8.84 ________________________________________________________________________________
Solve the following LP problem by the Simplex method and verify the solution graphically,
whenever possible.
Referring to Formulation 1 of Exercise 2.21, we have
Minimize f = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + y1 + y2 + y3 + y4 + y5 + y6 +
y7 + y8 + y9 + y10 + y11 + y12;
subject to 0.3081x1 + 0.3128x3 + 0.2847x5 + 0.3082x7 + 0.2886x9 + 0.285x11 + 0.3476y1 +
0.3264y3 + 0.3212y5 + 0.3216y7 + 0.3256y9 + 0.2976y11 = 330,000;
8.85 ________________________________________________________________________________
Solve the following LP problem by the Simplex method and verify the solution graphically,
whenever possible.
Obtain solutions for the three formulations of the “cabinet design” problem given in Section 2.6.
Compare solutions for the three formulations.
Chapter 8 Linear Programming Methods for Optimum Design
Section 8.7 Postoptimality Analysis
8.86/8.87____________________________________________________________________________
Formulate and solve the “crude oil” problem stated in Exercise 2.2. What is the effect on the cost
function if the market for lubricating oil suddenly increases to 12,000 barrels? What is the effect on
the solution if the price of Crude A drops to $110/bbl? Verify the solutions graphically.
1. If market for lubricating oil increases to 12,000 barrels, the cost function will not change (since
the Lagrange multiplier is zero).
8.88 ________________________________________________________________________________
Formulate and solve the problem stated in Exercise 2.6. What are the effects of the following
changes? Verify your solutions graphically.
1. The supply of material C increases to 120kg.
3. The market for product A decreases to 60.
4. The profit for A decreases to $8/kg.
Minimize = 3.61+ 3.0752+ 2.583+ 2.74
2. Supply of material D increases to 100 kg: Since the allowable increase is 17 kg, which is less
−
3. Market for product A decreases to 60 kg: Cost function will be increased by (70
−
60)(0.4) = $4,
Chapter 8 Linear Programming Methods for Optimum Design
4. Profit for A reduces to $8/kg: Since the allowable decrease is only $0.4 which is less than the
8.89 ________________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
8.90 ________________________________________________________________________________
8.91 ________________________________________________________________________________
For solution refer to Exercise 8.57
8.92 ________________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
8.93 ________________________________________________________________________________
For solution refer to Exercise 8.59
8.94 ________________________________________________________________________________
8.95 ________________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
8.96 ________________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
8.97 ________________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
Arora, Introduction to Optimum Design, 4e
8-118
8.98 ________________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
8.99 ________________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.100 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.101 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.102 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.103 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.104 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
8.105 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.106 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.107 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.108 _______________________________________________________________________________
Chapter 8 Linear Programming Methods for Optimum Design
Arora, Introduction to Optimum Design, 4e
8-119
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.109 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.110 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
8.111 _______________________________________________________________________________
Solve the following problem and determine Lagrange multipliers for the constraints at the optimum
point.
8.112 _______________________________________________________________________________
8.113 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.114 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.115 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.116 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.117 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.118 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.119 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.120 _______________________________________________________________________________
Chapter 8 Linear Programming Methods for Optimum Design
Solve the following problem and determine ranges for the right-side parameters.
8.121 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.122 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.123 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.124 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.125 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.126 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.127 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.128 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.129 _______________________________________________________________________________
8.130 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.131 _______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.132_______________________________________________________________________________
Solve the following problem and determine ranges for the right-side parameters.
8.133 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
Chapter 8 Linear Programming Methods for Optimum Design
8.134 _______________________________________________________________________________
8.135_______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.136 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.137 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.138 _______________________________________________________________________________
8.139 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.140_______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.141 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.142 _______________________________________________________________________________
8.143 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.144 _______________________________________________________________________________
8.145 _______________________________________________________________________________
8.146 _______________________________________________________________________________
Arora, Introduction to Optimum Design, 4e
8-122
8.147 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.148 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.149 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.150 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.151 _______________________________________________________________________________
8.152 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.153 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.154 _______________________________________________________________________________
Solve the following problem and determine ranges for the coefficients of the objective function.
8.155_______________________________________________________________________________
Formulate and solve the optimum design problem of Exercise 2.2. Determine Lagrange multipliers
for the constraints. Calculate the ranges for the right-side parameters, and the coefficients of the
objective function. Verify your results graphically.
8.156 _______________________________________________________________________________
Formulate and solve the optimum design problem of Exercise 2.6. Determine Lagrange multipliers
for the constraints. Calculate the ranges for the right-side parameters, and the coefficients of the
objective function. Verify your results graphically.
Chapter 8 Linear Programming Methods for Optimum Design
8.157 _______________________________________________________________________________
Formulate and solve the “diet” problem stated in Exercise 2.7. Refer to Exercise 8.79. Investigate
the effect on the optimum solution of the following changes:
1. Cost of milk increases to $1.20/kg;
2. Need for vitamin A increases to 6 units;
3. Need for vitamin reduces to 3 units;
8.158 _______________________________________________________________________________
1. Supply of empty bottles reduces to 750
1
2
2. Profit on a bottle of wine reduces to $0.80
3. Only 200 bottles of alcohol can be produced
1
2
8.159 _______________________________________________________________________________
1. Profit on margarine increases to $0.06/kg;
1
2
3
2. Supply of milk base substances increases to 2500 kg;
3. Supply of soybeans reduces to 220,000 kg;
Chapter 8 Linear Programming Methods for Optimum Design
8.160 _______________________________________________________________________________
2. Capacity of mill A reduces to 200 logs/day
Optimum solution is not changed.
3. Capacity of mill B reduces to 270 logs/day
1
2
3
4
8.161 _______________________________________________________________________________
2. Initial investment for truck B increases to $65,000; no change in the solution
3. Maintenance capacity reduces to 28 trucks;
8.162 _______________________________________________________________________________
2. Capacity of the reduction plant 2 reduces to 950,000;
4. Demand for product 2 increases to 130,000;
5. Demand for product 1 reduces to 280,000;
8.163 _______________________________________________________________________________
1. Bolting capacity is reduced to 5500/day;
−
−
−
3. The company must manufacture only 95 devices per day;
−
−
−
−
−
−
−
−
−
−
8.164 _______________________________________________________________________________
Minimize f=2x1−4x2
subject to g1=10x1+5x2 ≤ 15
Slack variables for g1 and g2 are x3 and x4, respectively. The final tableau for the problem is given
in Table E8.163. Using the given tableau:
2. Determine Lagrange multipliers for g1 and g2.
4. What is the smallest value that f can have, with the current basis, if the right side of g1 is
changed? What is the right side of g1 for that case?
1. x
*
1
= 0.0, x
*
2
= 3.0,
f*
=
−
12 ;
When ∆1 = 3.0, we have the smallest objective function value with current basis unchanged.