Assignment 1

1. Food A contains 20 units of vitamin x and 40 units of vitamin y per gram. Food B contains 30 units of

each of vitamins x and y. The daily minimum human requirements of vitamin x and y are 900 units and

1200 units respectively. How many grams of each type of food should be consumed so as to minimize

the cost, if food A costs 80 paisa per gram and food B 90 paisa per gram. Formulate the problem and solve by graphical method.

2. A company produces two types of products say type A and type B. Product A is of superior quality and product B is of a lower quality. Respective profits for the two types of products are Rs. 40/- and 30/-. The data on the resource required, availability of resources are given below:

Requirements

Product A Product B Capacity available per month

Raw materials (kg) 120 60 12000

Machining time (hrs/piece) 5 8 600

Assembly (man-hour) 4 3 500

Formulate the problem and solve by graphical method.

3. Dorian Auto manufactures luxury cars and trucks. The company believes that its most likely customers are high-income women and men. To reach these groups, Dorian Auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute commercial spots on two types of programs: comedy shows and football games. Each comedy commercial is seen by 7 million high-income women and 2 million high-income men. Each football commercial is seen by 2 million high-income women and 12 million high-income men. A 1-minute comedy ad costs $50,000, and a 1-minute football ad costs $100,000. Dorian would like the commercials to be seen by at least 28 million high-income women and 24 million high-income men. Use linear programming to determine how Dorian Auto can meet its advertising requirements at minimum cost.

4. Solve by simplex method

(a). Maximize Z = 3x + 7y

Subject to

x + 4y 20

2x + y 3 0

x + y 8

and x, y 0

(b). Maximize Z = 3x₁ + 2 x₂,

subject to

x₁ 4

x₁ + 3x₂ 15

2x₁ + x₂ 10

and

x₁ 0, x₂ 0.

(c). Maximize Z = 4x1 + 2x2 + x3

Subject to

x1 + x2 < 1

x1 + x3 < 1

x1,x2,x3 > 0

(d). Maximize Z = 2x + 3y

Subject to

-x + 2y 16

x + y 24

x + 3y > 45

-4x + 10y 20

and x, y 0

5. An advertising agency wishes to reach two types of audiences: Customers with annual income greater than Rs. 15,000 (target audience A) and customers with annual income less than Rs. 15,000 (target audience B). The total advertising budget is Rs. 2,00,000. One program of TV advertising costs Rs 50,000; one program on radio advertising costs Rs.20,000. For contract reasons, at least three program ought to be on TV and the number of radio programs must be limited to five. Surveys indicate that a single TV program reaches 4,50,000 customers in target audience A and 50,000 in target audience B. One radio program reaches 20,000 in target audience A and 80,000 in target audience B. Formulate the problem and determine the media mix to maximize the total reach.

6. A firm produces 3 products A, B and C using same type of materials L, M, and N. The specific

consumption of each material for unit production is given in the table. The profits of A, B and C are

respectively Rs. 70, Rs. 50 and Rs. 60.

Material Quantity required per unit of production Available Material

L 2 1 3 80

M 4 4 1 240

N 3 4 2 160

Find the suitable production program so as to maximize the profit.

7. A company produces three products P, Q and R form three raw materials A, B and C. One unit of

product P requires 2 units of A and 3 units B. A unit of product Q requires 2 units of B and 5 units of C and one unit of product R requires 3 units of A, 2 units of B and 4 units of C. The company has 8 units of

material A, 10 units of material B and 15 units of material C available to it. Profits per unit of products P, Q and R are Rs. 3, Rs. 5 and Rs. 4 respectively.

(a) Formulate the problem mathematically.

(b) How many units of each product should be produced to maximize profit?

8. Find the dual of the following Problems

(a) Maximize Z = x1 + 3x2

Subject to

6x1 + 19x2 100

3x1 + 5x2 40

x1 - 3x2 33

x2 25

x1 42

x1, x2 0

(b) Maximize P = x1 + 2x2

Where x1 0, x2 0

and x1 + 2x2 10

- x1 - x2 30

(c) Maximize W = 2x1 + x2 + x3 - x4

Subject to x1 - x2 + 2x3 + 2x4 3

2x1 + 2x2 -x3 = 4

x1 - 2x2 + 3x3 + 4x4 5

x1 - 2x2 + 3x3 + 4x4 5

x1, x2, x3 0

(d) Maximize Z = 4x1 + 5x2 + 3x3 + 6x4

Subject to

x1 + 3x2 + x3 + 2x4 2

3x1 + 3x2 + 2x3 +2x4 4

3x1 + 2x2 + 4x3 + 5x4 6

x1, x2, x3, x4 0

(e) Maximize Z = x1 + 1.5x2

Subject to

2x1 + 3x2 25

x1 + x2 1

x1 - 2x2 = 1

x1 , x2 0

9. Write the dual to the following problems, solve the dual and hence find the solution to the primal problem from the results of the dual.

(a) Minimize Z = 4x1 + 3x2 + 6x3

Subject to

x1 + x3 2

x2 + x3 5

x1 , x2 , x3 0

(b) Minimize W = 4 y₁ + 12 y₂ + 18 y₃,

subject to

y₁ + 3 y₃ 3

2 y₂ + 2 y₃ 5

y₁ 0, y₂ 0, y₃ 0.

(c) Minimize Z = 3x1 + 2.5 x2

Subject to

2x1 + 4x2 40

3x1 + 2x2 50

x1 , x2 0

10. Microcom is a growth oriented firm which establishes monthly performance goals for its sales force. It determines that the sales force has a maximum available hours per month for visits of 640 hours.Further, it is estimated that each visit to a potential new client requires 3 hours and each visit to a current client requires 2 hours.

Microcom establishes two goals for the coming month:

Contact at least 200 current clients

Contact at least 120 new clients

Overachieving either goal will not be penalized

Formulate the problem and solve by goal programming.

11. Formulate the goal programming from the following information of Dewrite Company.

Assignment 2

1. Mr. X flies quite often from town A to B. He can use the airport bus which costs Rs. 25 but if he takes it, there is a 0.08 chance that he will miss the flight. The stay in hotel costs Rs. 270 with a 0.96 chance of being on time for the flight. For Rs. 350 he can use a taxi which will make 99 percent chance of being on time for the flight. If Mr. X catches the plane on time, he will conclude a business transaction that will produce a profit of Rs. 10,000, otherwise he will lose it. Which mode of transportation should Mr. X use? Answer on the basis of the EMV criterion.

2. A retailer purchases cherries every morning at Rs. 50 a case and sells them for Rs. 80 a case. Any case that remains unsold at the end of the day can be disposed of the next day at a salvage value of Rs. 20 per case (thereafter they have no value). Past sales have ranged from 15 to 18 cases per day. The following is the record of sales for the past 120 days.

Cases sold: 15 16 17 18

Number of days: 12 24 48 36

Find out how many cases should the retailer purchase per day in order to maximize his profit.

3. A certain piece of equipment has to be purchased for a construction project at a remote location. This equipment contains an expensive part that is subject to random failure. Spares of this part can be purchased at the same time the equipment is purchased. Their unit cost is Rs. 1,500 and they have no scarp value. If the part fails on the job and no spare is available, the part will have to be manufactured on a special order basis. If this is required, the total cost including down time of the equipment, is estimated at Rs 9,000 for each such occurrence. Based on previous experience with similar parts the following probability estimates of the number of failures expected over the duration of the project are provided below.

Failure: 0 1 2

Probability: 0.80 0.15 0.05

a) Determine optimal EMV and optimal number of spares to purchase initially.

b) Based on opportunity losses, determine the optimal course of action and optimal value of EOL.

c) Determine the expected profit with perfect information and expected value of perfect information.

4. A food product company is contemplating the introduction of a revolutionary new product with new packaging or replacing the existing product at much higher price (S1). It may even make a moderate change in the composition of the existing product, with a new packaging at a small increase in price (S2), or may mall a small change in the composition of the existing product, backing it with the word New and a negligible increase in price (S3). The three possible states of nature or events are: i) high increase in sales (N1), ii) no change in sales (N2), and iii) 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. This is represented in the following table.

	State of Nature
Strategies	N1	N2	N3
S1	7,00,000	3,00,000	1,50,000
S2	5,00,000	4,50,000	0
S3	3,00,000	3,00,000	3,00,000

Which strategy should the concerned executive choose on the basis of

Maximax criterion,

Maximin criterion,

Minimax regret criterion,

Laplace criterion and

Hurwicz criterion for a .4.

5. Mr Chaudhary has Rs. 10,00,000 to invest in one of three options: A, B or C. The returns on his possible returns under each economic condition are given below:

State of Nature
Strategies	Inflation	Recession	No Change
A	20,000	12,000	15,000
B	30,000	8,000	10,000
C	25,000	10,000	18,000

What should he decide, using the pessimistic criterion, optimistic criterion, equally likely criterion and regret criterion?

6. A glass factory specializing in crystal is experiencing a substantial backlog, and the firm's management is considering three courses of action:

A) Arrange for subcontracting

B) Construct new facilities

C) Do nothing (no change)

The correct choice depends largely upon demand, which may be low, medium, or high. By consensus, management estimates the respective demand probabilities as 0.1, 0.5, and 0.4. The management also estimates the profits when choosing from the three alternatives (A, B, and C) under the differing probable levels of demand. These profits, in thousands of dollars are presented in the table below:

8. The following table is given the payoff table (in units of thousands of dollars) for a decision analysis problem:

	State of Nature
Strategies	N1	N2	N3
S1	7	2	2
S2	5	4	0
S3	3	3	3
Probability	0.3	0.4	0.3

(a) Which strategy should be chosen in order to get best profit by using decision tree?

(b) Find EVPI

(c) You are given the opportunity to spend $ 1,000 to obtain more information about which state of nature is likely to occur. Given your answer to part (b), might it be worthwhile to spend this money?

9. A grocery with a bakery department is faced with the problem of deciding how many cakes it should buy in order to meet the days demand. The grocer prefers not to sell day-old goods in competition with fresh products; leftover cakes are, therefore, a complete loss. On the other hand, if customer desires a cake and all of them have been sold, the disappointed customer will buy from elsewhere and sales will be lost. The grocer has, therefore, collected information on the past sales on a selected 100-day period as shown in the table.

Sales per Day	Number of Days
25	10
26	30
27	50
28	10

Construct the payoff table and opportunity loss table. What is the optimal number of cakes that should be bought each day? Also find and interpret EVPI. Given that, one cake costs $0.8 and sells for $ 1.

Assignment 3

1. The company manufactures around 200 mopeds. Depending upon the availability of raw materials and other conditions, the daily production has been varying from 196 mopeds to 204 mopeds, whose probability distribution is as given below:

Production/day: 196 197 198 199 200 201 202 203 204

Probability : 0.05 0.09 0.12 0.14 0.2 0.15 0.11 0.08 0.06

The finished mopeds are transported in a specially designed three-storied lorry that can accommodate only 200 mopeds. Using the following 10 random numbers : 82, 89, 78, 24, 53, 61, 18, 45, 23 and 50, simulate the mopeds waiting in the factory.

(i) What will be the average number of mopeds waiting in the factory?

(ii) What will be the average number of empty spaces in the lorry?

2. A bakery keeps stock of a popular brand of cake. Previous experience shows that the daily demand pattern for the item with associated probabilities, as given below:

Daily demand (number : 0 10 20 30 40 50

Probability : 0.01 0.20 0.15 o.50 0.12 0.02

Use the following sequence of random numbers to simulate the demand for next 15 days.

Random numbers: 25, 39, 65, 76, 12, 05, 73, 89, 19, 49, 81, 12, 57, 91, 34.

Also estimate the daily average demand for the cakes on the basis of the simulated data.

3. A book store wishes to carry a particular book in stock. Demand is not certain there is a lead-time of 2 days for stock replenishment. The probabilities of demand are given below:

Demand(units/day)	0	1	2	3	4
Probability	0.05	0.10	0.30	0.45	0.10

Each time an order is placed, the store incurs an ordering cost of NRs. 10 per order. The store also incurs a carrying cost is NRs. 0.5 per book per day. The inventory carrying cost is calculated on the basis of stock at the end of each day. The manager of the store wishes to find total coat, including ordering cost and Holding cost, for inventory decision.

Order 8 books when present inventory plus any outstanding order falls below 8 books.

Currently (beginning of the first day) the store has a stock of 8 books plus 6 books ordered two days ago and are expected to arrive next day. Carry out simulation runs for 10 days to find total cost. You may use random numbers in the given sequences (Using first number for 10 day one)

89, 34, 78, 63, 61, 81, 39, 16, 13, 73.

4. A dentist schedules all his patients for 30-minute appointments. Some of the patients take more than 30 minutes some less, depending on the type of dental work to be done. The following summary shows the various categories of work, their probabilities and time actually needed to complete the work:

he following summary shows the various categories of work, their probabilities and time actually needed to complete the work:

Category of Service	Time Required( minutes)	Probability of Category
Filling	45	0.40
Cleaning	15	0.15
Crown	60	0.15
Extraction	45	0.10
Check-up	15	0.20

Simulate the dentists clinic for four hours and determine the average waiting time for the patients as well as the idleness of the doctor. Assume that all the patients show up at the clinic at exactly their scheduled arrival time starting at 8.00 a.m. Use the following random numbers for handling the above problem: 40, 82, 11, 34, 25, 66, 17, 79.