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Indian Institute of Technology, Madras

Problem Sheet on Duality and Sensitivity Analysis

1. Consider the Linear Programming problem:

Maximize Z = -X1+ 7X2- 5X3+ 14X4

Subject to 3X1+ 4X2+ 5X3+ 6X4 < 24

-X1+ X2 - 2X3+ 2X4 < 4

Xj> 0

a) Write the dual to the given problem and verify that (1,4) is a feasible solution to the dual.

b) Use information in part (a) to obtain optimal solutions to primal and dual.

c) Suppose the constraint

X1+ X2+ X3+ X4< 8 is imposed on the problem, find the optimal solutions to primal and dual.

2. Consider the LPP

Maximize Z = 8X1+ 4X2+ 9X3

Subject to 7X1+ 6X2+ X3 < 20

4X1+ 3X2+ 2X3< 29

Xj> 0

Write the dual to this problem. Solve the dual optimally and obtain the optimal solution to the primal usingcomplementary slackness conditions.

3. Construct an LPP such that neither the primal nor the dual has feasible solutions.

4. The Shortest Path Problem:

A person located in City 1 wants to reach City 5. The network, connecting the cities is shown along withthe time taken to travel. The problem is to determine the path that takes the shortest time to travel from City1 to City 5.

a) Formulate the problem as an LPP by considering decision variables Xij= 1 if i-j is in the shortestpath and zero otherwise.

b) Verify that you have as many variables as the number of arcs and as many constraints as thenumber of nodes.

c) Rewrite your formulation such that every variable has a positive coefficient in one constraint anda negative coefficient in another.

41

2

3

5

8

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47

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d) Write the dual to part (c) considering all Xij> 0

e) Put Y5= 0 and solve the dual optimally by inspection.

f) Obtain the optimal solution to the primal using complementary slackness theorem.

5) A factory manufactures three products. Three resources -- technical services, labour and administration --are required to produce these products. The following table gives the requirements on each of the resourcesfor the three products:

PRODUCT RESOURCES (HOURS) UNIT

Technical Services Labour Administration PROFIT (Rs.)

1 1 10 2 10

2 1 4 2 6

3 1 5 6 4

There are 100 hours of technical services 600 hours of labour and 300 hours of administration available. In order to

determine the optimal product mix which maximizes the total profit, the following linear program was solved:

Maximize: Z = 10X1+ 6X2+ 4X3

Subject to X1+ X2+ X3 < 100 (Technical)

10X1+ 4X2+ 5X3 < 600 (Labour)

2X1+ 2X2+ 6X3 < 300 (Administration)

X1, X2, X3 > 0

X1, X2, and X3are the quantities of product 1, product 2 and product 3 produced. The optimal solution is given bythe following tableau, where X4, X5, and X6are the slack variables:

Basis X1 X2 X3 X4 X5 X6 b

X2 0 1 5/6 10/6 -1/6 0 420/6

X1 1 0 1/6 -4/6 1/6 0 200/6

X6 0 0 4 -2 0 1 100

Cj-Zj 0 0 -16/6 -20/6 -4/6 0 Z= 4400/6

Using sensitivity analysis, answer the following with respect to the above optimal tableau:

(a) What should be the profit of product 3 before it becomes worthwhile to manufacture? Find themost profitable product mix if the profit on product 3 were increased to 50/6

(b) What is the range on the profit of product 1 so that the current solution is still optimal?

(c) It is believed that the estimate of the available hours of technical services might be wrong. Thecorrect estimate is 100 + 10awhere a is some unknown parameter. Find the range of values of awithin which the given product mix is still optimal.

(d) Determine the shadow prices of all the resources.

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(e) The manufacturing department comes up with a proposal to produce a new product requiring 1

hour of technical services, 4 hours of labour, and 3 hours of administration. The marketing andsales department predict that the product can be sold at a unit profit of Rs.8/-. What should be themanagement's decision?

6. Messrs. Poise & Gass Chemicals (P&G) have to blend three types of raw materials say, P, Q and R,supplied respectively by Pai, Qadar Batcha and Rasayan Ram, in order to obtain their final product. Thecosts of the raw materials, P, Q, and R are Rs.5, Rs.3, and Re.1 per litre respectively. Each raw material

contains two essential ingredients, viz., sulphur and phosphorous, expressed in grams per litre. The finalproduct must contain a specified minimum amount of these ingredients in every barrel. Each barrelcontains one or more of the raw materials in sufficient quantities to fulfil the minimum specifications onsulphur and phosphorus. Once the specifications are met, the remaining capacity of the barrel is filled byanother cheap liquid at negligible cost.

The following L.P. Problem was formulated to determine the least cost solution.

Minimize w = 5Y1+ 3Y2+ Y3

Subject to 2Y1+ 2Y2+ Y3 > 8 (Sulphur)

3Y1+ Y2+ 2Y3 > 6 (Phosphorus)

Y1, Y2, Y3, > 0

The optimal simplex tableau without the artificial variables columns is given below:

Y1 Y2 Y3 Y4 Y5

Y3 2 2 1 -1 0 8

Y5 1 3 0 -2 1 10

3 1 0 1 0 w=8

Noting that SUBDIVISIONS ARE INDEPENDENT OF each other answer the following questions:

a) Qadar Batcha, who supplies the raw material Q, wants to lower his price so that he can manage tosell some quantity of Q to P&G Chemicals. What price of Q should be quote to strike a deal withP&G?

b) Pai, the guy who sells P, is a more shady character. He is trying to convince Rasayan Ram that heis being exploited by P&G and so he should raise his price. By how much can the price of Rincrease without changing the current optimal solution?

c) Suppose the B.I.S. changes the specification on sulphur content - what extent of change will alterthe current basis?

d) A fourth supplier, Sav Grakki, offers to sell a raw material `S' which contains 3 gms. of sulphurand 2 gms. of phosphorus per litre. What price of `S' would make P&G think of buying it?

e) Meanwhile, Pai is getting desperate. He now claims that he has increased the sulphur content in P.What should the new sulphur content in P be, so that Pai can succeed in selling it to P&G?

f) The BIS imposes one more specification on the final product with reference to a third ingredient.After some quick laboratory analyses, P&G frame the third constraint as:0.1 Y1+ 0.2 Y2+ 0.3 Y3 > 3What is the new optimal solution?