Unit purchase cost, P Results Economic production quantity, Q* 13333.33 Maximum inventory 10000.00 Average inventory 5000.00 Number of setups 5.63 Total holding cost $4,500.00 Total setup cost $4,500.00 Total production cost $0.00 Total cost, TC $9,000.00

Problem 12-26 Inventory Quantity Discount Model Input Data Demand rate, D 1000 Range 1 Range 2 Minimum quantity 0 200 Unit purchase cost, P $50.00 $48.50 Results Range 1 Range 2 Adjusted order quantity 80.00 200.00 Total holding cost 500.00 1,212.50 Total ordering cost 500.00 200.00 Total purchase cost 50,000.00 48,500.00 Total cost 51,000.00 49,912.50 Economic order quantity, Q* 80.00 81.23 Ordering cost, Co40 Carrying cost

Problem 12-28 Inventory Safety Stock (Normal DDLT) Inventory Safety Stock Input Data Input Data Mean DDLT, m 600 Mean DDLT, m 600 Std deviation of DDLT, s70 Std deviation of DDLT, s70 Service level desired 95.00% (percentage) Service level desired 99.00% Carrying cost, Ch$0.50 Carrying cost, Ch$0.50 Results Results Safety stock, SS 115.14 Safety stock, SS

Stockout cost per unit = $50 Alternative (ROP) 600 650 700 750 800 850 900 950 1000 1050 1100 EMV 600 $0 $12,500 $25,000 $37,500 $50,000 $62,500 $75,000 $87,500 $100,000 $112,500 $125,000 $32,875 650 $500 $0 $12,500 $25,000 $37,500 $50,000 $62,500 $75,000 $87,500 $100,000 $112,500 $23,625 700 $1,000 $500 $0 $12,500 $25,000 $37,500 $50,000

Problem 12-32 Inventory Quantity Discount Model Input Data Demand rate, D 500 Results Range 1 Range 2 Range 3 Range 4 Adjusted order quantity 173.21 173.21 350.00 600.00 Total holding cost 173.21 173.21 350.00 600.00 Total ordering cost 173.21 173.21 85.71 50.00 Total purchase cost 6,000.00 5,750.00 5,250.00 4,900.00 Total cost 6,346.41 6,096.41 5,685.71 5,550.00 Economic order quantity, Q* 173.21 173.21

Problem 12-34 Inventory Quantity Discount Model Discount at 200 = (10)x($9.90) + (1/1000)x($3,000) = $102.00 Discount per can = $0.51 Input Data Discount at 300 = (30)x($9.90) + (1/1000)x($3,000) Demand rate, D 1000 = $300.00 Discount at 300 = (30)x($9.90) + (1/1000)x($3,000) + (1/800)x($5,000) Range 1 Range 2 Range 3 Range 4 = $405.25 Minimum quantity

Problem 12-36 Inventory Safety Stock (Normal DDLT) Input Data Mean DDLT, m 56 Results Safety stock, SS 30.00 ROUND function used in cell B12 Reorder point, ROP 86.00 SS carrying cost $0.00 Std deviation of DDLT, s18 Service level desired 95.00% (percentage) Carrying cost, Ch

Stockout cost per unit = $80 Alternative (ROP) 100 200 300 400 500 EMV 200 $2,500 $0 $8,000 $16,000 $24,000 $8,750 300 $5,000 $2,500 $0 $8,000 $16,000 $4,950 400 $7,500 $5,000 $2,500 $0 $8,000 $4,300 500 $10,000 $7,500 $5,000 $2,500 $0 $5,750 Best Probability 0.3 0.1 0.3 0.2 0.1 100 $0 $8,000 $16,000 $24,000 $32,000 $13,600 Outcomes (DDLT) Problem

Inventory ABC Analysis Input Data Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % A1750 $10.00 $17,500.00 1.61% 1.61% B300 $1,500.00 $450,000.00 41.38% 42.99% ABC Analysis: Items sorted in descending order by dollar volume Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % Item B300.000 1500.000 $450,000.00 41.38% 41.38% A C500.000 500.000 $250,000.00 22.99% 64.37% A I3000.000 50.000 $150,000.00 13.79% 78.16% B F1500.000 45.000

Inventory ABC Analysis Input Data Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % Item 1 560 $150.00 $84,000.00 83.50% 83.50% Item 2 490 $4.10 $2,009.00 2.00% 85.49% ABC Analysis: Items sorted in descending order by dollar volume Volume Unit cost Dollar volume % Dollar volume Cumulative $- vol % Item Item 1 560.000 150.000 $84,000.00 83.50% 83.50% A Item 4 600.000 10.600

C-30 C-92 D-21 E-11 Constraints E-11 >= 15% 1 27.50 >= 7.5 C-92 & C-30 >= 45% 1 1 22.50 >= 22.5 D-21 & C-92 <= 30% 1 1 15.00 <= 15.0 LHS Sign RHS 50-lbs required 1 1 1 1 50.00 =50.0 Case: Golding Landscaping # of pounds 7.50 15.00 0.00 27.50 Cost $0.12 $0.09 $0.11 $0.04 $3.35

Ammonia Amm Phosphate Amm Nitrate Urea Hydro Acid Chlorine Caustic Soda Vinyl Chloride Urea 1 160.00 <= 160 Hydro acid 1 560.00 <= 560 Chlorine 1 1,200.00 <= 1200 Caustic soda 1 423.20 <= 1280 Vinyl chloride 1 840.00 <= 840 Natural gas usage 810 12 12 718 20 14 68,544.00 <= 68544 LHS Sign RHS Internet Case: Agri Chem Corp. Number of tons 1,200.00 540.00

X Y Solution 3.00 5.60 Obj coeff 3 7 48.20 Problem 2-17 Constraints: Constraint 1 9 3 43.80 >= 36 Constraint 2 4 5 40.00 >= 40 Constraint 3 1 -1 -2.60 <= 0 Constraint 3 2 6.00 <= 6 LHS Sign RHS

A B C Constraint 3 1 3.33 <= 5.00 Constraint 4 1 1.67 >= 1.67 LHS Sign RHS Constraint 2 0.5 0.5 0.5 3.33 <= 3.33 Problem 2-19 Solution 3.33 1.67 1.67 Obj coeff 20 25 30 158.33 Constraints: Constraint 1 10 15 -8 45.00 <= 45.00

X Y Constraint 3 2 0.00 <= 7 Constraint 4 2 6.00 >= 7 LHS Sign RHS Solver reports infeasible solution. Constraint 2 4 5 12.00 <= 20 Problem 2-20 (1) Solution 3.00 0.00 Obj coeff 3 7 9.00 Constraints: Constraint 1 2 1 6.00 <= 6 X Y Constraint 3 1 1.00 <= 4 Constraint 3 2 9.00 <= 9 LHS Sign RHS Constraint

A B Constraints: Wiring time 3 2 240.00 <= 240 Drilling time 2 1 120.00 <= 210 Assembly time 1.5 0.5 60.00 <= 120 LHS Sign RHS Problem 2-24 Number of units 0.00 120.00 Profit $22 $15 $1,800.00

A B Number of units 24.00 84.00 Profit $13 $15 $1,572.00 Problem 2-25 Constraints: Wiring trime 3 2 240.00 <= 240 Drilling time 2 1 132.00 <= 210 Assembly time 1.5 0.5 78.00 <= 120 Min A Motors 1 24.00 >= 24 LHS Sign RHS

Mission Rustic Problem 2-26 Number of units 80.00 0.00 Profit $910 $1,200 $72,800.00 Constraints: Materials 340 430 27200.00 <= 30000 Labor 15 20 1200.00 =1200 LHS Sign RHS

Pruned Regular Number of barrels 25.00 62.50 Problem 2-36 Revenue $20 $30 $2,375.00 Constraints: Labor hours 5 2 250.00 <= 250 Acres available 1 2 150.00 <= 150 Max pruned 1 25.00 <= 40 LHS Sign RHS Number of acres 25.00 125.00

Feed X Feed Y Feed Z # of pounds 16.00 16.00 0.00 Problem 2-37 Cost $3.00 $4.00 $2.25 $112.00 Constraints Nutrient A needed 3 2 4 80.00 >= 64 Nutrient B needed 2 3 1 80.00 >= 80 Nutrient C needed 1 2 16.00 >= 16 Nutrient D needed 6 8 4 224.00 >= 128 Feed Z max 1

Problem 2-40 Print Jet Print Desk Print Pro Min Omega 1 287.13 >= 170.79 Min Rocket & Omega 1 1 455.45 >= 455.45 LHS Sign RHS Number of units 168.32 287.13 683.17 Profit $60 $90 $73 $85,811.88 Constraints Assembly time 2.9 3.7 3.0 3600.00 <= 3600.00 Testing time 1.4 2.1 1.7 2000.00 <= 2000.00

Problem 2-42 Plain Mixed Premium Number of cans 375.00 400.00 100.00 Revenue $2.25 $3.37 $6.49 $2,840.75 Constraints: Peanuts 0.8 0.5 500.00 <= 500.00 Cashews 0.2 0.3 0.3 225.00 <= 225.00 Almonds 0.1 0.4 80.00 <= 100.00 Walnuts 0.1 0.4 80.00 <= 80.00 Plain vs Premium 1 375.00 >= 200.00 LHS Sign RHS

Problem 2-43 BBC CBC ABC $ invested $2,555.25 $1,139.50 $0.00 Investment 1 1 1 $3,694.75 Constraints Short term growth 0.39 0.26 0.42 $1,292.82 >= $1,000 Intermediate growth 1.59 1.70 1.45 $6,000.00 >= $6,000 Dividend income 0.08 0.04 0.06 $250.00 >= $250 LHS Sign RHS

solutions as valid), there will be an infinite number of feasible combinations of decision variable values (unless of course, only a single solution satisfies all the constraints). In most cases, only one of these feasible solutions yields the optimal solution. 2-3. A problem can have alternative optimal solutions if

Objective function: Maximize profit = 80X1 + 120X2 + 140X3 + 140X4 + 90X5 + 70X6 + 60X7 + 90X8 CHAPTER 2 LINEAR PROGRAMMING MODELS: GRAPHICAL AND COMPUTER METHODS Internet Case Study: Agri Chem Corporation This case demonstrates an interesting use of LP in a production

W75C W33C W5X W7X Packaging time 1 3 2 $2,300.0 <= 2300 W75C orders 1 $1,100.0 <= 1400 W33C orders 1 $250.0 <= 250 W5X orders 1 $0.0 <= 1510 W7X orders 1 $600.0 <= 1116 Minimum W75C 1 $1,100.0 >= 150 Minimum W7X 1 $600.0 >= 600 LHS Sign RHS Extrusion time 1 1 4 1 $1,950.0 <=

X Y Solution 2.30 3.90 Problem 2-13 Obj coeff 2 1 8.50 Constraints: Constraint 1 3 6 30.30 <= 32 Constraint 2 7 1 20.00 <= 20 Constraint 3 3 -1 3.00 >= 3 LHS Sign RHS

X Y Constraint 3 1 7.00 <= 7 Constraint 4 2 -1 12.50 >= 3 LHS Sign RHS Constraint 1 5 2 38.00 <= 40 Constraint 2 3 6 30.00 <= 30 Problem 2-14 Solution 7.00 1.50 Obj coeff 4 5 35.50 Constraints:

X Y Solution 22.50 6.75 Problem 2-15 Obj coeff 4 3 110.25 Constraints: Constraint 1 2 4 72.00 <= 72 Constraint 2 3 6 108.00 >= 27 Constraint 3 -3 10 0.00 >= 0 LHS Sign RHS

X Y Solution 24.00 4.00 Problem 2-16 Obj coeff 4 7 124.00 Constraints: Constraint 1 2 3 60.00 >= 60 Constraint 2 4 2 104.00 >= 80 Constraint 3 1 24.00 <= 24 LHS Sign RHS

X Y Solution 12.00 20.00 Obj coeff 4 7 188.00 Problem 2-18 Constraints: Constraint 1 3 6 156.00 >= 100 Constraint 2 10 2160.00 >= 160 Constraint 3 2 40.00 >= 40 Constraint 4 2 24.00 <= 75 LHS Sign RHS

Problem 2-21 Painting Glazing Constraints: Collate time, minutes 1.8 2.0 1723.2 <= 1800 Min painting to print 1 624.0 >= 400 Min glazing to print 1 300.0 >= 300 LHS Sign RHS Print time, minutes 2.5 2.0 2160.0 <= 2160 Number of copies 624.00 300.00 Revenue - cost $4.50 $3.50 $3,858.0

Problem 2-22 Carolina Other Aroma rating 2 1.2 99.00 >= 99.00 Min Carolina, pounds 1 33.75 >= 33.00 LHS Sign RHS Number of pounds 33.75 26.25 Cost $1.80 $0.60 $76.50 Constraints: Pounds per bag 1 1 60.00 =60.00

Internet Print Max Internet 1 10 <= 15 Print v Internet 1 50 <= 50 Min Internet 1 10 >= 5 LHS Sign RHS Problem 2-23 Number used 10.00 50.00 Exposure 200000 80000 6000000 Constraints: Budget available $3,000 $900 $75,000 <= $75,000

Carolina South West $ invested $401.72 $2,750.79 Problem 2-27 Investment 1 1 $3,152.52 Constraints Short term appr 0.46 0.26 900.00 >= 900 3-year appr 1.72 1.93 6000.00 >= 6000 Dividend income 0.09 0.13 393.76 >= 300 LHS Sign RHS

Double Single Constraints Labor hours 3.2 2.4 960.00 <= 960 Single limit 1 200.00 <= 200 LHS Sign RHS Total limit 1 1 350.00 <= 400 Problem 2-28(a) Number of units 150.00 200.00 Revenue $225.00 $175.00 Direct labor cost $101.25 $70.00 Production cost $38.75 $30.00 Storage bag cost $20.00 $20.00 Profit $65.00 $55.00 $20,750.00 Constraints Total limit 1 1 400.00 <= 400 Labor hours 3.2

Regular LowFat Number of units 52.00 48.00 Problem 2-29 Profit $1.58 $1.65 $161.36 Constraints Espresso 1.00 1.00 100.00 <= 100.0 Skim Milk 1.25 60.00 <= 60.0 2% Milk 1 52.00 <= 60.0 Whipped Cream 0.5 26.00 <= 30.0 LHS Sign RHS

Large Small Number of sheds 30.00 70.00 Problem 2-30 Rent $50 $20 $2,900.00 Constraints: Advt. budget $1 $1 $100.00 <= $100 Sq feet required 150 50 8000.00 <= 8000 Rental limit 1 30.00 <= 40 LHS Sign RHS

Problem 2-31 Small Large Min required, total 1 1 424.00 >= 350 Min required, large 1 80.00 >= 80 LHS Sign RHS Square feet available 0.50 0.85 240.00 <= 240 Number of boxes 344.00 80.00 Rent $30 $40 $13,520.00 Constraints:

Problem 2-32 T-notes M-bonds Max T-notes 1 $145,000 <= $175,000 Max risk score 2 3 605,000 <= 605,000 Min T-notes 1 $145,000 >= $125,000 LHS Sign RHS Amount available 1 1 $250,000 <= $250,000 Amount invested $145,000 $105,000 Return 8.00% 9.00% $21,050 Constraints:

Bench Table Number of units 175.00 50.00 Problem 2-33 Profit $9 $20 $2,575.00 Constraints: Labor hours 4 6 1000.00 <= 1000.00 Redwood 10 35 3500.00 <= 3500.00 Bench vs Table 1 175.00 >= 100.00 LHS Sign RHS

A B Steel available 120 100 24500.00 <= 24500.00 Zinc available 20 30 4550.00 <= 6000.00 Models A vs B 1 175.00 <= 175.00 LHS Sign RHS Problem 2-34 Number of units 175.00 35.00 Profit $90 $70 $18,200.00 Constraints:

Core Elective Courses 35.00 25.00 Wages $2,600 $3,000 $166,000 Problem 2-35 Constraints: Total courses 1 1 60.00 >= 60 Credit hours 3 4 205.00 >= 205 Min core 1 35.00 >= 20 Min elective 1 25.00 >= 20 LHS Sign RHS

V231 V242 V784 V906 Inspection time 0.25 0.25 0.25 0.25 408.13 <= 525 Minimum X409 1 332.50 >= 200 Minimum X3125 1 250.00 >= 250 Minimum X4950 1 600.00 >= 600 Minimum X2173 1 450.00 >= 450 LHS Sign RHS Lathe time 1.20 0.60 0.50 0.70 1164.00 <= 1200 Problem 2-38 # of valves 332.50 250.00 600.00 450.00 Profit $16

Problem 2-39 Small Large Mini Min required, large 1 80.00 >= 80 Min required, mini 1 100.00 >= 100 LHS Sign RHS Min required, total 1 1 1 464.00 >= 350 Number of boxes 284.00 80.00 100.00 Rent $30 $40 $17 $13,420.00 Constraints: Square feet available 0.50 0.85 0.30 240.00 <= 240 Max space, large & mini 0.85 0.30 98.00

EC221 EC496 NC455 NC791 Minimum XJ201 1 20650.00 >= 150 Minimum XM897 1 100.00 >= 100 Minimum TR29 1 2750.00 >= 300 Minimum BR788 1 400.00 >= 400 LHS Sign RHS Assembly time 0.2 4.0 1.0 2.0 8080.00 <= 10000 Inspection time 0.5 1.0 0.5 0.5 12000.00 <= 12000 Problem 2-41 # of units 20,650.00 100.00 2,750.00 400.00 Profit $9

Full time FT OT 5-6pm FT OT 6-7pm PT start @ 9am PT start @ 10am PT start @ 11am PT start @ 12n PT start @ 1pm PT start @ 2pm PT start @ 3pm PT leave @ 1pm PT leave @ 2pm PT leave @ 3pm PT leave @ 4pm PT leave @ 5pm PT leave @ 6pm No. of employees 28.6 9.0 9.0

Problem 3-1 Canvas Plastic Nylon Leather Constraints Cost Nylon available 2.10 52.50 <= 700.0 $7.65 Leather available 2.60 104.00 <= 550.0 $9.45 Canvas & plastic labor 1.50 1.50 90.00 <= 90.0 $7.00 Nylon labor 1.70 42.50 <= 42.5 $8.00 Leather labor 1.90 76.00 <= 80.0 $9.00 Max Canvas 1 20.00 <= 40.0 Max Plastic 1 40.00 <= 40.0 Max Nylon

Problem 3-10 Paper Magazine TV Max Paper spend $1,000 $9,000.00 <= $9,000 Max Magazine spend $300 $9,000.00 <= $9,000 Max TV spend $2,000 $2,000.00 <= $9,000 LHS Sign RHS City 3.5% 0.5% 5.0% 51.50% >= 50% Suburbs 3.0% 1.0% 3.0% 60.00% >= 60% Number of units 9.00 30.00 1.00 Cost $1,000 $300 $2,000 $20,000.00 Constraints:

$1.00 coupon $0.85 coupon $0.70 coupon $0.55 coupon $0.40 coupon Number issued 4000.00 6500.00 1500.00 1500.00 1500.00 Min $1.00 coupon 1 4000.00 >= 3000.00 Min $0.70 coupon 1 1500.00 >= 1500.00 Min $0.85 coupon 1 6500.00 >= 1500.00 Min $0.55 coupon 1 1500.00 >= 1500.00 Min $0.40 coupon 1 1500.00 >= 1500.00 LHS Sign RHS Problem 3-11 Incr. sales 1.50 1.40 1.25 1.00

Problem 3-13 Miami M- bonds American Smart Car Green Earth Energy Rosslyn Drugs RealCo (Real Estate) Real Estate 1 $83,333.33 >= $50,000.00 > 10% Pharma 1 $50,000.00 >= $50,000.00 > 10% Municipal bonds 1 $100,000.00 >= $100,000.00 > 20% Automobile Stock 1 $191,666.67 >= $75,000.00 > 15% Energy Stock 1 $75,000.00 >= $75,000.00 > 15% Energy and Auto 1 1 $266,666.67 >=

Low-inc mort Conv mort Govt mort Bond Stock Futures Max stock 1 $1,875.00 <= $1,875.00 Max futures 1 $1,250.00 <= $1,250.00 High risk 1 1 1 $6,250.00 <= $7,500.00 Non-mortgage 1 1 1 $4,375.00 >= $4,375.00 LHS Sign RHS Max govt mort 1 $3,125.00 <= $3,125.00 Max bond 1 $1,250.00 <= $1,500.00 Problem 3-14 $ invested ('000) $2,500.00 $2,500.00 $3,125.00

Problem 3-16 $1 Jack in 52 $2 Jack in 52 $4 Jack in 52 $1 Red Face in 52 $2 Red Face in 52 $4 Red Face in 52 $1 Face in 52 $2 Face in 52 $4 Face in 52 $1 Red in 52 $2 Red in 52 $4 Red in 52 Constraints Max total bets $ $1 $2 $4 $1 $2 $4

Start at 2am Start at 6am Start at 10am Start at 2pm Start at 6pm Start at 10pm 6pm-10pm needs 1 1 46.0 >= 29 10pm-2am needs 1 1 23.0 >= 23 2am-6am needs 1 1 15.0 >= 8 LHS Sign RHS 2pm-6pm needs 1 1 23.0 >= 23 Problem 3-18 (a) Number of attendants 15.0 12.0 0.0 23.0 23.0 0.0 Objective 1

Problem 3-19 A and A (alt) B and B (alt) C and C (alt) D and D (alt) 5pm-7pm needs 1 1 14.00 >= 14 7pm-5am needs 1 1 10.00 >= 10 LHS Sign RHS 7am-5pm needs 1 1 16.00 >= 8 Number assigned 12.00 4.00 10.00 0.00 Pay $756.0 $840.0 $882.0 $924.0 $21,252.00 Constraints 5am-7am needs 1 1 12.00 >= 12

New Yorker Garaboldi Kentuckian Russian Number of drinks 32.00 0.00 24.00 32.00 Problem 3-2 Obj coefficient 1 1 1 1 88.00 Constraints Bourbon 1 4 128.00 <= 128 Brandy 1 1 32.00 <= 128 Vodka 1 3 128.00 <= 128 Dry vermouth 1 1 32.00 <= 32 Sweet vermouth 1 2 32.00 <= 32 LHS Sign RHS

Cargo A Cargo B Cargo C Cargo D Cargo E Cargo F Cargo G Cargo C limit (lbs) 1 0.00 <= 90 Cargo D limit (lbs) 1 0.00 <= 120 Cargo E limit (lbs) 1 125.45 <= 130 Cargo F limit (lbs) 1 0.00 <= 340 Cargo G limit (lbs) 1 250.00 <= 250 LHS Sign

A in R- Fore B in R- Fore C in R- Fore A in R- Center B in R- Center C in R- Center A in R-Aft B in R- Aft C in R- Aft A in L- Fore B in L- Fore C in L- Fore A in L- Center B in L- Center C in L- Center A in L- Aft B in L- Aft C in L- Aft R-Aft vol 2000 3500 3000 6,805.56 <= 14000 L-Fore vol 2000 3500 3000 10,000.00 <=

Problem 3-24 A in starboard A in port B in starboard B in port Weight in tons 10.22 4.44 15.78 27.56 Value $85 $85 $79 $79 $4,670.00 Constraints: Available A 1 1 14.67 <= 15.00 Available B 1 1 43.33 <= 54.00 Startboard weight 1 1 26.00 <= 26.00 Port weight 1 1 32.00 <= 32.00 Startboard volume 675 450 14,000.00 <=

Corn Tomato Potato Okra Constraints: Tending time 2 8 2 12 365.00 <= 550 Harvest time 6 20 920 695.00 <= 775 Water 2.5 3 2 3 156.25 <= 300 Fertilizer 50 60 45 35 2500.00 <= 2500 LHS Sign RHS Total acreage 1 1 1 1 57.50 <= 60 Planting time 10 15 12 18 775.00 <=

Wheat SE Wheat N Wheat NW Wheat W Wheat SW Alfalfa SE Alfalfa N Alfalfa NW Alfalfa W Alfalfa SW Barley SE Barley N Barley NW Barley W Barley SW SW acreage 1 1 1 0.00 <= 500 Wheat acreage 1 1 1 1 1 2200.00 <= 2200 Alfalfa acreage 1 1 1 1 1 165.52 <= 1200 Barley acreage 1 1 1 1 1 1000.00 <=

Problem 3-29 Union Non- union Temp Number hired 15.00 12.00 3.00 Salary $154 $144 $60 $4,218.00 Constraints: Cells needed 70 80 30 2100.00 >= 2100.00 Union vs Non-union 1 12.00 <= 12.00 Union vs Temp 1 3.00 <= 3.00 LHS Sign RHS Average cost of fuel cell = $2.01

Beef Pork Corn Lamb Rice Chicken Number of pounds 0.167 0.000 0.000 0.459 0.066 0.309 Problem 3-31 Cost $0.52 $0.49 $0.20 $0.40 $0.17 $0.39 $0.40 Constraints: Protein (%) 16.9 12.0 8.5 15.4 8.5 18.0 16.00 >= 16 Fat (%) 26.0 4.1 3.8 6.3 3.8 17.9 13.00 >= 13 Fiber (%) 29.0 8.3 2.7 2.4 2.7 28.8 15.00

Beef Pork Chicken Turkey Number of pounds 0.0313 0.0313 0.0000 0.0625 Cost $0.76 $0.82 $0.64 $0.58 $0.086 Problem 3-35 Constraints: Calories 640 1055 780 528 85.969 <= 100.000 Fat 32.5 54.0 25.6 6.4 3.103 <= 6.000 Cholesterol 210 205 220 172 23.719 <= 27.000 At least 25% beef 1 0.031 >= 0.031 At least 25% pork 1 0.031 >=

Problem 3-37 A in Tuffcoat B in Tuffcoat A in Satinwear B in Satinwear Number of gallons 320.00 1,280.00 625.00 625.00 Cost $4.50 $6.50 $4.50 $6.50 $16,635.00 Constraints: Tuff Coat demand 1 1 1600.00 >= 1600.00 Satinwear demand 1 1 1250.00 >= 1250.00 Min X in Tuff Coat 0.25 0.35 528.00 >= 528.00 Min Y in Tuff Coat 0.34 0.42 646.40

Feed A Feed B Feed C Feed D Feed E Number of lbs 8.00 2.00 2.00 12.00 16.00 Problem 3-39 Cost $0.960 $0.850 $0.775 $0.450 $0.375 $22.33 Constraints: Max B, C, & E 1 1 1 20.00 <= 20.00 Min A 1 8.00 >= 8.00 Min B 1 2.00 >= 2.00 Min C 1 2.00 >= 2.00 Min D

IC341 Sep IC341 Oct IC341 Nov IC341 Dec IC256 Sep IC256 Oct IC256 Nov IC256 Dec Beginning inv 0.00 875.00 0.00 900.00 0.00 0.00 1639.42 900.00 Total inv Sign Cap Needed Min (>=) Max (<=) Oct storage 1,639.42 <= 1800 Oct labor 1,034.50 1000 1200 Nov storage 1,800.00 <= 1800 Nov labor 1,000.00 1000 1200 Dec storage 1,650.00 <= 1800 Dec labor

Agent Jan Agent Feb Agent Mar Agent Apr Agent May Trainee Jan Trainee Feb Trainee Mar Trainee Apr Trainee May January agents 1 175.00 = 175 February agents -0.95 1 -1 0.00 = 0 March agents -0.96 1 -1 0.00 = 0 April agents -0.97 1 -1 0.00 = 0 May agents -0.98 1 -1 0.00 = 0 LHS Sign RHS April requirement 160 -80 31944.00

Problem 3-45 Col to Sav Col to Mob Col to Roa Gre to Sav Gre to Mob Gre to Roa Number shipped 250.00 200.00 0.00 0.00 25.00 210.00 Shipping cost $13 $42 $38 $25 $48 $17 $16,420.00 Constraints: Columbia available 1 1 1 450.00 <= 450 Greensboro available 1 1 1 235.00 <= 290 Savannah required 1 1 250.00 >= 250 Mobile required

From/To C HS NW HS SE HS East 0.0 0.0 923.0 923.0 =923.0 South 704.0 0.0 37.0 741.0 =741.0 West 0.0 793.0 0.0 793.0 =793.0 1200.0 1200.0 960.0 <= <= <= 1200.0 1200.0 1200.0 From/To C HS NW HS SE HS North 8 4 16 South 5 17 6 East 5 15 6 West 11 618 Total Miles = 19,634.0 Problem

From/To C HS NW HS SE HS East 0.0 0.0 923.0 923.0 =923.0 South 544.0 0.0 197.0 741.0 =741.0 West 0.0 793.0 0.0 793.0 =793.0 1120.0 1120.0 1120.0 <= <= <= 1200.0 1200.0 1200.0 From/To C HS NW HS SE HS North 8 4 16 South 5 17 6 East 5 15 6 West 11 618 Total Miles = 20,114.0 Problem

Smartphone Blueberry MoPhone BoldPhone LuxPhone4G Tap3G Solution value 496.55 1241.38 0.00 0.00 0.00 0.00 Problem 3-6 Revenue $200.00 $120.00 $180.00 $130.00 $430.00 $260.00 $248,275.86 Material cost $35.00 $25.00 $40.00 $45.00 $170.00 $60.00 $48,413.79 Labor cost $3.65 $2.05 $4.50 $2.50 $10.20 $8.25 $4,357.24 Profit $161.35 $92.95 $135.50 $82.50 $249.80 $191.75 $195,504.83 Constraints Cost Test device #1 7 3 12

W111 to make W222 to make W333 to make W111 to buy W222 to buy W333 to buy Min V512 prodn 1 2250.00 >= 2250.00 Min V612 prodn 1 1020.00 >= 1020.00 V412 demand 1 1 2000.00 >= 2000.00 V512 demand 1 1 3750.00 >= 3750.00 V612 demand 1 1 1700.00 >= 1700.00 LHS Sign RHS Min V412 prodn 1 1864.00 >=

Small Large Bumper Mailings Ads Total budget $25 $60 $30 $81 $1,000 90500.00 <= $90,500 Min small 1 100.00 >= 100 Min large 1 300.00 >= 50 Min bumper 1 40.00 >= 40 Min mailings 1 800.00 >= 500 Min radio 1 4.00 >= 3 LHS Sign RHS Max mailings 1 800.00 <= 800 Max radio 1 4.00 <=

Problem 3-15 T-bills Intl mutual School bonds Cert of deposit Tax-Free munis Stock market Max total "liquid" 1 1 1 $70,000.00 <= $80,000.00 Max T-bills 1 $30,000.00 <= $50,000.00 Min T-bills 1 $30,000.00 >= $20,000.00 Min intl mutual 1 $60,000.00 >= $20,000.00 Min school bonds 1 $20,000.00 >= $20,000.00 Min cert of deposit 1 $20,000.00 >= $20,000.00 Min tax-free munis 1 $30,000.00

Start at 1am Start at 5am Start at 9am Start at 1pm Start at 5pm Start at 9pm 1pm-5pm needs 1 1 15.0 >= 10 5pm-9pm needs 1 1 12.0 >= 12 9pm-1am needs 1 1 5.0 >= 5 LHS Sign RHS 9am-1pm needs 1 1 17.0 >= 17 Problem 3-17 Number of Nurses 4.0 9.0 8.0 7.0 5.0 0.0 Objective 1 1

English A English B English C English D Bilingual A Bilingual B Bilingual C Bilingual D 6am-9am English 1 13.00 >= 6 9am-12n English 1 1 15.00 >= 10 12n-3pm English 1 1 11.00 >= 8 3pm-6pm English 1 1 18.00 >= 12 6pm-9pm English 1 9.00 >= 6 6am-9am Bilingual 1 3.00 >= 3 9am-12n Bilingual 1 1 5.00 >= 5 12n-3pm Bilingual 1

Problem 3-22 Cargo A Cargo B Cargo C Cargo D Cargo E Cargo D limit (tons) 1 460.00 <= 2300 Cargo E limit (tons) 1 720.00 <= 3600 Min cargo A (tons) 1 194.00 >= 194 Min cargo B (tons) 1 556.00 >= 170 Min cargo C (tons) 1 380.00 >= 380 Min cargo D (tons) 1

Corn Tomato Potato Okra Corn Tomato Potato Okra Number of acres 59.93 0.00 0.00 0.00 50.91 0.00 0.00 13.09 Min Grover farm 1 1 1 1 59.93 >= 48 Min Gupta farm 1 1 1 1 64.00 >= 64 LHS Sign RHS Water 2.5 3 2 3 2.5 3 2 3 316.36 <= 510 Fertilizer 50

Corn in A Corn in B Corn in C Wheat in A Wheat in B Wheat in C Soy in A Soy in B Soy in C Wheat acreage 1 1 1 700.00 <= 700 Soy acreage 1 1 1 400.00 <= 1000 Total water 200 200 200 300 300 300 150 150 150 450000.00 >= 450000 Plot A vs Plot B

Junior Travel Deluxe Constraints Cost Finishing 0.18 0.24 0.20 456.00 <= 600 $8.50 Packing 0.20 0.20 0.20 450.00 <= 450 $7.25 Max Junior 1 1200.00 <= 1200 Max Travel 1 750.00 <= 1200 Max Deluxe 1 300.00 <= 1200 Min Junior 1 1200.00 >= 300 Min Travel 1 750.00 >= 300 Min Deluxe 1 300.00 >= 300 LHS Sign RHS Cutting

E in X F in X G in X E in Y F in Y G in Y E in Z F in Z G in Z E in Y 1 1000.00 <= 1000 E in Z 1 2100.00 <= 2100 F in X 1 2000.00 >= 2000 F in Y 1 200.00 >=

Oat Grain Mineral Number of pounds 1.500 1.000 2.500 Problem 3-32 Cost $0.33 $0.44 $0.57 $2.36 Constraints: Nutrient A 2.0 3.0 1.0 8.50 >= 6 Nutrient B 0.5 1.0 0.5 3.00 >= 2 Nutrient C 3.0 5.0 6.0 24.50 >= 9 Nutrient D 1.0 1.5 2.0 8.00 >= 8 Nutrient E 0.5 0.5 1.5 5.00 >= 5 Feed needed 1

Problem 3-34 Alloy 1 Alloy 2 Alloy 3 Iron 1 Iron 2 Carbide 1 Carbide 2 Carbide 3 Number of pounds 0.000 0.000 0.000 896.629 1060.674 0.000 0.000 42.697 Cost $0.12 $0.13 $0.15 $0.09 $0.07 $0.10 $0.12 $0.09 $158.79 Constraints: % Min Manganese 0.700 0.550 0.120 0.010 0.050 62.00 >= 42 2.10% Min Silicon 0.150 0.300

Ital in Comm Ital in Virgin Ital in Extra Span in Comm Span in Virgin Span in Extra Problem 3-36 Number of gallons 245.00 2,200.00 630.00 455.00 0.00 770.00 Cost $5.75 $5.75 $5.75 $6.50 $6.50 $6.50 $25,643.75 Constraints: Comm demand 1 1 700.00 >= 700.00 Virgin demand 1 1 2200.00 >= 2200.00 Extra demand 1 1 1400.00 >= 1400.00 Min Span in Extra

A B Protein Number of ml 93.680 175.410 55.910 Problem 3-38 Cost $0.00325 $0.00450 $0.02800 $2.66 Constraints: Fat (g) 0.008 0.012 0.002 2.97 <= 3 Carbohydrate (g) 0.147 0.096 0.024 31.95 <= 38 Sodium (mg) 1.770 0.720 0.320 310.00 <= 310 Protein (g) 0.006 0.009 0.230 15.00 >= 15 325 ml size 1 1 1 325.00 =325 LHS Sign RHS

Cheese Pepperoni Veggie Solution value 60.00 150.00 90.00 Problem 3-4 Profit $1.45 $1.75 $1.98 $527.70 Constraints Total pizzas 1 1 1 300.00 <= 350 Max cheese 1 60.00 <= 120 Max pepperoni 1 150.00 <= 150 Max veggie 1 90.00 <= 100 Min cheese 1 60.00 >= 60 Max pepperoni 1 150.00 >= 150 LHS Sign RHS

Tech Aug Tech Sept Tech Oct Tech Nov Tech Dec Trainee Aug Trainee Sept Trainee Oct Trainee Nov Trainee Dec Dec need 130 -90 49000.00 >= 45000 Aug technicians 1 350.00 = 350 Sept technicians -0.98 1 -1 0.00 = 0 Oct technicians -0.98 1 -1 0.00 = 0 Nov technicians -0.98 1 -1 0.00 = 0 Dec technicians -0.98 1 -1 0.00 = 0 LHS

A-Y1 B-Y1 C-Y1 D-Y1 A-Y2 B-Y2 C-Y2 D-Y2 A-Y3 B-Y3 C-Y3 D-Y3 A-Y4 B-Y4 C-Y4 A-Y5 B-Y5 A-Y6 Y6 cash flow -1.32 -1.24 -1.16 -1.07 1 0.000 = 0.000 Y6 end need 1.32 1.24 1.16 1.07 500.000 = 500.000 Y1 cash flow 1 1 1 1 324.198 <= 356.000 Max invest <= 120.000 LHS Sign

1 x 4 + 2 x 4 + 2 x 4 + Problem 3-44 (a) Cutting pattern (number x width) 14-inch rolls 20-inch rolls 3 x 4 1 x 9 1 x 12 5 x 4 1 x 9 1 x 12 2 x 9 Number of rolls 0.00 500.00 1000.00 0.00 0.00 0.00 750.00 Cost $600 $600 $600 $1,100 $1,100

Itallian French Caribbean Number of units 116.67 60.00 350.00 Problem 3-5 Profit $72 $65 $78 $39,600.00 Constraints: Carpentry 3.00 2.25 2.50 1360.00 <= 1360 Painting 1.50 1.00 1.25 672.50 <= 700 Finishing 0.75 0.75 0.85 430.00 <= 430 Itallian 1 116.67 >= 60 French 1 60.00 >= 60 Caribbean 1 350.00 >= 60 LHS Sign RHS

Gear A to make Gear B to make Gear C to make Gear D to make Gear A to buy Gear B to buy Gear C to buy Gear D to buy Constraints: Cost/Hr Forming 0.30 0.36 0.38 0.45 388.50 <= 500 $9.00 Hardening 0.20 0.30 0.24 0.33 278.30 <= 300 $8.00 Deburring 0.30 0.30 0.35 0.25 310.00 <= 310 $7.50 Gear

M-Dem M-Rep M-Ind F-Dem F-Rep F-Ind Min M-Rep 1 1125.00 >= 450 Min M-Ind 1 450.00 >= 450 Min F-Dem 1 525.00 >= 450 Min F-Rep 1 675.00 >= 450 Min F-Ind 1 550.00 >= 450 Max Dem 1 1 1700.00 <= 1800 Max Rep 1 1 1800.00 <= 1800 Max M-Rep 1 1125.00 <= 1125 LHS Sign RHS Min

(b) If we increase the RHS of the first constraint to 90, the feasible region shrinks and the objective function value worsens. Since the increase of 30 is within the allowable increase, the same corner point (i.e., the intersection

X Y Solution 7.00 1.50 Obj coeff 4 5 35.50 Constraints: Problem 2-14 Constraint 1 5 2 38.00 <= 40 Constraint 2 3 6 30.00 <= 30 Constraint 3 1 7.00 <= 7 Constraint 4 2 -1 12.50 >= 3 LHS Sign RHS Worksheet: [P4-10.xls] Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease Constraints Final Shadow Constraint Allowable Allowable Cell

X Y Constraints: Constraint 2 3 6 108.00 >= 27 Constraint 3 -3 10 0.00 >= 0 LHS Sign RHS Constraint 1 2 4 72.00 <= 72 Problem 2-15 Solution 22.50 6.75 Obj coeff 4 3 110.25 Worksheet: [P4-12.xls] Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$4 Solution X 22.500 0.000 4.000 1E+30 2.500 Microsoft Excel 14.0

B1B2B3N1N2N3 <= 30 and border 31-50 and border >= 51 and border <= 30 and not border 31-50 and not border >= 51 and not border Border Mexico 1 1 1 740.00 >= 345 <= 30 and not border 1 1000.00 >= 500 >= 51 and border 1 140.00 <= 140 LHS Sign RHS <= 30 households 1 1 1000.00 >=

Oat Grain Mineral Number of pounds 1.500 1.000 2.500 Problem 3-32 Cost $0.33 $0.44 $0.57 $2.36 Constraints: Nutrient A 2.0 3.0 1.0 8.50 >= 6 Nutrient B 0.5 1.0 0.5 3.00 >= 2 Nutrient C 3.0 5.0 6.0 24.50 >= 9 Nutrient D 1.0 1.5 2.0 8.00 >= 8 Nutrient E 0.5 0.5 1.5 5.00 >= 5 Feed needed 1

X Y Solution value 2.00 4.00 Problem 4-20 Profit $4 $5 $28.00 Constraints Labor 1 2 10.00 <= 10 Material 6 6 36.00 <= 36 Storage 8 4 32.00 <= 40 LHS Sign RHS Problem 4-20 Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $D$7 Labor

Standard Deluxe Luxury Solution value 540.00 252.00 0.00 Good-to-Go Suitcase Company Selling price per unit $36.05 $39.50 $43.30 $29,421.00 Material cost per unit $6.25 $7.50 $8.50 $5,265.00 Labor cost per unit $19.80 $23.00 $25.30 $16,488.00 Profit $10.00 $9.00 $9.50 $7,668.00 Constraints Cost Cutting & Coloring 0.70 1.00 1.00 630.00 <= 630 $10 Assembly 0.50 0.83 0.67 480.00 <= 600

Oak tables Oak chairs Cherry tables Cherry chairs Pine tables Pine chairs Min cherry tables 1 3.00 >= 3 Min oak chairs 1 51.67 >= 10 Min cherry chairs 1 85.56 >= 10 Min pine chairs 1 33.08 >= 5 LHS Sign RHS Min oak tables 1 3.00 >= 3 Classic Furniture Company Number of units 3.00 51.67 3.00 85.56 42.26 33.08 Profit $75

breakeven analysis to test the model’s sensitivity to selling price, fixed cost, and variable cost. Likewise, it is important in inventory models in which we test the result’s sensitivity to changes in demand, lead- time, costs, and so on. decrease means fewer units are available. Obviously, if more units of a

Phosphoric acid Urea Ammonium phosphate Ammonium nitrate Chlorine Caustic soda Vinyl chloride Hydroflouric acid Max chlorine 1 480.00 <= 480 Max caustic soda 1 600.00 <= 600 Max vinyl chloride 1 300.00 <= 300 Max hydro acid 1 320.00 <= 320 Natural gas avail 5.5 7.0 8.0 10.0 15.0 16.0 12.0 11.0 32240.00 <= 36000 LHS Sign RHS Case: Coastal States Chemicals and Fertilizers

X Y Solution 24.00 4.00 Obj coeff 4 7 124.00 Problem 2-16 Constraints: Constraint 1 2 3 60.00 >= 60 Constraint 2 4 2 104.00 >= 80 Constraint 3 1 24.00 <= 24 LHS Sign RHS Worksheet: [P4-11.xls] Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase

T N P A TV spots Newspaper ads Prime-time radio spots Afternoon radio spots Constraints: Maximum newspaper 1 5.00 <= 5 Max prime-time radio 1 6.21 <= 25 Max afternoon radio 1 0.00 <= 20 Total budget $800 $925 $290 $380 $8,000.00 <= $8,000 Maximum radio $ $290 $380 $1,800.00 <= $1,800 Minimum radio spots 1 1 6.21 >= 5 LHS Sign

A B C Grain A Grain B Grain C Constraints: Riboflavin 16 14 25 2.35 >= 2 Phosphorus 8 7 9 1.00 >= 1 Magnesium 5 6 0.425 >= 0.425 Total Mix 1 1 1 0.125 =0.125 LHS Sign RHS Protein 22 28 21 3.00 >= 3 Whole Food Nutrition Center Number of pounds 0.025 0.050 0.050 Cost $0.33 $0.47 $0.38

Wheat SE Wheat N Wheat NW Wheat W Wheat SW Alfalfa SE Alfalfa N Alfalfa NW Alfalfa W Alfalfa SW Barley SE Barley N Barley NW Barley W Barley SW Number of acres 2,000.00 200.00 - - - -

X Y Solution value 15.00 10.00 Problem 4-21 Profit $5 $5 $125.00 Constraints Resource 1 2 3 60.00 <= 60 Resource 2 4 2 80.00 <= 80 Resource 3 1 15.00 <= 18 LHS Sign RHS Problem 4-21 Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease Microsoft Excel 14.0 Sensitivity Report $B$4 Solution value X 15.00 0.00 5.00

TiniTote TubbyTote ToddleTote Solution value 100.00 35.00 90.00 Selling price per unit $63.75 $82.50 $66.00 $15,202.50 Constraints Cost Sewing 2.0 1.0 2.0 415.00 <= 500 $8.50 Assembly 1.0 3.0 2.0 385.00 <= 480 $8.75 Tinitote demand 1.0 100.00 <= 180 Tubbytote demand 1.0 35.00 <= 70 Toddletote demand 1.0 90.00 <= 160 Toddletote max prod ratio -0.4 -0.4 0.6

Tuna Tuna/Ch Ham Ham/Ch Cheese Lettuce (oz.) 0.25 0.25 0.25 0.25 0.25 17.50 <= 20 Tomato (oz.) 0.5 0.5 0.5 0.5 0.5 35.00 <= 40 Package (unit) 1 1 1 1 1 70.00 <= 72 Labor (hrs) 0.08 0.08 0.08 0.08 0.08 5.60 <= 8 Min total 1 1 1 1 1 70.00 >= 70 Min

Node 3 20 20 15 Node 4 5 20 15 10 Node 5 30 15 15 25 20 Node 6 25 20 40 Node 7 20 20 20 15 Node 8 10 20 10 Node 9 15 10 15 Node 10 40 15 Least minutes = 60 Flows: From Node 1 Node 2 Node 3 Node 4 Node 5 Node

Shipments: From Chicago Milwaukee M'polis Detroit Flow out Costs: From Chicago Milwaukee M'polis Detroit Gary $10 $20 $40 $25 Fort Wayne $20 $30 $50 $15 Total cost = $9,000 Note: This existing shipping pattern is specified in the case and is NOT obtained using the transportation model. To To Case: Custom Vans, Inc. (Existing pattern) Gary 200.0 0.0 100.0 0.0

Assignments: From Prep Assem Finish Pack Flow out Node Flow in Flow out Net flow Sign RHS Tom 1.0 0.0 0.0 0.0 1.0 Tom 0 1 -1 =-1 Assem 1 0 1 = 1 Hours: Finish 1 0 1 = 1 From Prep Assem Finish Pack Pack 1 0 1 = 1 George 80 80 60

feet instead of 800 feet, the minimum total distance increases to 10,100 feet (101). A decision now has to be made about whether the increased distance and cost for the water and sewer system is worth the

CHAPTER 5 TRANSPORTATION, ASSIGNMENT, AND NETWORK MODELS Internet Case Study: Northwest General Hospital See file Ch05 Internet Case Northwest General.XLS. This is a transportation model. The total time is 4,825 minutes. One possible solution is shown below. Alternate solutions are possible. To From Wing 1

Cars: From Columbia Albany Springfield Pleasantburg Flow out Location Flow in Flow out Net flow Sign RHS Albany 45 045 =45 Distances: Springfield 25 025 =25 From Columbia Albany Springfield Pleasantburg Pleasantburg 20 020 =20 Parris 50 30 60 70 Butler 20 80 10 90 Century 100 40 80 30 Total car-miles = 3,100.0 To Problem 5-14 To Flow balance equations Parris 0.0

Problem 5-16 Number Windshields From M1 demand M2 demand M3 demand M4 demand Flow out Location Flow in Flow out Net flow Sign RHS M2 demand 140 0140 =140 Unit costs: M3 demand 260 0260 =260 From M1 demand M2 demand M3 demand M4 demand M4 demand 120 0120 =120 Beg inventory

Shipments: From Los Angeles New York Flow out Location Flow in Flow out Net flow Sign RHS Flow balance equations Problem 5-17 (with New Orleans) To Atlanta 0.0 600.0 600.0 Atlanta 0 600 -600 = -600 Tulsa 800.0 100.0 900.0 Tulsa 0 900 -900 = -900 New Orleans 0.0 500.0 500.0 New Orleans 0 500 -500 =

Shipments: From Atlanta Boston Charlestown Dover Flow out Location Flow in Flow out Net flow Sign RHS Eaglestown 8000.0 0.0 0.0 0.0 8000.0 Eaglestown 8000 -8000 = -8000 Boston 9000 9000 =9000 Unit costs: Charlestown 10000 10000 =10000 From Atlanta Boston Charlestown Dover Dover 5000 5000 =5000 Eaglestown $6.0 $7.0 $7.5 $7.5 Farrier $5.5 $5.5 $4.0 $7.0 Guyton

Shipments: From Charlotte Greenville Atlanta Wise Cleveland Flow out Location Flow in Flow out Net flow Sign RHS Greenville 200 200 =200 Unit costs: Atlanta 350 350 =350 From Charlotte Greenville Atlanta Wise Cleveland Wise $20 $25 $22 $0 $300 Humbolt $17 $27 $25 $300 $300 Cleveland $21 $20 $22 $300 $0 Brevard $29 $30 $30 $6 $5 Total

Shipments: From X Y Z J K L Flow out Site Flow in Flow out Net flow Sign RHS J0.0 325.0 0.0 0.0 0.0 0.0 325.0 J325 325 0 = 0 A0.0 0.0 0.0 100.0 0.0 300.0 400.0 A 0 400 -400 >= -400 C0.0 0.0 0.0 225.0 0.0 0.0 225.0 C 0 225

Assignments: Pitcher Des Moines Davenport Omaha Peoria Flow out Node Flow in Flow out Net flow Sign RHS Jones 0.0 1.0 0.0 0.0 1.0 Jones 0 1 -1 =-1 Davenport 1 0 1 = 1 Probabilities: Omaha 1 0 1 = 1 Pitcher Des Moines Davenport Omaha Peoria Peoria 1 0 1 = 1 Jones 0.40

Assignments: Ambulance site ENE SE CFlow out Node Flow in Flow out Net flow Sign RHS Site 1 0.0 0.0 1.0 0.0 1.0 Site 1 1 -1 =-1 Flow balance equations Problem 5-26 Population Center Site 2 0.0 1.0 0.0 0.0 1.0 Site 2 1 -1 =-1 Site 3 1.0 0.0 0.0 0.0 1.0 Site 3 1

Assignments: Professor Stat Mgmt Fin Econ Flow out Node Flow in Flow out Net flow Sign RHS Mgmt 1 1 = 1 Ratings: Finance 1 1 = 1 Professor Stat Mgmt Fin Econ Economics 1 1 = 1 Kelley 80 60 80 75 Davidson 65 55 80 60 Merkle 95 40 65 55 Total rating = 310 Strausbaugh 70

Flows: From Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Flow out Node Flow in Flow out Net flow Sign RHS Node 1 0.0 2.0 2.0 1.0 0.0 0.0 0.0 0.0 5.0 Node 1 5.0 5.0 0.0 = 0.0 Node 2 0.0 0.0 0.0 0.0

Minimal-spanning tree model Only optimal solution to this problem is shown by the bold arcs. Total length = 177 units Problem 5-32

Node 4 0 0 0 0 0 0 8 2 0 0 0 Node 5 4 0 0 0 0 0 0 14 0 0 0 Node 6 0 0 0 0 0 0 0 0 4 0 0 Node 7 0 0 1 0 0 0 0 0 0 8 0 Node 8

Node 5 0 0 0 0 0 0 0 0 4 0 0 0 0 0 Node 6 0 0 2 0 0 0 2 0 0 3 0 0 6 0 Node 7 0 0 1 0 0 1 0 0 0 0 2 0 0 0 Node 8 0 0 0

Node 3 1 0 0 0 0 2 2 0 0 0 0 0 0 0 Node 4 3 0 0 0 0 0 0 1 0 0 0 0 0 0 Node 5 0 0 0 0 0 0 0 0 4 0 0 0 0 0 Node 6 0 02000200 00

Node 3 200 10 200 200 200 200 16 200 200 200 200 200 200 200 Node 4 200 200 200 200 200 200 200 200 200 200 200 200 200 200 Node 5 26 200 200 200 200 22 200 18 200 200 200 200 200 200 Node 6 19 200 200

Minimal-spanning tree model Problem 5-40 2 5 6 3 8 7 9 1 6 3 4 7 4 8 3 5 6 3 9 8 8 10 9 2 4 6 4 6 An optimal solution to this problem is shown by the bold arcs. Minimum cost = 28 (= $28,000)

equal to total supply (from all sources). If a problem is unbalanced, either the demand or the supply constraints must be inequalities. number of possible assignments to be considered. In the 5 x 5 case, there are 5! = (5 x 4 x 3 x

#1 closed $188,360 $33,000 $221,360 (see sheet #1C) #2 closed $183,430 $34,000 $217,430 (see sheet #2C) #3 closed $188,930 $33,500 $222,430 (see sheet #3C) The lowest weekly total cost, with plant #2 closed, is $217,430. This is $3,300 per week

Flows: From Wing 1 Wing 2 Wing 3 Wing 4 Wing 5 Wing 6 Flow out Location Flow in Flow out Net flow Sign RHS Station 5A 0.0 0.0 120.0 0.0 0.0 80.0 200.0 Station 5A 200 -200 = -200 To Flow balance equations Internet Case: Northwest General Hospital Station 3G 80.0 0.0 0.0 85.0 60.0

Shipments: From WH 1 WH 2 WH 3 WH 4 WH 5 Flow out Location Flow in Flow out Net flow Sign RHS Plant 1 RT 0.0 13000.0 0.0 14000.0 0.0 27000.0 Plant 1 RT 27000 -27000 >= -27000 Plant 1 OT 0.0 0.0 0.0 0.0 0.0 0.0 Plant 1 OT 0 0

Students: From School B School C School E Flow out Location Flow in Flow out Net flow Sign RHS Sector A 500.0 300.0 0.0 800.0 Sector A 0 800 -800 = -800 School C 1100 01100 <= 1100 Distances: School E 800 0800 <= 1100 From School B School C School E Sector A 6 7

Shipments: From Job 1 Job 2 Job 3 Flow out Location Flow in Flow out Net flow Sign RHS Acme 2250.0 3750.0 0.0 6000.0 Acme 0 6000 -6000 >= -6000 Flow in 2500.0 3750.0 4850.0 Job 1 2500 0.0 2500 =2500 Job 2 3750 0.0 3750 =3750 Unit costs: Job 3 4850 0.0 4850 =4850 From

Loan: From Hill St. Banks St. Park Ave. Drury Ln. Flow out Location Flow in Flow out Net flow Sign RHS Banks St. 40000 040000 =40000 Interest rates: Park Ave. 130000 0130000 =130000 From Hill St. Banks St. Park Ave. Drury Ln. Drury Ln. 70000 070000 =70000 First Homestead 0.08 0.08 0.10 0.11 Commonwealth 0.09 0.09

Shipments: From Charlotte Greenville Atlanta Flow out Location Flow in Flow out Net flow Sign RHS Greenville 200 0200 =200 Unit costs: Atlanta 350 0350 =350 From Charlotte Greenville Atlanta Wise $20 $25 $22 Humbolt $17 $27 $25 Cleveland $21 $20 $22 Brevard $29 $30 $30 Total cost = $17,400.00 To To Flow balance equations Problem 5-19 (with Brevard) Wise 50.0 0.0 250.0 300.0

Shipments: From D E W X Y Z Flow out Site Flow in Flow out Net flow Sign RHS D0.0 0.0 0.0 300.0 0.0 400.0 700.0 D700 700 0 = 0 A450.0 0.0 0.0 0.0 0.0 0.0 450.0 A 0 450 -450 = -450 C250.0 130.0 0.0 0.0 0.0 0.0 380.0 C 0 380

Assignments: Job W X Y Z Flow out Node Flow in Flow out Net flow Sign RHS A - - 1 - 1.0 A 0 1 -1

Assignments: Nurses Urology Cardiology Orthopedics Pediatrics Flow out Node Flow in Flow out Net flow Sign RHS Morris - 1 -

Assignments: Squad A B C D E Flow out Node Flow in Flow out Net flow Sign RHS Squad 1 - - 1 - -

Minimal-spanning tree model An optimal solution is shown by the bold arcs. Total length = 45 (= 4,500 feet). Problem 5-29

Flows: From N-1 N-2 N-3 N-4 N-5 N-6 N-7 N-8 N-9 N-10 N-11 N-12 N-13 Flow out Node Flow in Flow out Net flow Sign RHS Node 1 0.0 1.0 0.0 1.0 Node 1 0.0 1.0 -1.0 = -1.0 Node 2 0.0 0.0 0.0 Node 2 0.0 0.0 0.0 = 0.0 Node 3 0.0

Flows: From Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Flow out Node Flow in Flow out Net flow Sign RHS Node 1 0.0 55.0 45.0 67.0 0.0 0.0 167.0 Node 1 167.0 167.0 0.0 = 0.0 Node 2 0.0 0.0 0.0 0.0 55.0 0.0 55.0 Node 2 55.0 55.0

Node 4 200 10 8200 12 200 12 200 200 200 200 200 200 200 Node 5 26 200 200 12 200 22 200 18 18 200 200 200 200 200 Node 6 19 200 200 200 22 200 200 200 200 20 200 200 200 200 Node 7 200 200 16 12

Problem 5-39 Minimal-spanning tree model An optimal solution to this problem is shown by the bold arcs. Minimum distance = 21 (= 2,100 yards)

1 2 3 4 5 6 LHS RHS 418.27 0.00 0.00 1.00 0.00 0.00 0.00 1.0 = 1.0 635.20 0.00 1.00 0.00 0.00 0.00 0.00 1.0 = 1.0 817.20 0.00 0.00

Bath & Body Beauty Parlor Children's Clothing Chinese Food Electronics Store Game Arcade Hardware Store Hobby Store Icecream Store Kitchen Store Lingerie Store Luggage Store Men’s Clothing Pizza Parlor Shoe Store Sports Bar Tex-Mex Food Toy Store Travel Agent Women’s Clothing At most 3 Food 1 1 1 1 1 3.00 <= 3 At most 3 Specialty 1 1 1 1 2.00 <= 3 Food&Service vs Rest 1 1 1 1 1

House Lawyer Doctor Problem 6-13 Number of units 8.0 10.0 Profit $90.00 $133 $250 $2,050.00 Constraints: Labor Hours 6 7 10 118.00 <= 120 Houses 1 8.00 >= 8 LHS Sign RHS

Problem 6-14 7A7s 7B7s 7C7s Number of units 9.0 8.0 0.0 Passengers 125000 95000 80000 1885000.0 Constraints: Total planes 1 1 1 17.0 =17.0 $ available (millions) $38 $27 $22 558.0 <= 700.0 $ maint (millions) $0.8 $0.6 $0.5 12.0 <= 12.0 Minimum 7A7s 1 9.0 >= 5.7 LHS Sign RHS

Problem 6-16 Single Double ICU CCU Constraints: Space (Sq. Ft.) 300 360 320 340 39960.0 <= 40000.0 Max Double 1 50.0 <= 50.0 Max ICU 1 20.0 <= 20.0 Max CCU 1 14.0 <= 20.0 Min Single 1 36.0 >= 35.0 Min ICU 1 20.0 >= 10.0 Min CCU 1 14.0 >= 10.0 LHS Sign RHS $ available (millions) $45 $54

$1 $0.25 $0.10 $0.05 $0.01 $1 available 1 5 <= 8 $0.25 available 1 2 <= 9 $0.10 available 1 0 <= 7 $0.05 available 1 1 <= 11 $0.01 available 1 2 <= 10 LHS Sign RHS Problem 6-17 Number used 5.0 2.0 0.0 1.0 2.0 Objective coeff 1 1 1 1 1 10.00 Constraints: Amount req'd 1 0.25 0.10

Problem 6-18 Andover Hamilton Easy Nebraska Southern Manuel Builders Rest Invest? (1 = Yes) 1.0 0.0 0.0 1.0 1.0 0.0 1.0 1.0 Exp return ($'000) $35 $100 $30 $65 $45 $20 $90 $50 $285.0 Constraints: Cost ($'000) $400 $1,000 $350 $700 $490 $270 $800 $500 $2,890.0 <= $3,000 No more than 1 bond 1 1 1.0

Problem 6-19 A B C D E F G Selected? (1 = yes) - 1 1 1 1 1

Problem 6-20 A B C D E F G Selected? (1 = yes) 1.0 1.0 1.0 0.0 0.0 1.0 0.0 Value $1,800 $1,400 $1,100 $900 $1,600 $1,100 $1,200 $5,400.0 Constraints: Capacity 700 600 450 400 650 350 600 2,100.0 <= 2200 B and G 1 1 1.0 <= 1 B, C, and D 1 1 1 2.0

Problem 6-22 P1 P2 P3 P4 P5 P6 Selected? (1 = yes) 0.0 0.0 0.0 1.0 1.0 1.0 NPV ($'000) $140 $260 $88 $124 $176 $192 $492.0 Constraints: Year 1 budget $80 $95 $58 $32 $115 $48 $195.0 <= $225 Year 2 budget $25 $40 $17 $24 $25 $20 $69.0 <= $80 Year 3 budget $22 $5

A B C D E F E requirements 1 1 1 2.0 >= 1 F requirements 1 1 1 1 1.0 >= 1 LHS Sign RHS Community Within 5 mts A A, B, D, E BA, B, C, F CB, C, E, F DA, D, F EA, C, E FB, C, D, F C requirements 1 1 1 1 1.0 >=

C1 C2 C3 T1 T2 T3 M1 M2 M3 B1 B2 B3 C T M B Memp capacity 1 1 1 7000.0 <= 7250 7250 Buff capacity 1 1 1 0.0 <= 07500 Buffalo : Tulsa 1 0.0 <= 1 Reg1 demand 1 1 1 1 3000.0 >= 3000 Reg2 demand 1 1 1 1

Assignment: Location A B C D E F G H Flow out Node Flow in Flow out Net flow Sign RHS 1 1 1 - 1 1 1 - - 5.0 1 5

ND NT NC D T C Solution values 625.0 0.0 0.0 1.0 0.0 0.0 Revenue $135 $110 $50 Problem 6-28 Regular variables Binary variables Decision variables: D = 1 if desks are made, = 0 if not T = 1 if tables are made, = 0 if not C = 1 if chairs are made, = 0 if not Note:

2-draw 3-draw d1-d1+d2-d3-d4- Solution value 650.00 300.00 0.00 0.00 25.00 0.00 100.00 Obj Coefficient 1 1 0.00 Constraints: Problem 6-30 (Rank R1 Goals Only) Profit 12 14 1-1 12000.00 =12000 Production limit 1.5 2 1 1600.00 =1600 Two-drawer sales 1 1 650.00 =650 Three-drawer sales 1 1 400.00 =400 LHS Sign RHS 2-draw 3-draw d1-d1+d2-d3-d4- Solution value 533.33 400.00 0.00 0.00

Begin Mon Begin Tue Begin Wed Begin Thu Begin Fri Begin Sat Begin Sun d1- d1+ d2+ d3+ Constraints: Sunday goal 0 0 1 1 1 1 1 -1 15 =15 Monday need 1 0 0 1 1 1 1 25 >= 23 Tuesday need 1 1 0 0 1 1 1 25 >= 16 Wednesday need 1 1 1 0 0

BKT BB PG SF d1- d2- d2+ d3- d3+ d4- d4+ Solution values 4.00 5.00 0.00 6.00 0.00 0.00 0.00 0.00 0.00 0.00 4.00 Objective coeff 1 0.00 Constraints: Budget goal 300 250 75 175 1 3500.00 = 3500 Problem 6-34 (Rank R1 Goals Only) Min usage goal 700 1000 800 1200 1-1 15000.00 = 15000 Max

A B G P S d1- d1+ d2- d2+ d3- d3+ Solution values 0.00 10.92 3.19 5.16 4.73 0.00 0.00 0.00 4.92 0.00 2.16 Objective coeff 1 0.00 Constraints: Problem 6-36 (Rank R1 Goals Only) Cost goal 1.49 0.49 1.69 0.99 2.99 1 -1 30.00 = 30 Banana goal 1 1 -1 6.00 = 6 Pear goal

X1X2X3 Solution value 2.80 4.92 3.37 X1X2X3X12X22X33 Variable terms 2.80 4.92 3.37 7.83 24.17 38.40 Objective coeff 10 15 25 186.07 Constraints: Problem 6-37 Constraint 1 1 1 1 11.09 <= 12 Constraint 2 1 1 32.00 <= 32 Constraint 3 2 1 44.00 <= 44 LHS Sign RHS

X1X2X3 X1X2X3X1X2X33 Variable terms 3.00 1.00 8.00 3.00 512.00 Objective coeff 2 1 -2 3 7 3,584.00 Constraints: Constraint 1 2 4 3 34.00 >= 34 Constraint 2 3 1 10.00 >= 10 Constraint 3 1 1 1 12.00 <= 12 LHS Sign RHS Problem 6-38 Solution value 3.00 1.00 8.00 X1X2X3 X1X2X3X1X2X33 Variable terms 3.00 1.00 8.00 3.00 512.00 Objective coeff 2 1

XJ6 XJ7 XJ8 Solution value 13.32 7.49 9.65 Problem 6-40 XJ6 XJ7 XJ8 XJ62XJ72XJ82 Variable terms 13.32 7.49 9.65 177.39 56.08 93.22 Objective coeff 4 5 6 -0.1 -0.2 -0.2 $101.05 Constraints: Testing hours 1 1.5 1.6 40.00 <= 40 LHS Sign RHS XJ6 XJ7 XJ8 XJ6 XJ7 XJ8 XJ62XJ72XJ82 Variable terms 13.32 7.49 9.65 177.39 56.08 93.22 Objective coeff 4 5 6 -0.1

QAQBPAPB QAQBPAPBQA*PAQB*PB Variable terms 275.00 212.50 26.13 19.06 7184.38 4049.31 Objective coeff -18 -12 1 1 $3,733.68 Constraints: Demand for A 1 200 5500.00 =5500 Demand for B 1 225 4500.00 =4500 A capacity 1 275.00 <= 275 B capacity 1 212.50 <= 350 Labor capacity 1 2 700.00 <= 700 LHS Sign RHS Profit function: Max Z = (PA-18)*QA+ (PB-12)*QB= QA*PA-18QA+

G to H G to N G to G G to C R to H R to N R to G R to C H to H H to N H to G H to C General 1 1 1 1.0 <= 1 CBT 1 1 1 0.0 <= 1 Gardener 1 1

6-2. The student should provide five realistic examples of IP. One good exercise would be to require students to find five articles and use those as examples. are integers. Mixed IP models are more common, as firms generally have only a few variables that must be integers. 6-5. Deviation

Sun A Mon A Tue A Wed A Thu A Fri A Sat A Sun B Mon B Tue B Wed B Thu B Fri B Sat B Sun C Mon C Tue C Wed C Th u C Fri C Sat C No. of collectors 0 3 5 0 6 0 4 0 1 3 3 3 3 3 4 3 0 5 0 6 0 Obj coefficient 1 1 1 1

Begin Mon Begin Tue Begin Wed Begin Thu Begin Fri Begin Sat Begin Sun Number scheduled 11.0 1.0 3.0 2.0 4.0 6.0 0.0 Objective coeff 1 1 1 1 1 1 1 27 Problem 6-15(a) Constraints: Monday need 1 1 1 1 1 23 >= 23 Tuesday need 1 1 1 1 1 22 >= 16 Wednesday need 1 1 1 1 1

Problem 6-21 A B C D E F G H Selected? (1 = yes) - 1 1 1 - 1

Problem 6-23 Atlanta Boston Chicago Dallas Denver LA Phila Seattle Selected? (1 = yes) 1 - - -

PN PS PW TN TS TW CN CS CW DN DS DW P T C D Constraints: Capacity Phil capacity 1 1 1 250.0 <= 250 250 TB capacity 1 1 1 0.0 <= 0260 Chi capacity 1 1 1 0.0 <= 0280 Den capacity 1 1 1 250.0 <= 270 270 TB or Chi 1

#B1 #B2 #B3 #B4 #B5 B1Pub B2Pub B3Pub B4Pub B5Pub Solution values 0.0 8000.0 5000.0 0.0 7000.0 0.0 1.0 1.0 0.0 1.0 Revenue $40 $60 $52 $34 $45 Problem 6-29 Binary variables Regular variables BK1 = # of Book 1 published BK2, BK3, Bk4, and Bk5 likewise BK1_P = 1 if Book 1 published; = 0 if not BK2_P,

TV News d1-d1+d2-d2+d3-d3+d4-d4+ Solution value 16.00 20.00 - - - 6.00 - - 1.20 - Problem 6-31 (Rank R1 Goal Only) Obj coefficient 1 0.00 Constraints: Budget 10000 4000 1-1 240000.00 =240000 TV spots 1 1 -1

CM CB SM SB HM HB d1- d1+ d2- d2+ d3- d3+ Constraints: Cost goal 2.60 3.12 1.80 2.16 1.40 1.68 1 -1 14000.00 = 14000 Fabrication goal 0.5 0.2 0.6 1 -1 3000.00 = 3000 Coils bought goal 1 1 -1 200.00 = 200 Molding capacity 0.4 0.7 0.3 600.00 <= 3000 Inspection capacity 0.2

A B C D E F d1- d1+ d2- d2+ d3- d3+ d4- d4+ d5- d5+ d6- d6+ d7- d7+ Solution values 0.0 1650.0 600.0 0.0 4500.0 2250.0 0.0 0.0 416.5 0.0 0.0 0.0 0.0 1350.0 0.0 0.0 0.0 2250.0 0.0 583.50$ Goal weights 25 20 15 15 10

4-cyl 6-cyl 8-cyl X1X2X3 X1X2X3X33 Variable terms 0.00 38.46 0.00 0.00 Fixed Costs 1800.00 2400.00 1950.00 Objective coeff -50 -70 -65 $89,615.38 Constraints: Prodn hours 100 130 140 5000.00 <= 5000 LHS Sign RHS Problem 6-39 Solution value 0.00 38.46 0.00

A B G A B G A2B2G2A*B A*G B*G Variable terms 0.21 0.33 0.46 0.04 0.11 0.22 0.07 0.10 0.15 Objective coeff 3.00 2.00 2.00 2.00 -1.10 -0.70 0.70 Constraints: Total invest 1 1 1 1.00 = 1.00 Min return 0.15 0.11 0.09 0.11 >= 0.10 Max Alpha 1 0.21 <= 0.70 Max Beta 1 0.33 <= 0.70 Max Gamma

1 2 3 4 5 6 LHS RHS 418.27 0.00 0.00 0.00 1.00 0.00 0.00 1.0 = 1.0 635.20 0.00 0.00 1.00 0.00 0.00 0.00 1.0 = 1.0 817.20 0.00 0.00

Activity Pred Optim time (a) Most likely time (m) Pessim time (b) Exp time Var Std dev EST EFT LST LFT Slack Critical? Critical path = A - 1.00 2.00 3.00 2.00 0.11 0.33 0.00 2.00 0.00 2.00 0.00 Y Project length = 35.50 days B - 1.00 1.50 5.00 2.00 0.44 0.67 0.00 2.00 1.00 3.00 1.00 N

Activity Pred Time EST EFT LST LFT Slack Critical? D B 4.0 4.0 8.0 6.0 10.0 2.0 N F C, E 2.0 8.0 10.0 8.0 10.0 0.0 Y End D, F 0.0 10.0 10.0 10.0 10.0 0.0 Y Critical path = Project length = 10.00 weeks E-F B - 4.0 0.0 4.0 1.0 5.0 1.0 N C A,

Activity Immed pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation I E 5 7 9 7.00 0.44 0.67 21.00 28.00 21.00 28.00 0.00 Y J D, I 10 11 12 11.00 0.11 0.33 28.00 39.00 28.00 39.00 0.00 Y K F, G, H, J 4 6 8 6.00 0.44 0.67 39.00 45.00

Activity Pred Optim time (a) Most likely time (m) Pessim time (b) Exp time Var Std dev C A 50.00 60.00 100.00 65.00 69.44 8.33 30.00 95.00 30.00 95.00 0.00 Y E B 25.00 30.00 35.00 30.00 2.78 1.67 90.00 120.00 119.90 149.90 29.90 N H G 10.00 20.00 30.00 20.00 11.11 3.33 180.00 200.00 180.00 200.00 0.00 Y J

Activity Pred Time EST EFT LST LFT Slack Critical? A—5 0 5 8 13 8 N B—7 0 7 12 19 12 N Case: Family Planning Research Center of Nigeria (a) C—5 0 5 0 5 0 Y D A 3 5 8 19 22 14 N E A 7 5 12 13 20 8 N F

However, $300 is saved by allowing activity A (not on any of the critical paths) to take 3 weeks, yielding a net cost increase of $1,050. weeks, with a critical path of E-F. See file Ch07 Internet Case Bay Community.XLS, sheet #1. CHAPTER 7 PROJECT MANAGEMENT Internet

these estimates, the expected completion time and completion variance can be determined. CPM allows the use of crashing. This technique allows a manager to reduce the total project completion time by expending additional resources on activities within the

Problem 7-15 H K C B D E J F A End G I Activity Time EST EFT LST LFT Slack Critical? A 4 0 4 0 4 0 Y B 6 4 10 410 0 Y C12 10 22 10 22 0 Y Problem 7-14 D11 10 21 13 24 3 N E 9 21 30 25 34 4 N F 8 21 29 26

Problem 7-16 A2 A5 A9 A3 A4 A6 A10 A7 A1 End A8 Activity Time EST EFT LST LST Slack Critical? A1 5 0 5 0 5 0 Y A2 6 5 11 612 1 N A3 2 5 7 16 18 11 N Problem 7-16 A4 9 5 14 514 0 Y A5 911 20 12 21 1 N A6 314 17 18 21

Problem 7-17 A H F C B I D Start E G J K End Activity Pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation EST EFT LST LFT Slack Critical? A - 3 6 9 6.00 1.00 1.00 0.00 6.00 7.00 13.00 7.00 N B - 2 4 6 4.00 0.44 0.67 0.00 4.00 5.00 9.00 5.00 N C- 1 2 3 2.00 0.11 0.33 0.00

Problem 7-18 B I E C D H F A G Activity Pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation EST EFT LST LFT Slack Critical? A - 3 6 6 5.50 0.25 0.50 0.00 5.50 0.00 5.50 0.00 Y B A 5 8 11 8.00 1.00 1.00 5.50 13.50 5.50 13.50 0.00 Y Problem 7-18 C A 5 6 10 6.50

Problem 7-19 F I A D K C B H G E Start J L End Activity Pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation EST EFT LST LST Slack Critical? A - 8 10 12 10.00 0.44 0.67 0.00 10.00 0.00 10.00 0.00 Y B - 6 7 8 7.00 0.11 0.33 0.00 7.00 23.00 30.00 23.00 N C - 3 3 6 3.50 0.25 0.50

Problem 7-20 C I G E B H F A D Activity Pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation EST EFT LST LST Slack Critical? A - 4 8 12 8.00 1.78 1.33 0.00 8.00 0.00 8.00 0.00 Y B A 4 10 13 9.50 2.25 1.50 8.00 17.50 8.00 17.50 0.00 Y Problem 7-20 C B 7 14 18 13.50 3.36

Problem 7-21 H A D C F B I G E J Start Activity Pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation EST EFT LST LST Slack Critical? A - 3 7 14 7.50 3.36 1.83 0.00 7.50 2.50 10.00 2.50 N B - 5 10 15 10.00 2.78 1.67 0.00 10.00 0.00 10.00 0.00 Y Problem 7-21 C A, B 3 5 10 5.50

Problem 7-22 Mean (months) 21 Variance (months^2) 6 Standard deviation (months) 2.4495 (a) P(need >= 17 months) = 0.9488 (b) P(need <= 20 months) = 0.3415 (c) P(need >= 23 months) = 0.2071 (d) P(need <= 25 months) = 0.9488

Activity Budgeted cost ($) % completed Value of work completed Actual cost ($) Difference ($) A$22,000 100% $22,000 $20,000 -$2,000 Problem 7-24 B$30,000 100% $30,000 $36,000 $6,000 C$26,000 100% $26,000 $26,000 $0 D$48,000 100% $48,000 $44,000 -$4,000 E$56,000 55% $30,800 $29,000 -$1,800 F$30,000 55% $16,500 $12,000 -$4,500 G$80,000 10% $8,000 $5,000 -$3,000 H$16,000 13% $2,080 $1,800 -$280 Total $183,380 $173,800 -$9,580 Underrun

Activity Budgeted cost ($) % completed Value of work completed Actual cost ($) Difference ($) A$10,000 100% $10,000 $13,000 $3,000 B$14,000 100% $14,000 $12,000 -$2,000 Problem 7-26 C$5,000 100% $5,000 $6,000 $1,000 D$6,000 100% $6,000 $6,000 $0 E$14,000 60% $8,400 $9,000 $600 F$13,000 10% $1,300 $800 -$500 G$4,000 80% $3,200 $3,600 $400 H$6,000 15% $900 $375 -$525 I$18,000 0% $0 $0 $0 J$12,000 0% $0 $0

Activity Budgeted cost ($) % completed Value of work completed Actual cost ($) Difference ($) A$2,200 100% $2,200 $1,900 -$300 Problem 7-28 B$5,100 100% $5,100 $5,300 $200 C$6,000 100% $6,000 $6,150 $150 D$3,600 40% $1,440 $1,800 $360 E$2,700 60% $1,620 $1,755 $135 F$1,800 0% $0 $0 $0 Total $16,360 $16,905 $545 Overrun

TATBTCTDTETFTGTHCACBCCCDCECFCGCH Start A Start B Start C Start D Start E Start F Start G Start H Crash A Crash B Crash C Crash D Crash E Crash F Crash G Crash H G --> H -1 1 1 5 >= 5 Crash limit A 1 1 <= 1 Crash limit B 1 0 <= 2 Crash limit C 1 1 <= 1 Crash limit D 1 1 <= 1 Crash limit

TATBTCTDTETFTGTEnd CACBCCCDCECFCG Start A Start B Start C Start D Start E Start F Start G Start End Crash A Crash B Crash C Crash D Crash E Crash F Crash G Crash limit C 1 0 <= 0 Crash limit D 1 4 <= 4 Crash limit E 1 2 <= 3 Crash limit F 1 0 <= 1 Crash limit G 1 0 <= 2 Project finish 1 10

Activity Normal time Crash time Normal cost Crash cost Crash cost per day A 7 5 $2,000 $3,500 $750 B10 8$3,000 $4,700 $850 Problem 7-32 C 8 7 $3,400 $3,700 $300 D 6 4 $1,600 $2,600 $500 E 7 4 $1,900 $4,000 $700 F 5 3 $1,200 $2,800 $800 G11 8$8,200 $10,900 $900 H 4 3 $2,600 $3,800 $1,200 TATBTCTDTETFTGTHCACBCCCDCECFCGCH Start A Start B Start C Start D Start E Start

Activity Immediate pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation EST EFT LST LST Slack Critical? 1 - 3 4 5 4.00 0.11 0.33 0.00 4.00 0.00 4.00 0.00 Y 2 1 4 5 12 6.00 1.78 1.33 4.00 10.00 6.00 12.00 2.00 N Problem 7-33 3 1 6 8 10 8.00 0.44 0.67

Activity Immediate predecessors Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation EST EFT LST LST Slack Critical? 1—1.00 4.00 7.00 4.00 1.000 1.00 0.00 4.00 7.50 11.50 7.50 N 2—2.00 3.00 4.00 3.00 0.111 0.33 0.00 3.00 15.50 18.50 15.50 N Problem 7-34 3—3.00 4.00 5.00 4.00 0.111 0.33 0.00 4.00 13.50 17.50 13.50 N 4—7.00

Activity Immediate pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation G B 2.0 4.0 6.0 4.00 0.44 0.67 3.50 7.50 15.00 19.00 11.50 N I C 9.0 10.0 14.0 10.50 0.69 0.83 12.00 22.50 14.50 25.00 2.50 N K D 2.0 4.0 6.0 4.00 0.44 0.67 5.50 9.50 19.50 23.50 14.00 N M

Activity Immed pred Optim time (a) Most likely time (m) Pessim time (b) Expected time Variance Standard deviation R L, M 1 2 3 2.00 0.11 0.33 15.00 17.00 26.50 28.50 11.50 N S N 6 6 9 6.50 0.25 0.50 28.50 35.00 28.50 35.00 0.00 Y T O 3 4 5 4.00 0.11 0.33 21.00 25.00 31.00 35.00 10.00

CHAPTER 7 PROJECT MANAGEMENT Internet Case Study: Shale Oil Company 1. The project network is shown below. See file Ch07 Internet Case Shale Oil.XLS. The expected shutdown time is 45.75 days. There are 9

total cost of $15,000. To further crash the project to 240 days from the current 250 days, we crash activity D by 10 days at $1,900/day, for a total cost of $19,000. Hence, the total additional cost to crash the project to 240 days is $34,000.

the same as before, but has an expected length of only 31.0 days with a variance of 0.6944. This yields a standard normal value of 4.8 implying that it is almost certain that the issues will be sent out on time. analysis could include the alum’s status with the fraternity,

Problem 7-12 B C D E A F G Start Activity Time EST EFT LST LFT Slack Critical? A 2 0 2 13 15 13 N B 5 0 5 0 5 0 Y Problem 7-12 C 1 0 1 11 12 11 N D10 515 515 0 Y E 3 15 18 15 18 0 Y F 6 1 7

Problem 7-13 A D G B C E H F Start End Activity Time EST EFT LST LFT Slack Critical? A 4 0 4 0 4 0 Y B 6 0 6 1 7 1 N Problem 7-13 C 3 4 7 4 7 0 Y D 4 4 8 11 15 7 N E 8 7 15 715 0 Y F

Problem 7-14 A C G B D E H F Start End Activity Time EST EFT LST LFT Slack Critical? A 5 0 5 2 7 2 N B 4 0 4 0 4 0 Y Problem 7-15 C 2 5 7 7 9 2 N D 1 5 6 11 12 6 N E 5 4 9 4 9 0 Y F

Problem 7-23 Mean (weeks) 25 (a) P(need <= 30 weeks) = 0.8944 (b) P(need <= 22 weeks) = 0.2266 (c) 99% completion time = 34.31 (d) 85% completion time = 29.15 Standard deviation (weeks) 4

Using EST: Act Time Cost Cost/P d EST LST 12345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 A 6 ###### ##### 0 0 ##### $1,667 $1,667 $1,667 $1,667 $1,667 B 2 ###### ##### 1 4 $7,000 $7,000 C 7 $5,000 $714 3

Problem 7-27 B D F A C E Using EST: Act Pred Time Cost Cost/P d EST EFT LST LFT 1 2 3 4 5 6 7 8 9 10 11 12 Using LST: A##### ##### B $1,700 $1,700 $1,700 C##### ##### $1,500 $1,500 D $1,800 $1,800 E$900 $900 $900 F$600 $600 $600 Budget per period = ##### ##### ##### ##### $3,200 $3,200 $2,600 $2,700 $2,700 $600

Activity Normal time Crash time Normal cost Crash cost Crash cost per day A 4 2 $2,000 $2,600 $300 B 6 5 $3,500 $4,300 $800 Problem 7-31 C 8 6 $3,300 $3,900 $300 D 5 4 $1,200 $1,800 $600 E 3 2 $1,700 $2,200 $500 F 7 5 $2,200 $3,600 $700 G 5 4 $900 $1,550 $650 H 4 3 $1,200 $1,700 $500 TATBTCTDTETFTGTHCACBCCCDCECFCGCH Start Start

The overall purpose of Problem 7-35 is to have students use a project approach in attempting to solve a problem that almost all students face. The first step is for students to list all courses that they must take, including possible electives, to get a degree from their particular college

Activity Predecessors Optim time (a) Most likely time (m) Pessim time (b) Exp time Var Std dev EST EFT LST LFT Slack Critical? Critical path = 1 - 1.0 2.0 2.5 1.92 0.06 0.25 0.00 1.92 0.00 1.92 0.00 Y Project length = 45.75 days 2 1 1.5 2.0 2.5 2.00 0.03 0.17 1.92 3.92 1.92 3.92 0.00 Y

0.8 0.6 Comp 12-months Moderate $100,000 $0 $95,000 $300,000 $100,000 0.3 Low 0.9 $50,000 Success (Continue) $250,000 $50,000 -$100,000 $154,000 0.1 Substantial $600,000 $800,000 $600,000 0.2 0.6 No comp 12-months Moderate 12-months normal $400,000 $0 $390,000 $600,000 $400,000 -$100,000 $128,600 0.3 Low $300,000 $500,000 $300,000 0.1 Failure (Stop) -$100,000 $0 -$100,000 0.6 Comp 8-months (Stop) -$80,000 $0 -$80,000 0.1 Substantial $160,000 $400,000 $160,000 0.8 0.6 Comp 12-months Moderate $60,000 $0 $55,000 $300,000 $60,000 8-months & crash 0.3 -$80,000 -$10,160 Low 0.9 $10,000 Success (Continue) $250,000 $10,000 -$160,000 $114,000 0.1 Substantial $560,000 $800,000 $560,000 0.2 0.6 0.4 No comp 12-months Moderate No comp 8-months $360,000 $0 $350,000 $600,000 $360,000 $0 $94,600 0.3 Low $260,000 $500,000 $260,000 0.1 Failure (Stop) -$80,000 $0 -$80,000 0.1 Substantial 1 $210,000 $128,600 $400,000 $210,000 0.5 0.6 Comp 6-months (Copy) Moderate $110,000 -$130,000 $105,000 $300,000 $110,000 0.3 Low $60,000 $250,000 $60,000 0.6 Comp 8-months (Stop) 6-months program -$80,000 $0 -$80,000 -$60,000 $47,420 0.1 Substantial $160,000 $400,000 $160,000 0.8 0.6 Comp 12-months Moderate $60,000 0.5 $0 $55,000 $300,000 $60,000 No comp 6-months 0.3 -$20,000 -$10,160 Low 0.9 $10,000 Success (Continue) $250,000 $10,000 -$160,000 $114,000 0.1 Substantial $560,000 $800,000 $560,000 0.2 0.6 0.4 No comp 12-months Moderate No comp 8-months $360,000 $0 $350,000 $600,000 $360,000 $0 $94,600 0.3 Low $260,000 $500,000 $260,000 0.1 Failure (Stop) -$80,000 $0 -$80,000 Do nothing $0 $0 $0 Internet Case: Drink At Home, Inc. 0.1 Adopt 12-month normal research program. Substantial Expected profit = $128,600 $200,000 $400,000 $200,000

Cost per case (with overtime) = $1.50 + $1.75 + $0.90 = $4.15. Cost of machine = $350,000. Cost to modify machine = $20,000. Volume Profit (Excluding Cost of Machine) 120,000 ($6 – $3.25) × 120,000 = $330,000 130,000 ($6

CHAPTER 8 DECISION ANALYSIS Internet Case Study: Ruth Jones’ Heart By-Pass Operation Expected survival rate with surgery = 0  0.05 + 1  0.45 + 5  0.20 + 10  0.13 + 15  0.08 + 20  0.05 + 25  0.04

alternatives, (3) identify the possible outcomes, (4) evaluate the consequences, and (5) select an evaluation criterion and make the appropriate decision. The first four steps or procedures are common for all decision-making problems. Step 5, however, depends

PAYOFFS Alternatives Poor Average Good Excellent EMV Choice PP -$5,000 -$2,000 $2,000 $5,000 $700 LB and PP -$10,000 -$4,000 $6,000 $12,000 $2,600 Best Outcomes Case: Ski Right TR and PP -$15,000 -$10,000 $7,000 $13,000 $900 CC and PP -$30,000 -$20,000 $10,000 $30,000 $1,000 LB, CC, and TR -$60,000 -$35,000 $20,000 $55,000 $2,500 Probability 0.1 0.3 0.4 0.2 Best outcome -$5,000 -$2,000

0.7 Succ venture 0.50 Unsucc venture Fav survey -$600,000 1 $0 -$600,000 $0 $800,000 No product -$100,000 $0 -$100,000 Hire MAI 0.4 -$100,000 $500,000 Succ venture $1,400,000 Introduce product $2,000,000 $1,400,000 -$500,000 $200,000 0.6 0.50 Unsucc venture Unfav survey -$600,000 1 $0 -$600,000 $0 $200,000 No product -$100,000 $0 -$100,000 Survey 1 0.87 $0 $500,000 Succ venture $1,200,000 Introduce product $2,000,000 $1,200,000 -$500,000 $940,000 0.13 0.62 Unsucc venture Fav survey -$800,000 1 $0 -$800,000 $0 $940,000 No product -$300,000 $0 -$300,000 Hire I&K 0.16 -$300,000 $468,800 Succ

PAYOFFS Alternatives Fav market Unfav market Maximum Choice Minimum Choice EMV Choice Acme $400,000 -$175,000 $400,000 Best -$175,000 $262,000 Best Problem 8-13 Outcomes (a) Maximax (b) Maximin (c) EMV Standard $280,000 -$90,000 $280,000 -$90,000 $191,200 High Pro $95,000 -$15,000 $95,000 -$15,000 Best $68,600 Probability 0.76 0.24

PAYOFFS Alternatives Good Average Poor EMV Choice Best outcome $90,000 $41,000 -$4,000 Best Expected Monetary Value (EMV) = $32,550 Expected Value OF Perfect Information (EVPI) = $7,200 Expected Value WITH Perfect Information (EVwPI) = $39,750 REGRET Alternatives Good Average Poor EOL Choice Move $0 $0 $24,000 $7,200 Best No change $50,000 $26,000 $0 $24,200 Probability 0.25 0.45 0.30 Expand $14,000 $11,000 $13,000

PAYOFFS Alternatives Strong Average Weak EMV Choice Stocks 12% 6% -10% 0.00% Best Expected Monetary Value (EMV) = 3.25% Expected Value OF Perfect Information (EVPI) = 2.15% Expected Value WITH Perfect Information (EVwPI) = 5.40% REGRET Alternatives Strong Average Weak EOL Choice Bonds 5% 2% 1% 2.15% Best Money Market 8% 3% 0% 2.65% Probability 0.20 0.35 0.45 Stocks 0% 0% 12%

Selling price $2.50 a = 0.45 REGRET Alternatives 100 300 500 Maximum Choice 100 $0 $250 $500 $500 300 $250 $0 $250 $250 Best 500 $500 $250 $0 $500 (e) Minimax Problem 8-19 Demand Variable cost $1.25 PAYOFFS Alternatives 100 300 500 Maximum Choice Minimum Choice Average Choice Realism Choice 100 $125 $125 $125 $125 $125 Best $125 $125 Best 300 -$125 $375 $375 $375 -$125 $208

Risk avoiders. -$45,000 -$40,000 -$5,000 $0 $95,000 $100,000 Monetary value Dollar Utility -$45,000 0.00 -$40,000 0.10 -$5,000 0.70 $100,000 1.00 Problem 8-27 (a) 0.60 Utility 0.20 0.00 0.40 0.80 1.00 Utility Curve $0 0.90 $95,000 0.99 0.82 High demand 0.55 Low demand Positive result 0.000 0.812 Not Construct 0.700 Study 0.761 High demand 0.109 0.89 20.000 0.900 0.700 0.50 High demand 0.900 0.550 0.50 Low demand No Study 0.100 20.100 0.812 0.18 10.000 0.700 0.11 0.990 Construct 0.990 0.45 Low demand Negative result 0.000 0.700 2 Not Construct 0.700 1.000 Construct 1.000 0.900 Not Construct 0.900 Problem 8-27 (b) Do not conduct study. 0.990 Do not construct

Selling price $90.00 PAYOFFS Alternatives 6 7 8 9 EMV Choice 7$155 $245 $245 $245 $236 Best 8$100 $190 $280 $280 $217 9$45 $135 $225 $315 $171 Probability 0.10 0.50 0.30 0.10 6$210 $210 $210 $210 $210 Best outcome $210 $245 $280 $315 Best Expected Monetary Value (EMV) = $236 Expected Value OF Perfect Information (EVPI) = $23 Expected Value WITH Perfect

Selling price $30.00 PAYOFFS Alternatives 50 100 150 200 EMV Choice 50 $500 $500 $500 $500 $500 100 -$300 $1,000 $1,000 $1,000 $740 Best 150 -$1,100 $200 $1,500 $1,500 $525 200 -$1,900 -$600 $700 $2,000 -$15 Probability 0.20 0.35 0.25 0.20 Best outcome $500 $1,000 $1,500 $2,000 Expected Value WITH Perfect Information (EVwPI) = $1,225 Best Expected Monetary Value (EMV) =

TIME Alternatives None Mild Severe EMV Choice Broad 10 21 30 18.40 Worst outcome 10 20 20 Expected Value WITH Perfect Information (EVwPI) = 16.00 Expected Value OF Perfect Information (EVPI) = 0.60 Best Expected Time (EMV) = 16.60 REGRET Alternatives None Mild Severe EOL Choice Drexel 3 0 3 1.80 Best Expressway 10 4 0 5.60 Probability 0.40 0.40 0.20 In this

0.6 1% defect 0.25 Buy from A 3% defect -$180 $0 -$126 -$180 -$180 0.15 5% defect -$300 -$300 -$300 1 -$126 0.3 1% defect -$23 -$60 -$23 0.4 Buy from B 3% defect -$143 $37 -$143 -$180 -$143 0.3 5% defect -$263 -$300 -$263 All costs have been expressed as negative amounts. Hence, select highest expected values at all nodes. Sample calculation in cell H4: (-$0.60 x 1% x 10,000 units) = -$60 Problem

0.83 Sunny (a) Hire Elmo. $16 If he says "good chance", take lemonade. Cocoa $20 $16 If he says "bad chance", take cocoa. 0.3 $76 Bad chance $80 $76 1 $0 $61.00 0.25 Sunny $56 Lemonade $60 $56 1 $50.36 $0 $11.00 0.75 Rainy -$4 $0 -$4 0.6 Sunny $20 Cocoa $20 $20.00 $0 $44.00 0.4 Rainy $80 No Susan $80 $80.00 1 $0 $44.00 0.6 Sunny $60 Lemonade $60 $60.00 $0 $36.00 0.4 Rainy $0 $0 $0.00 Problem 8-31 Expected profit = $50.36 $0 $26.20

Problem 8-33 0.75 Win point Accept the wager. 0.65 $100 0.75 Win point 0.65 $100 Hard 1st serve Serve good $100 $100 $0 $31.63 $0 $50 0.25 Lose point Hard 2nd serve -$100 -$100 -$100 $0 -$3 0.35 Serve bad -$100 0.35 -$100 -$100 Serve bad 1 0.27 $0 -$2.50 Win point 0.88 $100 Serve good $100 $100 $0 -$46 0.73 Lose point Soft 2nd serve -$100 -$100 -$100 1 $0 -$52 $31.63 0.12 Serve bad -$100 -$100 -$100 0.27 Win

Conditional probabilities Prior Probabilities P(Pilot works | Facility works) = 0.80 P(Facility works) = 0.60 P(Pilot fails | Facility works) = 0.20 P(Facility fails) = 0.40 Problem 8-35 P(Pilot works | Facility fails) = 0.15 P(Pilot fails | Facility fails) = 0.85 Posterior probabilities GIVEN pilot works Conditional prob Prior Joint Posterior Outcome P(Pilot works | Outcome) prob prob prob Facility works 0.80

Test land. If test is positive, drill. If test is negative, sell. Expected profit = $565,000. Sell land 0.3 $650 $550 Positive 2 0.65 $0 $2,000 Oil $3,400 Drill Land $4,000 $3,400 -$500 $2,000 0.35 Test land Dry -$600 -$100 $565 $0 -$600 2 $565 Sell land -$50 0.7 $50 -$50 Negative 1 0.05 $0 -$50 Oil $3,400 Drill land $4,000 $3,400 -$500 -$400 0.95 Dry -$600 $0 -$600 0.2 Oil $3,500 Drill Land $4,000 $3,500 -$500 $300 0.8 Dry -$500 $0

Dollar Utility -$600 0.00 -$500 0.03 $550 0.17 $3,400 0.90 $3,500 1.00 0.75 Risk seeker. -$600 $0 $600 $1,200 $1,800 $2,400 $3,000 $3,600 Monetary value Problem 8-39 (b) 0.00 0.25 0.50 1.00 Utility Utility Curve -$50 0.10 $400 0.15

0.78 Good market 0.45 Poor market Positive result 0.500 1 0.500 0.851 Not Produce 0.800 0.800 Questionnaire 0.27 0.823 Good market 0.950 Produce 0.950 0.622 0.73 0.55 Poor market Negative result 0.500 2 0.500 0.800 Not Produce 0.800 0.800 0.89 Good market 0.900 Produce 0.900 0.801 0.11 0.45 Poor market Positive pilot 0.000 1 0.000 0.00 0.801 1 Not Produce 0.823 0.700 0.700 Pilot study 0.18 0.745 Good market 0.900 Produce 0.900 0.162 0.82 0.55 Poor market Negative pilot 0.000 2 0.000 0.700 Not Produce 0.700 0.700 0.50 Good market 1.000 Produce 1.000 0.775 0.50 Poor market Neither test 0.550 2 0.550 0.810 Not produce 0.810 0.810 Problem 8-41

Lock-in now 0.150 Lock-in 1 month increases after one month, lock-in then. If rate is unchanged or decreases 0.100 0.100 and accept current rate at settlement. Expected utility = 0.300 0.500 Increase 0.5% Increase 0.5% 0.261. 1 0.000 0.100 0.000 0.250 Don't lock-in 1 month Unchanged 2 0.120 0.261 0.083 0.120 0.250 Decrease 0.5% 0.210 0.210 Lock-in 1 month 0.190 0.190 0.500 0.250 Don't lock-in now Unchanged Increase 0.5% 2 0.120 0.261 0.210 0.120 0.500 Don't

Build 1.00 0.62 0.40 Failure No info 0.05 1 0.05 0.62 Not build 0.20 0.20 0.90 Success 0.40 Build 0.40 1 0.62 0.36 0.10 0.60 Failure Fav info 0.00 1 0.00 0.36 Not build 0.10 0.10 Get info 0.20 0.26 Success 0.40 Build 0.40 0.08 0.80 0.40 Failure Unfav info 0.00 2 0.00 0.10 Not build 0.10 0.10 Problem 8-45 (a) Get info. If favorable, build. 0.60 If unfavorable, don't build. Success Expected utility = 0.62. 1.00 Dollar Utility -$20,000 0.10 $0 0.20 $80,000 0.40 $100,000 1.00 0.75 Risk seeker. -$100,000 -$50,000 $0 $50,000 $100,000 Monetary value Problem

Conditional probabilities Prior probabilities P(Favorable survey | Success) = 0.75 P(Sucess) = 0.50 P(Unfavorable survey | Success) = 0.25 P(Failure) = 0.50 Problem 8-47 P(Favorable survey | Failure) = 0.35 P(Unfavorable survey | Failure) = 0.65 Posterior probabilities GIVEN favorable survey Conditional prob Prior Joint Posterior Outcome P(Favorable | Outcome) prob prob prob Success 0.75 0.50 0.375 0.682 Failure 0.35 0.50 0.175 0.318 P(Favorable survey)

Conditional probabilities Prior probabilities P(Growth prediction | Actual growth) = 0.75 P(Actual growth) = 0.30 P(Growth prediction | Actual steady) = 0.18 P(Actual steady) = 0.45 Posterior probabilities GIVEN growth prediction Conditional prob Prior Joint Posterior Outcome P(Growth pred | Outcome) prob prob prob Actual growth 0.75 0.30 0.225 0.6955 Actual steady 0.18 0.45 0.081 0.2504 Actual

CHAPTER 8 DECISION ANALYSIS Internet Case Study: Starting Right This is an example of decision making under uncertainty. There are two outcomes: a favorable market and an unfavorable market (event 2). There are four alternatives: do nothing, invest in corporate bonds, invest in preferred stock, and invest

CHAPTER 8 DECISION ANALYSIS Internet Case Study: Drink At Home, Inc. See file Ch08 Internet Case Drink At Home.XLS for the TreePlan solution. This is a fairly elaborate decision tree since we have to consider the possibility that competition may exist at the end of 6-months, 8-months.,

PAYOFFS Alternatives Fav Mkt Unfav Mkt Max payoff Choice Min payoff Choice Do nothing $0 $0 $0.00 $0 Best a = 0.11 REGRET Alternatives Fav Mkt Unfav Mkt Max regret Choice Do nothing $210,000 $0 $210,000.00 Corp bonds $184,727 $10,000 $184,727.00 Pref stock $120,000 $15,000 $120,000.00 Com stock $0 $30,000 $30,000.00 Best Maximin Internet Case: Starting Right Outcomes Maximax Minimax Outcomes Corp bonds $25,273 -$10,000 $25,273.00 -$10,000 Pref stock

PAYOFFS Alternatives Good Average Poor Maximum Choice Minimum Choice Average Choice Realism Choice Expand $76,000 $30,000 -$17,000 $76,000 -$17,000 $29,667 $20,200 Best REGRET Alternatives Good Average Poor Maximum Choice Expand $14,000 $11,000 $21,000 $21,000 Best Move $0 $0 $32,000 $32,000 No change $50,000 $26,000 $0 $50,000 Outcomes (d) Hurwicz (e) Minimax Problem 8-14 (a) Maximax (b) Maximin (c) Equally Likely Outcomes Move $90,000 $41,000 -$28,000 $90,000 Best -$28,000 $34,333

PAYOFFS Alternatives Strong Average Weak Maximum Choice Minimum Choice Average Choice Realism Choice REGRET Alternatives Strong Average Weak Maximum Choice Stocks 0% 0% 12% 12% Bonds 5% 2% 1% 5% Best Money Market 8% 3% 0% 8% (c) Equally Likely (d) Hurwicz Problem 8-16 Outcomes Outcomes Minimax (a) Maximax (b) Maximin Stocks 12% 6% -10% 12% Best -10% 3% 1% Bonds 7% 4% 1% 7% 1%

PAYOFFS Alternatives Grow Same Average Choice Realism Choice Large wing $225,000 -$125,000 $50,000 Best $120,000 Best (c) Hurwicz Problem 8-18 (b) Equally Likely Outcomes Small wing $90,000 -$65,000 $12,500 $43,500 No wing $0 $0 $0 $0 a = 0.7

PAYOFFS Alternatives 100 300 500 EMV Choice 300 -$125 $375 $375 $175 Best 500 -$375 $125 $625 $50 Probability 0.40 0.35 0.25 Best outcome $125 $375 $625 100 $125 $125 $125 $125 Best Expected Monetary Value (EMV) = $175 Expected Value OF Perfect Information (EVPI) = $163 Expected Value WITH Perfect Information (EVwPI) = $338 REGRET Alternatives 100 300 500 EOL Choice 300

PAYOFFS Alternatives 50 100 150 200 Maximum Choice Minimum Choice Average Choice Realism Choice 50 $500 $500 $500 $500 $500 $500 Best $500 $500 100 -$300 $1,000 $1,000 $1,000 $1,000 -$300 $675 Best $610 150 -$1,100 $200 $1,500 $1,500 $1,500 -$1,100 $525 $720 200 -$1,900 -$600 $700 $2,000 $2,000 Best -$1,900 $50 $830 Best a = 0.7 REGRET Alternatives 50 100

PAYOFFS Alternatives Good Fair Poor Maximum Choice Minimum Choice Average Choice Realism Choice Small $70,000 $28,000 -$14,000 $70,000 -$14,000 Best $28,000 $53,200 REGRET Alternatives Good Fair Poor Max regret Choice Small $350,000 $14,000 $0 $350,000 Medium $308,000 $0 $14,000 $308,000 Large $280,000 $0 $42,000 $280,000 Very large $0 $7,000 $210,000 $210,000 Best Problem 8-24 (c) Equally Likely (d) Hurwicz (e) Minimax Outcomes Outcomes (a) Maximax (b) Maximin Medium

0.82 High demand (a) Construct clinic; EMV = $30,000 $95,000 (b) Conduct study. Construct clinic if positive result Construct $100,000 $95,000 Do not construct clinic if negative result. EMV = $36,140 $0 $69,800 0.18 (c) EVSI = $11,140 0.55 Low demand Construct $100,000 $100,000 $0 $30,000 0.50 Low demand No Study -$40,000 1 -$40,000 -$40,000 $0 $30,000 Not Construct $0 $0 $0 $95,000 Construct $100,000 $95,000 $0 -$29,600 0.89 0.45

0.59 Good market (a) Conduct survey $60,000 If favorable, build large shop Large shop $60,000 $60,000 If unfavorable, don't build any shop 0.90 Good market $55,000 Large shop $60,000 $55,000 $0 $45,000 0.10 2Bad market $25,000 -$45,000 -$40,000 -$45,000 0.90 0.60 Good market Favorable $25,000 1Small shop $30,000 $25,000 $0 $45,000 $0 $21,000 0.10 Bad market -$15,000 -$10,000 -$15,000 No shop -$5,000 $0 -$5,000 Survey 0.12 -$5,000 $25,000 Good market $55,000 Large shop $60,000 $55,000 $0 -$33,000 0.88 Bad market -$45,000 -$40,000 -$45,000 0.12 0.40

P8-30 0.4 EVSI = $101.88 Heavy snow EVwPI $340.00 $700 Max EMV = $125.00 $700 EVPI = $215.00 Efficiency = 47.38% 0.3 New blower Moderate snow $200 Dont hire Adams $0 $70.00 $200 3 $0 $125.00 0.3 Light snow -$900 -$900 0.4 Heavy snow $350 $350 0.3 Repair blower Moderate snow $100 $0 $125.00 $100 0.3 Light snow -$150 -$150 Neither -$50 $0 -$50.00 2 0.7 $176.88 Heavy snow $650 $700 0.25 New blower Moderate snow $0 $150 Seasonably cold $0 $445.00 $200 2 $445.00 0.05 Light snow -$950 -$900 0.7 Heavy snow $300 $350 0.25 Repair

0.30 Competitor High 0.55 Our price high Competitor Medium $170,000 $0 $181,250 $250,000 $170,000 0.15 Competitor Low -$55,000 $25,000 -$55,000 0.20 Competitor High $395,000 475000 $395,000 0.75 0.70 Competitor Our price medium Competitor Medium 2 $245,000 $0 $260,000 $0 $260,000 $325,000 $245,000 0.10 Competitor Low $95,000 $175,000 $95,000 0.15 Competitor High $270,000 $350,000 $270,000 0.25 Market Our price low Competitor Medium $170,000 -$80,000 $325,000 $0 $110,000 $250,000 $170,000 0.60 Competitor Low $45,000 $125,000 $45,000 Our price high $520,000 $600,000 $520,000 1 $325,000 0.25 No Competitor Our price medium 1 $420,000 $0

0.85 Facility works 0.60 Facility fails Pilot works -$885,000 1 -$825,000 -$885,000 $0 $390,000 Don't build -$60,000 $0 -$60,000 Build pilot 0.20 -$60,000 $210,000 Facility works $615,000 Build facility $675,000 $615,000 $0 -$585,000 0.80 0.40 Facility fails Pilot fails -$885,000 2 -$825,000 -$885,000 $0 -$60,000 Don't build -$60,000 1 $0 -$60,000 $210,000 0.60 Facility works $675,000 Build facility $675,000 $675,000 $0 $75,000 0.40 Facility fails -$825,000 -$825,000 -$825,000 Do nothing $0 $0 $0 Problem 8-34 Build the pilot plant. $615,000 If pilot plant succeeds, build facility.

0.75 Win point 0.00 0.00 0.000 0.75 Win point 0.65 1.00 Hard 1st serve Serve good 1.00 1.00 0.00 0.658 0.00 0.750 0.25 Lose point Hard 2nd serve 0.00 0.00 0.00 0.00 0.488 0.35 Serve bad 0.00 0.35 0.00 0.000 Serve bad 1 0.27 0.00 0.488 Win point 0.88 1.00 Serve good 1.00 1.00 0.00 0.270 0.73 Lose point Soft 2nd serve 0.00 0.00 0.00 1 0.00 0.238 0.658 0.12 Serve bad 0.00 0.00 0.000 0.27 Win point 0.88 1.00 Serve good 1.00 1.000 0.00

Test land. If test is positive, drill. If test is negative, sell. Expected profit = $427,300. Sell land 0.37 $650 $550 Positive 2 0.46 $0 $1,240 Oil $3,400 Drill Land $4,000 $3,400 -$500 $1,240 0.54 Test land Dry -$600 -$100 $427.30 $0 -$600 2 $427.30 Sell land -$50 0.63 $50 -$50 Negative 1 0.05 $0 -$50 Oil $3,400 Drill land $4,000 $3,400 -$500 -$400 0.95 Dry -$600 $0 -$600 0.2 Oil $3,500 Drill Land $4,000 $3,500 -$500 $300.00 0.8 Dry -$500 $0

0.78 Good market (a) Conduct questionnaire. $135,000 If positive, produce razor. Produce $140,000 $135,000 0.27 -$5,000 $35,824 Good market $135,000 Produce $140,000 $135,000 $0 -$28,520 0.73 0.55 Poor market Negative result -$89,000 2 -$84,000 -$89,000 $0 -$5,000 Not Produce -$5,000 $0 -$5,000 0.89 Good market $120,000 Produce $140,000 $120,000 $0 $95,360 0.11 0.45 Poor market Positive pilot -$104,000 1 -$84,000 -$104,000 $0 $95,360 1 Not Produce $35,824 -$20,000 $0 -$20,000 Pilot study 0.18 -$20,000 $31,912 Good market $120,000 Produce $140,000 $120,000 $0 -$63,680

Lock-in now -$1,000 0.3 0.5 Increase 0.5% Increase 0.5% 1 -$4,800 -$2,900 -$4,800 0.25 Don't lock-in 1 month Unchanged 2 -$2,400 -$270 -$3,000 -$2,400 0.25 Decrease 0.5% $0 $0 Lock-in 1 month -$500 -$500 0.5 0.25 Don't lock-in now Unchanged Increase 0.5% 2 -$2,400 -$270 $0 -$2,400 0.5 Don't lock-in 1 month Unchanged $0 $0 $0 0.25 Decrease 0.5% $2,400 $2,400 Lock-in 1 month $1,900 $1,900 0.2 0.25 Decreases 0.5% Increase 0.5% 2 $0 $3,000 $0 0.25 Don't lock-in 1 month Unchanged $2,400 $3,000 $2,400 0.5 Decrease 0.5% $4,800 $4,800 Problem 8-42 $270. -$1,000 Lock-in 1 month -$2,900 -$2,900 Don't

(a) Get info. If favorable, build. 0.6 If unfavorable, don't build. Success Expected return = $29,200. $100 0.9 Success $80 Build $100 $80 2 $29.20 $0 62 0.1 0.6 Failure Favorable -$100 1 -$80 -$100 $0 $62 Not build -$20 $0 -$20 Get info 0.2 -$20 $29.20 Success $80 Build $100 $80 $0 -64 0.8 0.4 Failure Unfavorable -$100 2 -$80 -$100 $0 -$20 Not build -$20 $0 -$20 Problem 8-44 (b) EVSI = $21.20 Build $100 $100 EVPI = $32.00 Efficiency

Conditional probabilities Prior probabilities P(Low survey | Low demand) = 0.70 P(Low demand) = 0.15 P(Fair survey | Low demand) = 0.20 P(Fair demand) = 0.40 P(High survey | Low demand) = 0.10 P(High demand) = 0.45 Fair demand 0.40 0.40 0.160 0.516 High demand 0.10 0.45 0.045 0.145 P(Low survey) = 0.310 Posterior probabilities GIVEN fair

Conditional probabilities Prior probabilities P(Fav survey | Fav market) = 0.875 P(Fav mkt) = 0.550 Posterior probabilities GIVEN favorable survey Conditional prob Prior Joint Posterior Outcome P(Fav survey | Outcome) prob prob prob Fav market 0.875 0.550 0.481 0.855 Unfav market 0.182 0.450 0.082 0.145 P(Favorable survey) = 0.563 Posterior probabilities GIVEN unfavorable survey Conditional prob Prior Joint Posterior Outcome P(Unfav survey

Expressway -30 -30.0 2 0.5 -27.5 Congested -40 Broad street -40 -40 0 -27.5 0.5 Not congested -15 -15 -15 Use Broad street. Expected travel time = 27.5 minutes. Problem 8-50 (a) -30 Expressway 0.7 0.70 1 0.5 0.70 Congested 0.2 Broad street 0.2 0.2 0 0.55 0.5 Not congested 0.9 0.9 0.9 Use Expressway. Expected utility = 0.70. Problem 8-50 (b) 0.7 Time Utility 15 0.90 40 0.20 Risk avoider. Note: We have drawn the X-axis in reverse order so that

0.15 0.25 Sales 130,000 $10,000 $260,000 $10,000 0.4 Sales 140,000 $30,000 Machine 1 $280,000 $30,000 (Semi-Automated) -$250,000 $21,450 Overtime 0.15 $38,000 Sales 150,000 $288,000 $38,000 1 $0 $38,000 Modify $35,000 $285,000 $35,000 Overtime 0.05 $46,000 Sales 160,000 $296,000 $46,000 2 $0 $55,000 2 Modify $26,300 $55,000 $305,000 $55,000 0.15 Sales 120,000 -$20,000 $330,000 -$20,000 0.25 Sales 130,000 $7,500 $357,500 $7,500 0.4 Machine 2 Sales 140,000 (Automated) $35,000 $385,000 $35,000 -$350,000 $26,300 0.15 Sales 150,000 $62,500 $412,500 $62,500 Overtime 0.05 $81,000 Sales 160,000 $431,000 $81,000 1 $0 $81,000 Modify $70,000 $420,000 $70,000 Internet Case: Toledo Leather Company Use automated machine. Sales 120,000 Expected profit = $26,300 -$10,000 $240,000 -$10,000

1 2 3 4 5 6 7 8 9 15 16 17 18 20 00.1765 0.1765 22 20.1729 0.5965 24 40.0847 0.8023 25 26 50.0593 0.8616 60.0415 0.9031 28 80.0203 0.9525 33 34 35 36 37 38 39 40 13 0.0034 0.9920 14 0.0024 0.9944 15 0.0017 0.9961 16 0.0012 0.9973 17 0.0008 0.9981 18 0.0006 0.9987 19 0.0004 0.9991 20 0.0003 0.9993 19 21 23 27 29 30 31 32 A B C D E F G H I J Queuing Model M/M/s (Exponential Service Times) Input Data Operating Characteristics Arrival rate (l)7Average server utilization (r)0.7000 Probabilities Number of Units Probability Cumulative Probability 10.2471 0.4235 30.1211

1 2 3 4 17 18 19 20 21 22 23 24 25 Probabilities Number of Units Probability Cumulative Probability 0 0.1000 0.1000 1 0.0900 0.1900 2 0.0810 0.2710 3 0.0729 0.3439 4 0.0656 0.4095 5 0.0590 0.4686 14 15 16 31 32 33 34 35 36 37 38 39 11 0.0314 0.7176 12 0.0282 0.7458 13 0.0254 0.7712 14 0.0229 0.7941 15 0.0206 0.8147 16 0.0185 0.8332 17 0.0167 0.8499 18 0.0150 0.8649 19 0.0135 0.8784 26 27 28 29 30 A B C D E F G H I J Queuing Model M/M/s (Exponential Service Times) Probability (%

1. Arrivals are FIFO. 3. Arrivals are independent. 5. Service times are independent. 2. Average time spent in the system (W) 4. Average time in the queue (Wq) 6. Percent idle time (Po) 9-4. If the service rate is not greater than the arrival rate (that is, μ < λ),

1 2 3 4 17 18 19 20 21 22 23 24 25 Probabilities Number of Units Probability Cumulative Probability 0 0.0016 0.0016 1 0.0095 0.0111 2 0.0285 0.0395 3 0.0569 0.0964 4 0.0854 0.1818 5 0.1024 0.2842 14 15 16 31 32 33 34 35 36 37 38 39 11 0.0473 0.7165 12 0.0406 0.7570 13 0.0348 0.7918 14 0.0298 0.8216 15 0.0255 0.8471 16 0.0219 0.8690 17 0.0187 0.8877 18 0.0161 0.9038 19 0.0138 0.9175 26 27 28 29 30 A B C D E F G H I J Queuing Model M/M/s (Exponential Service Times) Probability (%

current service times, even though a UPC reader is going to be installed. This means that our waiting times are an upper bound for the new, better system (the M/M/s model). We begin with a rough analysis (one that is going to have a very interesting feature, by the

1 2 3 4 5 14 15 16 17 18 19 20 21 22 23 24 25 26 Probabilities Number of Units Probability Cumulative Probability 0 0.1667 0.1667 1 0.1389 0.3056 2 0.1157 0.4213 3 0.0965 0.5177 4 0.0804 0.5981 5 0.0670 0.6651 6 0.0558 0.7209 32 33 34 35 36 37 38 39 12 0.0187 0.9065 13 0.0156 0.9221 14 0.0130 0.9351 15 0.0108 0.9459 16 0.0090 0.9549 17 0.0075 0.9624 18 0.0063 0.9687 19 0.0052 0.9739 27 28 29 30 31 A B C D E F G H I J Problem 9-11 Queuing Model M/M/s (Exponential Service Times) 7

1 2 3 4 5 14 15 16 17 18 19 20 21 22 23 24 25 26 Probabilities Number of Units Probability Cumulative Probability 0 0.6250 0.6250 1 0.2344 0.8594 2 0.0879 0.9473 3 0.0330 0.9802 4 0.0124 0.9926 5 0.0046 0.9972 6 0.0017 0.9990 32 33 34 35 36 37 38 39 12 0.0000 1.0000 13 0.0000 1.0000 14 0.0000 1.0000 15 0.0000 1.0000 16 0.0000 1.0000 17 0.0000 1.0000 18 0.0000 1.0000 19 0.0000 1.0000 27 28 29 30 31 A B C D E F G H I J Problem 9-12 Queuing Model M/M/s (Exponential Service Times) 7

1 2 3 4 17 18 19 20 21 22 23 24 25 Probabilities Number of Units Probability Cumulative Probability 0 0.2000 0.2000 1 0.1600 0.3600 2 0.1280 0.4880 3 0.1024 0.5904 4 0.0819 0.6723 5 0.0655 0.7379 14 15 16 31 32 33 34 35 36 37 38 39 11 0.0172 0.9313 12 0.0137 0.9450 13 0.0110 0.9560 14 0.0088 0.9648 15 0.0070 0.9719 16 0.0056 0.9775 17 0.0045 0.9820 18 0.0036 0.9856 19 0.0029 0.9885 26 27 28 29 30 A B C D E F G H I J Problem 9-13 Queuing Model M/M/s (Exponential Service

1 2 3 4 5 14 15 16 17 18 19 20 21 22 23 24 25 26 Probabilities Number of Units Probability Cumulative Probability 0 0.4286 0.4286 1 0.2449 0.6735 2 0.1399 0.8134 3 0.0800 0.8934 4 0.0457 0.9391 5 0.0261 0.9652 6 0.0149 0.9801 32 33 34 35 36 37 38 39 12 0.0005 0.9993 13 0.0003 0.9996 14 0.0002 0.9998 15 0.0001 0.9999 16 0.0001 0.9999 17 0.0000 1.0000 18 0.0000 1.0000 19 0.0000 1.0000 27 28 29 30 31 A B C D E F G H I J Problem 9-14 Queuing Model M/M/s (Exponential Service Times) 7

1 2 3 4 5 6 7 8 9 15 16 17 18 31 32 33 34 35 36 37 38 39 11 0.0039 0.9923 12 0.0026 0.9949 13 0.0017 0.9966 14 0.0011 0.9977 15 0.0008 0.9985 16 0.0005 0.9990 17 0.0003 0.9993 18 0.0002 0.9995 19 0.0002 0.9997 21 22 23 24 25 26 1 0.2222 0.5556 0.4444 1+ 2 0.1481 0.7037 0.2963 2+ 3 0.0988 0.8025 0.1975 3+ 4 0.0658 0.8683 0.1317 5 0.0439 0.9122 0.0878 6 0.0293 0.9415 0.0585 19 20 27 28 29 30 A B C D E F G H I J Queuing Model

1 2 3 4 5 14 15 16 17 18 19 20 21 22 23 24 25 26 Probabilities Number of Units Probability Cumulative Probability 0 0.1429 0.1429 1 0.1224 0.2653 2 0.1050 0.3703 3 0.0900 0.4602 4 0.0771 0.5373 5 0.0661 0.6034 6 0.0567 0.6601 32 33 34 35 36 37 38 39 12 0.0225 0.8652 13 0.0193 0.8845 14 0.0165 0.9010 15 0.0141 0.9151 16 0.0121 0.9272 17 0.0104 0.9376 18 0.0089 0.9465 19 0.0076 0.9542 27 28 29 30 31 A B C D E F G H I J Problem 9-16 Queuing Model M/M/s (Exponential Service Times) 7

1 2 3 4 5 6 7 8 17 18 20 00.2500 0.2500 22 20.1406 0.5781 24 40.0791 0.7627 26 60.0445 0.8665 28 80.0250 0.9249 33 34 35 36 37 38 39 40 13 0.0059 0.9822 14 0.0045 0.9866 15 0.0033 0.9900 16 0.0025 0.9925 17 0.0019 0.9944 18 0.0014 0.9958 19 0.0011 0.9968 20 0.0008 0.9976 19 21 23 25 27 29 30 31 32 A B C D E F G H I J Queuing Model M/M/s (Exponential Service Times) Input Data Operating Characteristics Probabilities Number of Units Probability Cumulative Probability 10.1875 0.4375 30.1055 0.6836 50.0593 0.8220 70.0334 0.8999 90.0188 0.9437 10 0.0141 0.9578 11

1 2 3 4 5 14 15 16 17 18 19 20 21 22 23 24 25 26 Probabilities Number of Units Probability Cumulative Probability 0 0.0377 0.0377 1 0.1132 0.1509 2 0.1698 0.3208 3 0.1698 0.4906 4 0.1274 0.6179 5 0.0955 0.7134 6 0.0716 0.7851 32 33 34 35 36 37 38 39 12 0.0128 0.9617 13 0.0096 0.9713 14 0.0072 0.9785 15 0.0054 0.9839 16 0.0040 0.9879 17 0.0030 0.9909 18 0.0023 0.9932 19 0.0017 0.9949 27 28 29 30 31 A B C D E F G H I J Problem 9-18 Queuing Model M/M/s (Exponential Service Times) 7

1 2 3 4 17 18 19 20 21 22 23 24 25 Probabilities Number of Units Probability Cumulative Probability 0 0.2000 0.2000 1 0.1600 0.3600 2 0.1280 0.4880 3 0.1024 0.5904 4 0.0819 0.6723 5 0.0655 0.7379 14 15 16 31 32 33 34 35 36 37 38 39 11 0.0172 0.9313 12 0.0137 0.9450 13 0.0110 0.9560 14 0.0088 0.9648 15 0.0070 0.9719 16 0.0056 0.9775 17 0.0045 0.9820 18 0.0036 0.9856 19 0.0029 0.9885 26 27 28 29 30 A B C D E F G H I J Queuing Model M/M/s (Exponential Service Times) Probability (%

1 2 3 4 5 6 7 8 9 17 18 22 20.1764 0.3140 24 40.1633 0.6733 26 60.0726 0.8548 27 28 70.0484 0.9032 80.0323 0.9355 33 34 35 36 37 38 39 40 13 0.0042 0.9915 14 0.0028 0.9943 15 0.0019 0.9962 16 0.0013 0.9975 17 0.0008 0.9983 18 0.0006 0.9989 19 0.0004 0.9993 20 0.0002 0.9995 19 20 21 23 25 29 30 31 32 A B C D E F G H I J Queuing Model M/M/s (Exponential Service Times) Input Data Operating Characteristics Arrival rate (l)20 Average server utilization (r)0.6667 Probabilities Number of Units Probability Cumulative Probability 00.0318 0.0318 10.1058

1 2 3 4 5 6 7 8 17 18 20 00.1667 0.1667 22 20.1157 0.4213 24 40.0804 0.5981 26 60.0558 0.7209 28 80.0388 0.8062 33 34 35 36 37 38 39 40 13 0.0156 0.9221 14 0.0130 0.9351 15 0.0108 0.9459 16 0.0090 0.9549 17 0.0075 0.9624 18 0.0063 0.9687 19 0.0052 0.9739 20 0.0043 0.9783 19 21 23 25 27 29 30 31 32 A B C D E F G H I J Problem 9-22 (a) Queuing Model M/M/s (Exponential Service Times) Input Data Operating Characteristics Probabilities Number of Units Probability Cumulative Probability 10.1389 0.3056 30.0965 0.5177 50.0670 0.6651 70.0465 0.7674 90.0323 0.8385 10 0.0269

1 2 3 4 5 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Number of Units Probability Cumulative Probability 0 0.4118 0.4118 1 0.3431 0.7549 2 0.1430 0.8979 3 0.0596 0.9574 4 0.0248 0.9823 5 0.0103 0.9926 6 0.0043 0.9969 7 0.0018 0.9987 33 34 35 36 37 38 39 40 13 0.0000 1.0000 14 0.0000 1.0000 15 0.0000 1.0000 16 0.0000 1.0000 17 0.0000 1.0000 18 0.0000 1.0000 19 0.0000 1.0000 20 0.0000 1.0000 28 29 30 31 32 A B C D E F G H I J Problem 9-23 Queuing Model M/M/s (Exponential

1 2 3 4 5 6 7 8 9 15 16 17 18 32 33 34 35 36 37 38 39 12 0.0000 1.0000 13 0.0000 1.0000 14 0.0000 1.0000 15 0.0000 1.0000 16 0.0000 1.0000 17 0.0000 1.0000 18 0.0000 1.0000 19 0.0000 1.0000 22 23 24 25 26 2 0.1535 0.9393 3 0.0435 0.9828 4 0.0123 0.9951 5 0.0035 0.9986 6 0.0010 0.9996 19 20 21 27 28 29 30 31 A B C D E F G H I J Problem 9-26 Queuing Model M/M/s (Exponential Service Times) Input Data Operating Characteristics Arrival rate (l)30 Average server utilization

1 2 3 4 5 6 7 8 9 17 18 32 33 34 35 36 37 38 39 40 12 0.0126 0.9732 13 0.0086 0.9818 14 0.0058 0.9876 15 0.0040 0.9916 16 0.0027 0.9943 17 0.0018 0.9961 18 0.0012 0.9973 19 0.0008 0.9982 20 0.0006 0.9988 21 22 23 24 25 26 27 28 1 0.0373 0.0452 2 0.0889 0.1340 3 0.1411 0.2751 4 0.1679 0.4430 5 0.1599 0.6030 6 0.1269 0.7299 7 0.0864 0.8163 8 0.0587 0.8750 19 20 29 30 31 A B C D E F G H I J Problem 9-27 Queuing Model M/M/s (Exponential Service Times) Input

Problem 9-28 Queuing Model M/G/1 (General Service Times) Input Data Operating Characteristics Arrival rate (l)1Average server utilization (r)0.5000 Service rate (m)2Average number of customers in the queue (Lq)0.3125 Standard deviation (s)0.2500 Average number of customers in the system (L) 0.8125 Average waiting time in the queue (Wq)0.3125 Average time in the system (W) 0.8125 Probability (% of time)

Queuing Model M/D/1 (Constant Service Times) Input Data Operating Characteristics Arrival rate (l)4Average server utilization (r)0.6667 Problem 9-30 Service rate (m)6Average number of customers in the queue (Lq)0.6667 Average number of customers in the system (L) 1.3333 Average waiting time in the queue (Wq)0.1667 Average time in the system (W) 0.3333 Probability (% of time) system is empty

Queuing Model M/G/1 (General Service Times) Input Data Operating Characteristics Arrival rate (l)0.9 Average server utilization (r)0.9000 Problme 9-32 (m = 1) Service rate (m)1Average number of customers in the queue (Lq)5.0625 Standard deviation (s)0.5000 Average number of customers in the system (L) 5.9625 Average waiting time in the queue (Wq)5.6250 Average time in the system (W)

Problem 9-34 Queuing Model M/M/s with a finite population Input Data Operating Characteristics Arrival rate per customer (l) 0.7058824 Average server utilization (r)0.6635 Probabilities 2.971843 Number of Units Probability Cumulative Probability Number waiting Arrival rate(n) Term 1 Sum term 1 Term 2 Sum term 2 Decum term 2 P0(s) 0 0.3365 0.3365 0 3.529412 1 1 1 1 1.971843 1 0.2969 0.6334 0 2.823529 0.882353 1.882353 0.882353

Problem 9-36 Queuing Model M/M/s with a finite population Input Data Operating Characteristics Arrival rate per customer (l) 0.03 Average server utilization (r)0.6899 Service rate (m)0.2 Average number of customers in the queue (Lq)0.7105 Probabilities 3.225126 Number of Units Probability Cumulative Probability Number waiting Arrival rate(n) Term 1 Sum term 1 Term 2 Sum term 2 Decum term 2 P0(s) 0 0.3101 0.3101 0 0.18 1 1

1 2 3 4 17 18 19 20 21 22 23 24 25 Desired waiting time of 6 seconds, or 0.1 minutes Probabilities Desired queue length of 8 cummuters Number of Units Probability Cumulative Probability 0 0.0502 0.0502 1 0.1406 0.1908 2 0.1968 0.3876 3 0.1837 0.5713 4 0.1286 0.6999 5 0.0900 0.7900 14 15 16 31 32 33 34 35 36 37 38 39 11 0.0106 0.9753 12 0.0074 0.9827 13 0.0052 0.9879 14 0.0036 0.9915 15 0.0025 0.9941 16 0.0018 0.9958 17 0.0012 0.9971 18 0.0009 0.9980 19 0.0006 0.9986 26 27 28 29 30 A B

Problem 9-38 Queuing Model M/G/1 (General Service Times) Input Data Operating Characteristics Arrival rate (l)8Average server utilization (r)0.8000 Service rate (m)10 Average number of customers in the queue (Lq)1.7778 Standard deviation (s)0.0333 Average number of customers in the system (L) 2.5778 Average waiting time in the queue (Wq)0.2222 Average time in the system (W) 0.3222 Probability (% of

Problem 9-10 1 2 3 4 Number of customers 300 300 300 300 Cost per waiting hour $13 $13 $13 $13 Number of checkout clerks Checkout clerk hourly salary $10 $10 $10 $10 Average waiting time per customer (minutes) 10 643 Total customer waiting time (hours) 50 30 20 15 Total waiting costs $650 $390 $260 $195 Total pay

1 2 3 4 5 17 18 19 20 Probabilities Number of Units Probability Cumulative Probability 0 0.2500 0.2500 This system is now 4 independent M/M/1 queues with arrival rate = 12/4 = 3 per hour. Answers above are per queue. So multiply L and Lq by 4 to get system averages. (a) (3 customers) x (4 queues) = 12 customers total (b) 1.0 hour (c)

1 2 3 4 5 21 22 23 24 25 26 27 28 29 30 1 0.2479 0.7022 2 0.1353 0.8375 3 0.0738 0.9113 4 0.0403 0.9516 5 0.0220 0.9736 6 0.0120 0.9856 7 0.0065 0.9921 8 0.0036 0.9957 9 0.0019 0.9977 10 0.0011 0.9987 minutes per customer 36 37 38 39 16 0.0000 1.0000 17 0.0000 1.0000 18 0.0000 1.0000 19 0.0000 1.0000 19 20 31 32 33 34 35 A B C D E F G H I J Problem 9-24 Queuing Model M/M/s (Exponential Service Times) Number of Units Probability Probability 0 0.4543

Problem 9-25 Queuing Model M/G/1 (General Service Times) Input Data Operating Characteristics Arrival rate (l)10 Average server utilization (r)0.8333 Service rate (m)12 Average number of customers in the queue (Lq)2.2708 Standard deviation (s)0.0250 Average number of customers in the system (L) 3.1042 Average waiting time in the queue (Wq)0.2271 Average time in the system (W) 0.3104 Probability

Problem 9-29 Queuing Model M/G/1 (General Service Times) Input Data Operating Characteristics Arrival rate (l)1Average server utilization (r)0.3797 Service rate (m)2.633836 Average number of customers in the queue (Lq)0.1666 Standard deviation (s)0.2500 Average number of customers in the system (L) 0.5462 Average waiting time in the queue (Wq)0.1666 Average time in the system (W) 0.5462 Probability (% of

Problem 9-31 Queuing Model M/G/1 (General Service Times) Input Data Operating Characteristics Arrival rate (l)0.125 Average server utilization (r)0.6250 Service rate (m)0.2 Average number of customers in the queue (Lq)0.5417 Standard deviation (s)1.0000 Average number of customers in the system (L) 1.1667 Average waiting time in the queue (Wq)4.3333 Average time in the system (W) 9.3333 Probability (%

Problem 9-33 Queuing Model M/M/s with a finite population Input Data Operating Characteristics Arrival rate per customer (l) 0.1 Average server utilization (r)0.5155 Service rate (m)1Average number of customers in the queue (Lq)0.3297 Probabilities 2.06392 Number of Units Probability Cumulative Probability Number waiting Arrival rate(n) Term 1 Sum term 1 Term 2 Sum term 2 Decum term 2 P0(s) 0 0.4845 0.4845 0 0.6 1 1 1

Problem 9-35 Queuing Model M/M/s with a finite population Input Data Operating Characteristics Arrival rate per customer (l) 0.5 Average server utilization (r)0.6918 Service rate (m)4Average number of customers in the queue (Lq)0.7735 Probabilities 3.245018 Number of Units Probability Cumulative Probability Number waiting Arrival rate(n) Term 1 Sum term 1 Term 2 Sum term 2 Decum term 2 P0(s) 0 0.3082 0.3082 0 3.5 1 1 1

Problem 9-39 Queuing Model M/M/s with a finite population Input Data Operating Characteristics Arrival rate per customer (l) 0.9090909 Average server utilization (r)0.7190 Probabilities 8.143153 Number of Units Probability Cumulative Probability Number waiting Arrival rate(n) Term 1 Sum term 1 Term 2 Sum term 2 Decum term 2 P0(s) 0 0.1400 0.1400 0 6.363636 1 1 2 2 6.143153 1 0.2821 0.4221 0 5.454545 2.015152 3.015152 2.015152