5 Practice Questions on Linear Programming (Part 4)


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16.       Zigma Electronics produces two models of electronic products using Resistors, Capacitors and chips. The following table gives the entire Technological and other details in this regard:

Resource

Unit resource requirement

Maximum Availability

Model 1

Model 2

Resistor

2

3

1200

Capacitor

2

1

1000

Chips

0

4

800

Unit Profit (`)

3

4

After formulating the above problem as a Linear Programming Problem the following optimal Simplex Solution table is obtained.

Profit Coefficient

Basic Variables

Solution Values

C: 3

4

0

0

0

Cb

Xb

b

X: X1

X2

S1

S2

S3

3

X1

450

1

0

– ¼

¾

0

0

S3

400

0

0

– 2

2

1

4

X2

100

0

1

1/3

– ½

0

Z = ` 175

Z

3

4

5/4

1/4

0

D = C – Z

0

0

– 5/4

– 1/4

0

(i)          Determine the value of each resource.

(ii)         In terms of optimal profit, determine the worth of one Resistor, one Capacitor and one Chip.

(iii)        Determine the range of the applicability of the shadow prices (dual prices) for each resource.

(iv)             If the available number of chips is reduced to 350 units, will you be able to determine the new optimum solution directly from the given information? Explain.

17.       Mr. U is a production manager of Sai and Associates Production Company, which has a daily budget of 320 hours of labors and 350 units of raw material to manufacture two products. If necessary the company can employ upto 10 hours daily of overtime labor hours at the additional cost of ` 2 per hour. It takes one labor hour and 3 units of raw materials to produce A and 2 labor hours and one unit of raw material to produce one unit of product B, the profit per unit of product A is ` 10 and that of product B is ` 12. After formulating this problem as a LPP to maximize total profit, the following optional simplex solution is obtained.

Profit Coefficient

Basis Variable

Solution Values

C: 10

12

-2

0

0

0

Cb

Xb

b

X: X1

X2

X3

S1

S2

S3

12

X2

128

0

1

0

3/5

-1/5

3/5

10

X1

74

1

0

0

-1/5

2/5

-1/5

-2

X3

10

0

0

1

0

0

1

2256

Z

10

12

-2

26/5

8/5

16/5

D = C- Z

0

0

0

-26/5

-8/5

-16/5

On the basis of the above information answer the following questions:

(i)           Obtain the mathematical formulation of the above LPP

(ii)          Examine the shadow prices (dual prices) for labor hours (constraint 1) and overtime hours. Shouldn’t these two values be the same? Explain.

(iii)         Sai and associates currently pays on additional ` 2 per hour overtime charges. What is the most the company should be willing to pay?

(iv)         If Sai and Associates can acquire additional 100 units of raw material daily @ ` 1.5 per unit, would you advise the company to do so? What if the cost of raw material is ` 2 per unit?

(v)          Suppose that the company is experiencing shortage in raw material and that it cannot acquire more than 200 units, determine the associated optimal solution.

 

18.       Padma Ltd makes 3 different types of boats. All boats can be made profitably in this company but the company’s monthly production is constrained by the limited amount of labour, wood and screws available each month. The director will choose the combination of boats that maximizes his revenue, in the view of the information given in the following table.

Input

Row boat

Canoe

Kayak

Monthly availability

Labour (hrs)

12

7

9

1260 hrs

Wood (board feet)

22

18

16

19008 board feet

Screws (kg)

2

4

3

396 kg

Selling price (`)

4000

2000

5000

1)         Formulate the above as a LPP & solve it by Simplex Method

2)         From the optimal table of the solved LPP answer the following questions

(i)         How many boats of each type will be produced and what will be the resulting revenue?

(ii)        Which, if any, of the resources are not fully utilized? If so, how much of spare capacity is left? How much wood will be used to make all of the boats given in the optimal solution

 

19.       A business problem is formulated and expressed below as an LPP. (Profit is in ` and Resources are in units).

Objective Function: Maximize Z= 5X1 + 10X2 + 8X3

Subject to resource constraints,

3X1 + 5X2+ 2X3   £ 60……….. (Resource 1)

4X1 + 4X2 + 4 X3 £ 72……….. (Resource 2)

2X1 + 4X2 + 5X3  £ 100……….. (Resource 3)

X1, X2, X3 ≥ 0

Simplex algorithm of LPP, applied to the above problem yielded solution.

Basis

 

 

Cb

Xb

X1

X2

X3

S1

S2

S3

b

10

X2

1/3

1

0

1/3

-1/6

0

8

8

X3

2/3

0

1

-1/3

5/12

0

10

0

S3

-8/3

0

0

1/3

-17/12

1

18

 

Cj

5

10

8

0

0

0

Dj =Cj – Zj

-11/3

0

0

-2/3

-5/3

0

(i)         Is the above solution optimal and unique? Justify your answer.

(ii)       A customer needs the product denoted by decision variable X1. By what margin the price of this product should increase so that it becomes the, part of the optimal solution? Justify your answer.

(iii)      If the three resources are available in market what price you are willing to pay for individual resources? Justify your answer.

(iv)       If the coefficient of X2 in objective function changes to 12, will the solution above remain optimal?

(v)       There is a plan to introduce a new product whose profit margin is ` 5 and
resource requirements are 2 units of resource 1, 3 units of resource 2 and 3 units of resource 3. Should the plan be implemented? Justify your advice.

20.       A business problem is formulated and expressed below as an LPP. X1 & X2 are the production volumes of products of products A & B respectively. The resources required for producing these products are R1 & R2. Total profit is Z.

Objective function: Maximize Z = 10X1 + 4X2

Subject to the resource constraints,

20X1 + 10X2 < 1200………….R1

40X1 + 10X2 < 1600………….R2

X1, X2 > 0

Simplex algorithm of LPP, applied to the above problem yielded the following feasible solution.

CJ

10

4

0

0

C

V

X1

X2

S1

S2

bi

0

S1

0

5

1

-1/2

400

10

X1

1

¼

0

1/40

40

(I)      Please improve the above solution to optimality.

(II)     Study the solution found by you and answer the following questions with
justification:

(i)        Is the solution found by you infeasible?

(ii)       Is this a case of multiple optimal solutions (alternate optima)?

(iii)      What is the product mix and the maximum profit?

(iv)      Calculate the percent utilization of resources R1 & R2?

(v)       If one unit of R2 becomes unavailable what is the reduction in maximum profit?

 

 


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