Duality theorem

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Fnebbbu1v

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Mathematics

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problems

5.1

5.5

5.9

5.1

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LINEAR PROGRAMMING o ind VAŠEK CHVÁTAL PRE strate Theorem 5.1 on each of the three LP problems in problem 2.1. 2 Mauimur subject to 4xS 4 ** wiu -* - 2x -3x + xs-1 * - *; SI - 2x, + 7x, s6 - 5x + 2x₂ 3-3 7x, - 3x, 5 6 *.*, 2 0. proposed solution As for each of the two problems below, use Theorem 5.3 to check the optimality of the * Maximize 7x + 6x₂ + 5x,- 2x + 3x, subject to x + 3x₂ + 5x,- 2x + 2x₂ 5 4 4x + 2x₂ - 2x + x + x, 53 2x + 4x₂ + 4x - 2x + 5x, ss 3x + x₂ + 2x - x - 2x, S1 X, X, X, X. x, 20. Proposed solution: *+-0, x - 39 x = 0. h Maximize 4x + 5x₂ + x + 3x - 5x + 8x subject to - 4x + 3x + x + xos1 5x₂ + 3x₂ + x, - 5x + 3x S 4 4x + 5x₂ – 3x₂ + 3x - 4x + x 54 - X2 + 2x + x - 5x55 - 2x + x₂ + x₂ + x + 2x + 2x 5 7 2x - 3x₂ + 2x₂ - xy + 4x + 5x 55 *. *, *. *, X5, X6 20. 5 1 Proposed solution: *, - 0, * = 0. *3 = 2 * = 2 *s = 0, ** 2 54 In problem 16, one of the possible strategies is as follows: * Smoke all 400 bellies on regular time. * Smoke 20 picnics on regular time and 210 on overtime. • Smoke 40 hams on overtime and sell 440 fresh. Use Theorem 53 to find out whether this strategy is optimal or not. $$ In problem 1.7. one of the possible strategies is as follows: 7 Bled 3.754 barrels of alkylate, 2,666 barrels of catalytic, 920 barrels of straight-run, and 3 barrels of isopentane into 7.883 barrels of Avgas A. Blend 0 harrels of alkylate, 3.096 barrels of straight-run, and 672 barrels of isopentane mo 32 barrels of Avgas B. • Sel s barrels of isopentane raw. Use Theorem 53 to find out whether this strategy is optimal or not. 70 5 The Duality Theorem In the optimal solution to problem 16. all the bellies and picnics are smoked. However suficiently drastic changes in market prices might provide an incentive to change thi policy. Assume that the market price of fresh bellies increases by x dollars, while all the other prices remain fixed at their original levels. How large would x have to be in order to make it profitable for the plant to sell fresh bellies? Ask and answer a similar question for picnics. How would the sales of small amounts of fresh bellies and picnics affect the rest of the operation? What precisely do "small amounts" mean in this context? 7 in the optimal solution to problem 1.7, 85 barrels of isopentane are sold raw at 8483 per barre. Find the break-even selling prices for raw alkylate, catalytic, and straight-run . Next, assume there is a demand for Avgas C with PN at least 80 and RVP at most 7. Find the break-even selling price of this gasoline 5* Can you interpret the complementary slackness conditions in economic terms? 59 Lets* be the optimal value of (5.24) and let yi...... be any optimal solution of the dual problem. Prove that ΣΑ, ς * + Στε for every feasible solution x, Xy.... X, of (5.25). 5.10 Construct an example showing that the conclusion of Theorem 5.5 may fail if the hypothesis that (524) has a nondegenerate basic optimal solution is omitted.
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Explanation & Answer

attached is my solution to 5.1, 5.5 and 5.9

Problem 5.1:
a) We maximize P  3x1  2 x2  4 x3 subjected to

x1  x2  2 x3  4
2 x1  3x3  5
2 x1  x2  3x3  7
x1 , x2 , x3  0
The optimal solution to the maximizing problem, by the method of Solver in Excel, is

x1* 

5 * 3
, x2  and x3*  0
2
2

Hence, the dual to this linear programming (LP) problem is
Minimize D  4 y1  5 y2  7 y3 subjected to

y1  2 y2  2 y3  3
y1  y3  2
2 y1  3 y2  3 y3  4
y1 , y2 , y3  0
The optimal solution to the dual problem, by the method of Solver in Excel, is

y1*  0, y2* 

As demonstrated in Theorem 5.1, we check for

n

m

j 1

i 1

5
2
, and y3* 
6
3
n

m

j 1

i 1

 c j x*j   bi yi* where

 c j x*j   bi yi*
5
3
5
2
 3  2   4  0  4  0  5  7 
2
2
6
3
21 21


2
2
Hence, the validity of Theorem 5.1 is checked.
b) We maximize P  5 x1  6 x2  9 x3  8 x4 subjected to

x1  2 x2  3x3  x4  5
x1  x2  2 x3  3x4  3
x1 , x2 , x3 , x4  0
The optimal solution to the maximizing problem, by the method of Solver in Excel, is

x1*  1, x2*...