Q.1 A company makes two products X and Y. Mr. Girish, Production Manager has estimated a unit profit of Rs. 124 and Rs. 80 on these products X and Y respectively. Both the products pass through two departments for processing and the times in minutes per unit taken by X and Y are
Product X Product Y
Department 1 150 90
Department 2 100 120
Currently the production manager is provided with 225 hours per week in department 1 and 200 hours in department 2. The company is selling its products to countries of EEC, and can sell all it can produce of X, but the EEC quotas restrict the sale of Y to a maximum of 75 units per week.
The production manager currently is manufacturing and selling 30 units of X and 75 units of Y per week.
The company wishes to maximize contribution margin and therefore has directed Mr. Girish to review his production plan. Mr. Girish is considering several possibilities including.
i. altering the production plan if it could be proved that there is a better plan than the current one.
ii. Increasing the availability of hours for either department 1 or department 2. The extra costs involved in increasing the capacity are Re. 0.5 per hour for each department.
iii. Transfer some of their allowed sales quota for product Y to another company. Because of the commitment, the company could always retain a minimum sales level of 30 units.
You are required to help Mr. Girish in the following
i. Calculate the optimum production plan using the existing capacities and state the extra contribution that would be achieved compared with the existing plan.
ii. Advise Mr. Girish whether the company should increase the capacity of either department 1 or department 2 and, if so, by how many hours and what the resulting increase should in contribution would be over that, calculated in the improved production plan.
iii. Calculate the minimum price per unit for which the company could sell the rights to their sales quota, down to the minimum level, given the plan in
(a) as a starting point.
Q. 2 A cement company has three factories A, B & C which manufacture cement which is then transported to four distribution centers. The quantity of monthly production of each factory, the demand of each distribution center and the associated transportation cost per quintal are given below.
Factory Distribution center Monthly production
W X Y Z
A 10 8 5 4 7,000
B 7 9 15 8 8,000
C 6 10 14 8 10,000
Monthly demand 6,000 6,000 8,000 5,000
(a) Suggest the optimal transportation schedule
(b) Is there any other transportation schedule which is equally attractive? If so write that.
(c) If the company wants at least 5000 quintals of cement to be transported from factory C to distribution center Y, will the transportation schedule be any different? If so, then what is the new optimal schedule and the effect on the cost?
(d) Suppose the company desires to send at most 500 quintals of cement from factory C to distribution center Y, what will the new optimal schedule be? Also obtain the transportation cost in this case.
Q.3 A firm uses three machines in the manufacture of three products. Each unit of product A requires 3 hours on machine I, 2 hours on machine II and 1 hour on machine III. Each unit of product B requires 4 hours on machine I, 1 hour on machine II and 3 hours on machine III. While unit of product C requires 2 hours on each of the three machines. The contribution of each product is Rs.30, Rs. 40 and Rs 35 respectively. The machine hours available on the three machines are 90, 54 & 93 respectively.
Formulate the above as LPP
1. Obtain the optimal solution by simplex method. Which of these three products are manufactured by the firm?
2. Calculate the percentage of capacity utilization in optimal solution
3. What are the shadow prices of machine hours?
4. Is the optimal solution degenerate?
Q.4 A small garment –making unit has five tailors stitching five different types of garments. All the five tailors are capable of stitching all the five types of garments. The output per tailor and the profit in rupees for each type of garment are given below
Garments
Tailors 1 2 3 4 5
A 7 9 4 8 6
B 4 9 5 7 8
C 8 5 7 9 8
D 6 5 8 10 10
E 7 8 10 9 9
Profit (Rs) 2 3 2 3 4
Per garment
i. Which type of garment should be assigned to which tailor in order to maximize profit assuming that there are no other constraints?
ii. If tailor D is not available during the specific period and no other substitute tailor is available, what should be the optimal assignment?
Q.5 The following table shows all the necessary information on the availability of supply to each warehouse, the requirement of each market and unit transportation cost from each warehouse to each market.
Market
P Q R S Supply
A 6 3 5 4 22
B 5 9 2 7 15
C 5 7 8 6 8
Demand 7 12 17 9
The shipping clerk has worked out the following schedule from experience: 12 units from A to Q, 1 unit from A to R, 9 units from A to S, 15 units from B to R, 7 units from C to P, and 1 unit from C to R. Check whether the clerk has optimal schedule.
Q.6 The following table gives details of a project
Normal Crash
Activity Predecessor Time Cost(Rs’000) Time Cost(Rs’000)
A - 10 20 7 30
B - 8 15 6 20
C B 5 8 4 14
D B 6 11 4 15
E B 8 9 5 15
F E 5 5 4 8
G A, D, C 12 3 8 4
Indirect cost is Rs. 400 per day. Find the optimal duration and the associated project cost