3.27
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A B C D E F G H I
Unit Cost
1 2 3 4
1 $500 $600 $400 $200
Plant 2 $200 $900 $100 $300
3 $300 $400 $200 $100
4 $200 $100 $300 $200
Shipments
1 2 3 4 Total Shipped Supply
1 0 0 0 10 10 =10
Plant 2 20 0 0 0 20 =20
3 0 0 10 10 20 =20
4 0 10 0 0 10 =10
Total Received 20 10 10 20
= = = = Total Cost
Demand 20 10 10 20 $10,000
Retail Outlet
Retail Outlet
3.28
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A B C D E F G H I
1 2 3 4
1 800 1,300 400 700
Plant 2 1,100 1,400 600 1,000
3 600 1,200 800 900
Fixed Cost $100
Cost per Mile $0.50
Unit Cost
1 2 3 4
1 $500 $750 $300 $450
Plant 2 $650 $800 $400 $600
3 $400 $700 $500 $550
Shipments
1 2 3 4 Total Shipped Supply
1 0 0 2 10 12 =12
Plant 2 0 9 8 0 17 =17
310 1 0 0 11 =11
Total Received 10 10 10 10
Distribution Center
Distribution Center
Distribution Center
3.32
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A B C D E F G H
Unit Cost Job
1 2 3
A$5 $7 $4
Person B $3 $6 $5
C$2 $3 $4
Assignments Job Total
1 2 3 Assignments Supply
A 0 0 1 1 = 1
Person B 1 0 0 1 = 1
C 0 1 0 1 = 1
Total Assigned 1 1 1
= = = Total Cost
Demand 1 1 1 $10
3.33 a) This problem fits as an assignment problem with ships as assignees and ports as
assignments.
b)
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A B C D E F G H I
Unit Cost
1 2 3 4
1 $500 $400 $600 $700
Ship 2 $600 $600 $700 $500
3 $700 $500 $700 $600
4 $500 $400 $600 $600
Assign ments Total
1 2 3 4 Assignments Supply
1 0 1 0 0 1 = 1
Ship 2 0 0 0 1 1 = 1
3 0 0 1 0 1 = 1
4 1 0 0 0 1 = 1
Total Assigned 1 1 1 1
= = = = Total Cost
Demand 1 1 1 1 $2,100
Port
Port
3-26
3.36 a)
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A B C D E F
Model A Model B
(high speed) (lower speed)
Unit Cost $6,000 $4,000
Total Capacity
Capacity Needed
Capacity 20,000 10,000 80,000 >= 75,000
Model A Model B
(high speed) (lower speed) Total Total Cost
Purchase 2 4 6 $28,000
>= >=
1 Min Needed 6
Copies per Day
b) Let A = the number of Model A (high-speed) copiers to buy
B = the number of Model B (lower-speed) copiers to buy
Minimize Cost = $6,000A + $4,000B
3.37 a)
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A B C D E F G
Long–Range Medium-Range Short-Range
Jets Jets Jets
Annual Profit ($million) 4.2 3 2.3
Resource Resource
Resource Used Per Unit Produced Used Available
Budget 67 50 35 1498 <= 1500
Maintenance Capacity 1.667 1.333 1 39.333 <= 40
Pilot Crews 1 1 1 30 <= 30
Long–Range Medium-Range Short–Range Total Annual
Jets Jets Jets Profit ($million)
Purchase 14 016 95.6
b) Let L = the number of long-range jets to purchase
M = the number of medium-range jets to purchase
3-27
Cases
Cases
3.1 Option 1 (Shipping by Rail):
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A B C D E F G H I
Shipping Cost ($thousands)
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 61 72 45 55 66
Source 2 69 78 60 49 56
Source 3 59 66 63 61 47
Shipment Quan tity (million board feet) Total Total
Market 1 Market 2 Market 3 Market 4 Market 5 Shipped Available
Source 1 6 0 9 0 0 15 <= 15
Source 2 2 0 0 10 820 <= 20
Source 3 3 12 0 0 0 15 <= 15
Total Received 11 12 910 8
= = = = = Total Cost ($thousands)
Total To Sell 11 12 910 8 2816
Range Name Cells
ShipmentQuantity B10:F12
ShippingCost B3:F5
TotalAv ailable I10:I12
TotalCost I15
TotalReceived B13:F13
TotalShipped G10:G12
TotalToSell B15:F15
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Total
Shipped
=SUM(B10:F10)
=SUM(B11:F11)
=SUM(B12:F12)
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Total Cost ($thousands)
=SUMPRODUCT(ShippingCost,ShipmentQuantity)
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A B C D E F
Total Received =SUM(B10:B12) =SUM(C10:C12) =SUM(D10:D12) =SUM(E10:E12) =SUM(F10:F12)
3-28
Option 2 (Shipping by Ship):
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A B C D E F G H I
Shipping Cost ($thousan ds)
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 31 38 24 55 35
bold = rail cost
Source 2 36 43 28 24 31 (only rail feasible)
Source 3 59 33 36 32 26
Ship Investment ($tho usand)
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 275 303 238 0285
Source 2 293 318 270 250 265
Source 3 0283 275 268 240
Equivalent Annual Cost ($thousand s) Equivalent Annual
Market 1 Market 2 Market 3 Market 4 Market 5 Investment Cost Factor
Source 1 58.5 68.3 47.8 55 63.5 10%
Source 2 65.3 74.8 55 49 57.5
Source 3 59 61.3 63.5 58.8 50
Shipment Quantity (millio n board-feet) Total T otal
Market 1 Market 2 Market 3 Market 4 Market 5 Shipped Available
Source 1 6 0 9 0 0 15 <= 15
Source 2 5 0 0 10 520 <= 20
Source 3 0 12 0 0 3 15 <= 15
Total Received 11 12 910 8
= = = = = Total Cost ($thousands)
Total To Sell 11 12 910 8 2770.8
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A B C D E F
Equivalent Annual Cos
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 =B3+ CostFactor*B9 =C3+ CostFactor*C9 = D3+ CostFactor*D9 = E3+CostFactor*E9 =F3+CostFactor*F9
Source 2 =B4+ CostFactor*B10 =C4+ CostFactor*C10 = D4+CostFactor*D10 =E4+CostFactor*E10 =F4+CostFactor*F10
Source 3 =B5+ CostFactor*B11 =C5+ CostFactor*C11 = D5+CostFactor*D11 =E5+CostFactor*E11 =F5+CostFactor*F11
Range Name Cells
CostF actor I15
EquivalentCost B15:F17
ShipInvestment B9:F11
ShipmentQuantity B21:F23
ShippingCost B3:F5
TotalAvailable I21:I23
TotalReceived B24:F24
TotalShipped G21:G23
TotalToSell B26:F26
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Total Cost ($thousands)
=SUMPRODUCT(EquivalentCost,ShipmentQuantity)
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Source 2 =MIN(B4,B22) =MIN(C4,C22) =MIN(D4,D22) =MIN(E4,E22) = MIN(F4,F22)
Source 3 =MIN(B5,B23) =MIN(C5,C23) =MIN(D5,D23) =MIN(E5,E23) = MIN(F5,F23)
Option 3 (Shipping by Best Available for each Route):
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A B C D E F G H I
Shipp ing Cost (Rail) ($tho usands)
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 61 72 45 55 66
Source 2 69 78 60 49 56
Source 3 59 66 63 61 47
Shipp ing Cost (Ship) ($thousand s)
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 31 38 24 55 35
Source 2 36 43 28 24 31
Source 3 59 33 36 32 26
Ship Investment ($thousands)
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 275 303 238 0285
Source 2 293 318 270 250 265
Source 3 0283 275 268 240
Equivalent Annu al Cost ($th ousands) Equivalent Annual
Market 1 Market 2 Market 3 Market 4 Market 5 Investment Cost Factor
Source 1 58.5 68.3 47.8 55 63.5 10%
Source 2 65.3 74.8 55 49 57.5
Source 3 59 61.3 63.5 58.8 50
Ann u al Cost (Best Metho d) ($thousands)
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 58.5 68.3 45 55 63.5
Source 2 65.3 74.8 55 49 56
Source 3 59 61.3 63 58.8 47
Shipment Q u antity (million board feet) Total Total
Market 1 Market 2 Market 3 Market 4 Market 5 Shipped Available
Source 1 6 0 9 0 0 15 <= 15
Source 2 5 0 0 10 520 <= 20
Source 3 0 12 0 0 3 15 <= 15
Total Received 11 12 910 8
= = = = = Total Cost ($000)
Total T o Sell 11 12 910 8 2729.1
Metho d of Sh ipment
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 Ship Rail
Source 2 Ship Ship Rail
Source 3 Ship Rail
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A B C D E F
Annual Cost (Best Method
Market 1 Market 2 Market 3 Market 4 Market 5
Source 1 =MIN(B3,B21) =MIN(C3,C21) =MIN(D3,D21) =MIN(E3,E21) = MIN(F3,F21)
3-30
When comparing the three options, it is best to use the combination plan, while shipping
entirely by rail leads to the highest costs.
3.2 a) With this approach, we need to formulate an integer program for each month and
optimize each month individually.
In the second month she must buy computers to ensure that the Sales Department can
start the intranet. Emily can formulate her decision problem as an integer problem (the
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A B C D E F G H
Standard Enhanced SGI Sun
Intel Intel Workstation W orkstation
Original Cost $2,500 $5,000 $10,000 $25,000
Discount 0% 0% 10% 25%
Unit Cost $2,500 $5,000 $9,000 $18,750
Total Support
Support Needed
Support 30 80 200 2,000 80 >= 60
Budget Budget
Spent Available
Budget $2,500 $5,000 $9,000 $18,750 $5,000 <= $9,500
Standard Enhanced SGI Sun Total
Servers Intel Intel Workstation W orkstation Cost
Purchased 0 1 0 0 $5,000
Number of Employees Server Supports
Budget Spent per Server Purchased
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A B C D E
Unit Cost =(1-Discount)*OriginalCost =(1-Discount)*OriginalCost =(1-Discount)*OriginalCost =(1-Discount)*OriginalCost
Range Name Cells
Budget B8:E8
BudgetAvailable F8
BudgetSpent H8
Discount B4:E4
OriginalCost B3:E3
ServersPurchased B12:E12
Support B8:E8
SupportNeeded H8
TotalCost H12
TotalSupport F8
UnitCost B4:E5
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Total
Support
= SUMPRO DUCT (Support ,ServersPurchased)
Budget
Spent
= SUMPRO DUCT (B12:E12,ServersPurchased)
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Total
Cost
3-32
For the third month Emily needs to support 260 users. Since she has already computing
power to support 80 users, she now needs to figure out how to support additional 180
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A B C D E F G H
Discount 0% 0% 0% 0%
Unit Cost $2,500 $5,000 $10,000 $25,000
Total Support
Support Needed
Support 30 80 200 2,000 200 >= 180
Standard Enhanced SGI Sun Total
Servers Intel Intel W orkstation Workstation Cost
Purchased 0 0 1 0 $10,000
Number of Employees Server Supports
Emily decides to buy one SGI Workstation in month 3. The network is now able to
support 280 users.
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A B C D E F G H
Discount 0% 0% 0% 0%
Unit Cost $2,500 $5,000 $10,000 $25,000
Total Support
Support Needed
Support 30 80 200 2,000 30 >= 10
Standard Enhanced SGI Sun Total
Servers Intel Intel W orkstation Workstation Cost
Purchased 1 0 0 0 $2,500
Number of Employees Server Supports
Emily decides to buy a standard PC in the fourth month. The network is now able to
support 310 users.
3-33
Finally, in the fifth and last month Emily needs to support the entire company with a
total of 365 users. Since she has already computing power to support 310 users, she
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A B C D E F G H
Discount 0% 0% 0% 0%
Unit Cost $2,500 $5,000 $10,000 $25,000
Total Support
Support Needed
Support 30 80 200 2,000 80 >= 55
Standard Enhanced SGI Sun Total
Servers Intel Intel W orkstation Workstation Cost
Purchased 0 1 0 0 $5,000
Number of Employees Server Supports
Emily decides to buy another enhanced PC in the fifth month. (Note that again she
could have also bought two standard PC’s, but clearly the enhanced PC provides more
b) Due to the budget restriction and discount in the first two months Emily needs to
distinguish between the computers she buys in those early months and in the later
months. Therefore, Emily uses two variables for each server type.
Emily essentially faces four constraints. First, she must support the 60 users in the sales
department in the second month. She realizes that, since she no longer buys the
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A B C D E F G H
Standard Enhanced SGI Sun
Intel Intel W orkstation W orkstation
Month 3-5 Cost $2,500 $5,000 $10,000 $25,000
Month 2 Discount 0% 0% 10% 25%
Month 2 Cost $2,500 $5,000 $9,000 $18,750
Total Support
Support Support Needed
Month 2 30 80 200 2,000 200 >= 60
Month 3-5 30 80 200 2,000 400 >= 365
Budget Budget
Budget Spent Available
Month 2 $2,500 $5,000 $9,000 $18,750 $9,000 <= $9,500
Server Purchases Standard Enhanced SGI Sun
Intel Intel W orkstation W orkstation
Month 2 0 0 1 0 Month 2 Cost $9,000
Month 3-5 0 0 1 0 Month 3-5 Cost $10,000
Total Purchases 0 0 2 0 Total Cost $19,000
Number of Employees Server Supports
Budget Spent per Server Purchased
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A B C
Month 2 Cost =(1-Month2Discount)*Month3to5Cost =(1-Month2Discount)*Month3to5Cost
Range Name Cells
AdvancedServersNeeded E22
BudgetAvailable H13
BudgetSpent F13
Month2Budget B13:E13
Month2Cost B5:E5
Month2Discount B4:E4
Month2Purchases B17:E17
Month2Support B8:E8
Month3to5Cost B3:E3
Month3to5Purchases B18:E18
Month3to5Support B9:E9
SupportNeeded H8
TotalAdvancedServers C22
TotalCost H19
TotalPurchases B19:E19
TotalSupport F8
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Total
Support
= SUMPRO DUCT(Month2Support,Mont h2Purchases)
= SUMPRO DUCT(Month3to5Support,T ot alPurchases)
Budget
Spent
= SUMPRO DUCT(Month2Budget,Mont h2Purchases)
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Month 2 Cost =SUMPRODUCT(Month2Cost,Month2Purchases)
Month 3-5 Cost =SUMPRODUCT(Month3to5Cost,Month3to5Purchases)
3-35
d) Installing the intranet will incur a number of other costs. These costs include:
Training cost,
Labor cost for network installation,
e) The intranet and the local area network are complete departures from the way business
has been done in the past. The departments may therefore be concerned that the new
technology will eliminate jobs. For example, in the past the manufacturing department
3.3 a) The fixed design and fashion costs are sunk costs and therefore should not be
considered when setting the production now in July. Since the velvet shirts have a
positive contribution to covering the sunk costs, they should be produced or at least