Or Problem Set

7/27/2019 Or Problem Set

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Operations Research problems by Devendra Jaiswal

Q1 Reddy Mikks produces both interior and exterior paints from two raw materials, M1 and

M2. The following table provides the basic data of the problem: refer Data Table.

A market survey indicates that the daily demand for interior paint cannot exceed that for

exterior paint by more than 1 ton. Also the maximum daily demand for interior paint is 2 tons.

Reddy Mikks wants to determine the optimal product mix of interior and exterior paints thatmaximizes the total daily profit.

Tons of raw material per

ton of

Exterior

Paint

Interior

Paint

Maximum daily

availability

Raw material M1 6 4 24

Raw material M2 1 2 6

Profit Per Ton 5000 4000

Q2 A firm makes two products X, Y and has a total production capacity of 9 tons per day, X and

Y requiring the same production capacity. The firm has a permanent contract to supply at least

2 tons of X and atleast 3 tons of Y per day to another company. Each ton of X requires 20

machine hours of production time and each ton of Y requires 50 machine hours of production

time. The daily maximum possible number of machine hours is 360. All the firms output can be

sold, and the profit made is Rs.80 per ton of X and Rs.120 per ton of Y. it is required to

determine the production schedule for the maximum profit and to calculate this profit.

Q3 A manufacturing company is engaged in producing 3 types of products- A, B and C. the

production department produces, each day, components sufficient to make 50 units of A, 25

units of B and 30 units of C. The management is confronted with the problem of optimizing the

daily production of products in assembly department where only 100 man-hours are available

Type of productProfit contribution per unit of product (Rs.) Assembly time per product (hrs)

A 12 0.8

B 20 1.7

C 45 2.5


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daily to assemble the products. The following additional information is available.

The company has a daily commitment for 20 units of products A and a total of 15 units of

product B and C. Formulate this problem as an LP model so as to maximize the total profit.

Q4 An electronic company produces 3 types of parts for automatic washing machines. Itpurchases casting of the parts from local foundry and then finishes the parts on Drilling,

Shaping and Polishing machines.

The selling prices of parts A, B and C, respectively are Rs.8, Rs.10 and Rs.14. All parts made

can be sold . Casting of parts A, B, and C cost Rs.5, Rs.6 and Rs.10.

The shop processes only one of each type of machine. Costs per hour to run each of the 3

machines are Rs.20 for Drilling, Rs.30 for Shaping and Rs.30 for Polishing. The capacities (parts

per Hour) for each part on each machine are shown in the following table-

Machine

Capacity per hour

Part A Part B Part C

Drilling 25 40 25

Shaping 25 20 20

Polishing 40 30 40

The management of the shop wants to know how many parts of each type it should produce

per hour in order to maximize profit for an hours run. Formulate this problem as an LP modelso as to maximize profit to the company.

Q5 A company, engaged in producing tinned food, has 300 trained employees on rolls, each of

whom can produce one can of food in a week. Due to the developing taste of the public for this

kind of food, the company plans to add to the existing labor force by employing 150 people, in a

phased manner, over the next five weeks. The newcomers would have to undergo a 2-week

training program before being put to work. The training is to be given by employees from

among the existing ones and it is known that one employee can train 3 trainees. Assume that

there would be no production from the trainers and the trainees during the training period as

the training is off-the-job. However, the trainees would be remunerated at the rate of Rs.300per week, the same rate as for the trainers.


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The company has booked the following orders to supply during the next 5 weeks.

Week 1 2 3 4 5

No. of cans 280 298 305 360 400

Assume that the production in any week would not be more than the number of cans

ordered for so that every delivery of the food would be fresh.

Formulate this problem as an LP model to develop a training schedule that minimizes the

labor cost over the five-week period.

Q6 A company makes two kinds of leather belts. Belt A is a high quality belt and belt B is of

lower quality. The respective profits are Rs.4 and Rs.3 per belt. The production of each of type A

requires twice as much time as a belt of type B, and if all belts were of type B, the company

could make 1000 per day. The supply of leather is sufficient for only 800 belts per day ( both A

and B combined). Belt A requires a fancy buckle and only 400 per day are available. There are

only 700 buckles a day available for belt B. What should be the daily production of each type of

belt? Formulate this problem as an LP model and solve it by simplex method.

Some More Examples

Q1 A garment manufacturer has a production line making two styles of shirts. Style I needs 200

g of cotton thread, 300 g of Dacron thread and 300 g of linen thread. Corresponding

requirements of style II are 200g, 200g and 100g. The net contributions are Rs. 19.50 for style I

and Rs. 15.90 for style II. The available inventory of cotton thread, Dacron thread and linen

thread are, respectively, 24 kg, 26 kg and 22 kg.

The manufacturer wants to determine the number of each style to be produced with

the given inventory. Formulate the LPP model.

Q2 An animal feed company must produce 200 kg of a mixture consisting of ingredients A and B

daily. A costs Rs. 3 per kg and B costs Rs. 8 per kg. Not more than 80 kg of A can be used and at

least 60 kg of B must be used.

The company wants to know how much of each ingredient should be used to minimize

cost. Formulate the LPP.

Q3 A farmer has a 125 acre farm. He produces radish, mutter and potato. Whatever he raises is

fully sold. He gets Rs. 5 per kg for radish, Rs. 4 per kg for mutter and Rs. 5 per kg for potato. The

average yield per acre is 1500 kg for radish, 1800 kg for mutter and 1200 kg for potato. Cost of

manure per acre is Rs. 187.50, Rs. 225 and Rs. 187.50 for radish, mutter and potato

respectively. Labour required per acre is 6 mandays each for radish and potato and 5 man days

for mutter. A total of 500 mandays of labour is available at the rate of Rs. 40 per manday.


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Formulate this as an LPP model to maximise the profit.

Q4 Jindal manufactures a type of sofa set containing seven components: one sofa, two centre

tables and four chairs.

These can either be manufactured in-house or sub-contracted as per the data given in thetable:

Per component Sofa Table Chair

Direct Material Rs. 1,000 Rs. 500 Rs. 550

Direct Labour hours 100 50 10

Sub-contract price Rs. 2,500 Rs. 1,000 Rs. 750

Sales of sofa sets are 8,000 per period, each selling for Rs.7,500. A capacity constraint of

500,000 direct labour hours obliges the company to sub-contract some components.

The variable overheads vary with direct labour hours at Rs. 2 per hour. Fixed costs are Rs.

1,750,000 per period and labour costs Rs. 5.50 per hour.

Formulate LPP to minimise costs.

Q5 A mutual fund has Rs. 2 million available for investment in Government bonds, blue chipstocks, speculative stocks and short-term bank deposits. The annual expected return and the

risk factor are as shown:

Investment Return% Risk factor (0- 100)

Bonds 14 12

Blue Chip 19 24

Speculative 23 48

Short-term 12 6


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The fund is required to keep at least Rs. 200,000 in short-term deposits and not to exceed an

average risk factor of 42. Speculative stocks must not exceed 20% of the money invested.

Formulate the LPP maximizing expected annual return.

Q6 The vitamins V and W are found in two different foods, F1and F2. The respective prices per

unit of each food are Rs. 3 and Rs. 2.5. One unit of F1contains 2 units of vitamin V and 3 units of

vitamin W. One unit of F2contains 4 units of vitamin V and 2 units of vitamin W. The daily

requirements of V and W are at least 60 units and 75 units respectively.

Formulate an LPP to meet the daily requirement of the vitamins at minimum cost

Transportation problem

Q1 Hi-Fi Ltd. has 3 production shops supplying a product to 5 warehouses. The cost of

production varies from shop to shop and cost of transportation from one shop to a warehouse

also varies. Each shop has a specific production capacity and each warehouse has certain

amount of requirement. The costs of transportation are given as:

Ware House

I II III IV V SS

Shop

A 6 4 4 7 5 100

B 5 6 7 4 8 125

C 3 4 6 3 4 175

DD 60 80 85 105 70 400

The cost of manufacturing the product at different production shops is

Shop Variable cost Fixed Cost

A 14 7000

B 16 4000

C 15 5000

Find the optimum quantity to be supplied from each shop to different warehouses at minimum

total cost.


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Q2 Sun ray transport company ships truckloads of grain from three silos to four mills. The

supply (in truckloads) and the demand (in truckloads) together with the unit transportation

costs per truckload on the different routes are summarized in the transportation model below.

The unit transportation costs are in hundreds of dollars. The model seeks minimum cost

shipping schedule between silos and mills.

M1 M2 M3 M4

S1

10 2 20 11

15

S2

12 7 9 20

25

S3

4 14 16 18

10

5 15 15 15 50

Q3 Gammon India is interested in taking loans from banks for some of its projects P, Q, R, S, T.

the rate of interest and the lending capacity differ from bank to bank. All these projects are to

be completed. The relevant details are provided in the following table. Assuming the role of a

consultant, advise the company as to how it should take the loans so that the total interest

payable will be least. Are there alternate optimum solutions? If so, indicate such solutions.

Bank Interest rate in percentage for projects Max. Credit

P Q R S T (in thousands)

ICICI 20 18 18 17 17 Any Amount

SBI 16 16 16 15 16 400

Cooperative bank 15 15 15 13 14 250

Amount required 200 150 200 125 75

(in thousands)


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Q4 Champatlal & Co. wishes to develop a monthly production schedule for the next three

months. Depending upon the sales commitments, the company can either keep the production

constant, allowing fluctuations in inventory, or inventories can be maintained at a constant

level, with fluctuating production. Fluctuating production necessitates, working overtime, the

cost of which is estimated to be double the normal production cost of Rs.12 per unit.Fluctuating inventories results in inventory carrying cost of Rs.2 per unit. If the company fails to

fulfill its sales commitment, it incurs a shortage cost of Rs.4 per unit per month. The production

capacities for the next three months are shown in the table:

Production Capacities

Month Regular Overtime Sales

M1 50 30 60

M2 50 0 120

M3 60 50 40

Determine the optimal production schedule

Decision Analysis

Q1 A food products company is contemplating the introduction of a revolutionary new

product with new packaging or replace the existing product at much higher price (S1) or a

moderate change in the composition of the existing product with a new packaging at a smallincrease in price (S2) or a small change in the composition of the existing product except the

word New with a negligible increase in price (S3). The three possible states of nature or

events are;

a) high increase in sales (N1),

b) No change in sales (N2), and

c) decrease in sales (N3).

The marketing department of the company worked out the payoffs in terms of yearly net

profits for each of the strategies of three events (expected sales). This is represented in the

table. Which strategy should the concerned executive choose on the basis of

1) Maximin criterion 2) Maximax criterion

3) Minimax criterion 4) Laplace criterion 5) Hurwicz criterion given =0.45


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Q2 A toy manufacturer is considering a project of manufacturing a dancing doll with three

different movement designs. The doll will be sold at an average of Rs.10. the first movement

design using gears and levels will provide the lowest tooling and setupcost of Rs.1,00,000 and

Rs.5 per unit of variable cost. A second design with spring action will have a fixed cost of

Rs.1,60,000 and a variable cost of Rs. 4 per unit. Yet another design with weights and pulleyswill have a fixed cost of Rs.3,00,000 and variable cost Rs.3 per unit. One of the following

demand events can occur for the doll with the probabilities:

Construct a pay-off table for the above project

Which is the optimal design?

How much can the decision-maker afford to pay to obtain perfect information about the

demand?

Demand (units) Probability

Light demand 25,000 0.1

Moderate demand 1,00,000 0.7

Heavy demand 1,50,000 0.2

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