Operations research

Day 1

Assignment problem

• A solicitors’ firm employs typists on hourly piece-rate basis for their daily work. There are five typists and their charges and speed are different. According to an early understanding, only one job is given to one typist and the typist is paid for a full hour even when he works for a fraction of an hour. Find the least cost allocation for the data

Data

Typist Rate/hour(rs)

No. of pages typed/hr

Job No. of pages

A 5 12 P 199

B 6 14 Q 175

C 3 8 R 145

D 4 10 S 298

E 4 11 T 178

Question 2

• A company plans to assign 5 salesmen to 5 districts in which it operates. estimates of sales revenue in thousands of rupees for each salesman in different districts are given below. In your opinion, what should be the placement of the salesmen if the objective is to maximize the expected sales revenue?

Data

District

salesmen

S1 40 46 48 36 48

S2 48 32 36 29 44

S3 49 35 41 38 45

S4 30 46 49 44 44

S5 37 41 48 43 47

Home work

• Q1. Question No. 10 page no.314 Book. Quantitative techniques in management N.D. Vohra

• Q2. Question 11 page no.314 Book. Quantitative techniques in management N.D. Vohra

Day 2 – transportation Problem

• Q1. priyanshu Enterprise has three factories at locations A,B, and C which supplies three warehouse e located at D , e, and F monthly factory capacities are 10, 80 and 15 units respectively. Monthly warehouse requirements are 75, 20, and 50 respectively. Unit shipping costs (in Rs) are given below.

Data warehouse

Factory D E F

A 5 1 7

B 6 4 6

C 3 2 5

The penalty cost for not satisfying demand at the warehouses D,E, and F are 5, 3, 2 rupees. Per unit respectively. Determine the optimal distribution of Priyanshu

Question 2

• A company has four factories situated in different locations in the country and four sales agencies located in four other locations in the country. The cost of production (Rs per unit), the sale price (Rs. Per unit) shipping cost (Rs. Per unit) in the cells of matrix, monthly capacities and monthly requirements are given below.

Data Factory Sales agency Monthly

CapacityCost of production

1 2 3 4

A 7 5 6 4 10 10

B 3 5 4 2 15 15

C 4 6 4 5 20 16

D 8 7 6 5 15 15

Monthly Req.(units)

8 12 18 22

Sales Price 20 22 25 18

Find the monthly production distribution schedule which will maximize profits

Home work

• Q1). Q10 page 262

• Q2) . Q11 page 263

• Book – Quantitative techniques in management N.D.

Vohra

What is Game Theory?

Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome

Day 3- Game theory

• There are several game theory models which can be classified on the basis of factors like the number of players involved, the sum of gains or losses and the number of strategies employed in the game.

Cont..

• Two person zero sum game- one which involves two persons and where the gain made by one equals the loss incurred by the other.

The Payoff Matrix

Player #1

Player #2

Strategy #1 Strategy #2

Strategy #1

Strategy #2

Payoff (1,1) Payoff (1,2)

Payoff (2,1) Payoff (2,2)

Saddle point

• The point of equilibrium is known as saddle point. ie the point at which the maximin value is equal to minimax value and the game is solved.

Problem 1

• Suppose that there are two firms A&B in an area which, for a long period in the past, have been selling a competing product and are now engaged in struggle for a large share of the market.

• Assume that both these firms are considering the same three strategies in a bid to gain the share in the market; low advertising, high advertising, and quality improvement.

• Assume also that currently they are sharing the market equally and each of the firm can employ only one of the strategies at a time.

Cont..

• The pay-off matrix is given below

B’s strategy

A’s strategy

B1 B2 B3

A1 12 -8 -2

A2 6 7 3

A3 -10 -6 2

The pay off matrix is drawn from A’s point of view.A +ve pay off indicates that the firm A has gained the market share at the expense of firm B. while –Ve imply B’s gain at A’s expense

Cont..

• Determine the best strategy for A&B assuming that they both are acquainted with the information contained in the pay-off matrix and each one is not aware of the move the other is likely to take.

Solution

B’s strategy

A’s strategy

B1 B2 B3

A1 12 -8 -2

A2 6 7 3

A3 -10 -6 2

-83*-10

12 7 3*

Row Minima

3*Column Maxima

Saddle point

Problem 2

• Two leading firms, Nirmala Textiles Ltd. And Swati Rayons Ltd., for years have been selling shirting, which is but a small part of both firms’ total sales. The marketing Director of Nirmala Textiles raised the question. “What should the firm’s strategies be in terms of advertising for the product in question?” The system group of Nirmala Textiles developed the following data for varying degrees of advertising:

• No advertising, medium advertising and heavy advertising for both firms will result in equal market share.

• Nirmala Textiles with no advertising: 40 % of the market with medium advertising by Swati Rayons and 28% of the market with heavy advertising by Swati Rayons.

• Nirmala Textiles using medium advertising: 70% of the market with no advertising by Swati an 45%of the market with heavy advertising by Swati

• Nirmala Textiles using heavy advertising: 75% of the market with no advertising by Swati an 52.5%of the market with medium advertising by Swati

• Based upon the above information, answer the marketing directors question

When no saddle point exists

• For solving such problems – Mixed strategies are used- a combination of

two or more strategies that are selected one at a time, according to pre-determined probabilities.

– Ie. A player decides to mix his choices among several alternatives in a certain ratio.

Cont..

• Suppose that A plays strategy a1 with probability x and plays strategy a2 with probability 1-x.

• If B plays strategy b1 , then A’s expected pay-off can be determined in reference to the figures given in the first column of the payoff matrix as follows

• Expected payoff (given that B plays b1)=• Expected payoff (given that B plays b2)=

Cont..

• Determine a value x so that the expected pay-off for A is the same irrespective of the strategy adopted by B

• Obtain this value by equating the two equations

• A shall net a gain/loss of x per play in the long run.

Dominance rule

• In a game, sometimes a strategy available to a player might be found to be preferable to some other strategy/ strategies. Such a strategy is said to dominate the other one(s).

• Used in simplifying the games

Home work

• A company is currently involved in negotiations with its union on the upcoming wage contract. With the aid of n outside mediator, the table below was constructed by the management group. The pluses are to be interpreted as proposed wage increases while a minus figure indicates that a wage reduction is proposed.

Cont…• The mediator informs the management group

that he has been in touch with the union and that they have constructed a table that is comparable to the table developed by the management .

• Both the company and the union must decide on an overall strategy before negotiations begin. The management group understands the relationship of company strategies to union strategies in the following table but lacks specific knowledge of game theory to select the best strategy for them. Assist the management on this problem. What game value and strategies are available to the opposing group

Conditional cost to company

Company strategies

Union strategies

U1 U2 U3 U4

C1 +0.25 +0.27 +0.35 -0.02

C2 +0.20 +0.16 +0.08 +0.08

C3 +0.14 +0.12 +0.15 +0.13

C4 +0.30 +0.14 +0.19 0.00

Day 4 –Formulation & Graphical

• A local travel agent is planning a charter trip to a major sea port. The eight day and seven night package includes the fare for round trip, surface transportation, board and lodging and selected tour options. The charter trip is restricted to 200 persons and past experience indicates that there will not be any Problem for getting 200 clients.

• The problem for the travel agent is to determine the number of Deluxe, Standard and Economy tour packages to offer for this charter. These three plans differ according to seating and service for the flight, quality of accommodations, meal plans and tour options. The following table summarises the estimated prices for the three packages and the corresponding expenses for the travel agent. The travel agent has hired an aircraft for the flat fee of Rs. 2,00,000 for the entire trip.

Cont..

In planning the trip, the following considerations must be taken into account:

1. at least 10% of the packages must be deluxe type.2. At least 35% but not more than 70% must be standard

type.3. At least 30% must be economy type.4. The maximum No. of deluxe available in any aircraft is

restricted to 60.5. The hotel desires that at least 120 of the tourists

should on the deluxe & standard packages taken together.

Prices and costs for tour packages per person

Tour Plan Price Hotel costs Meal & other expenses

Deluxe 10,000 3,000 4,750

Standard 7,000 2,200 2,500

Economy 6,500 1,900 2,200

Cont..

The travel agent wishes to determine the number of packages to offer in each type so as to maximize the total profit.

1. Formulate this as a LPP

2. Restate the above LPP in terms of two decisions variables

3. Find the optimal strategy.

Example 2• A manufacturer employs three inputs: man hours,

machine-hours and cloth material to manufacture two types of dresses. Type A dress fetches him a profit of 160 per piece, while type B, that of Rs.180 per piece. The manufacturer has enough man hours to manufacture 50 pieces of type A or 20 pieces of type B dresses per day while the machine hours he possesses suffice only for 36 pieces of type A or for 24 pieces for type B dresses. Cloth material available per day is limited but sufficient enough for 30 pieces of either type of dresses. Formulate LPP & solve graphically.

Day 5

Example 1• A firm produces three products A, B and C, each

of which passes through three departments. Fabrication, Finishing and Packing. Each unit of product A requires 3,4 and 2 units. Product B requires 5, 4, 4 and a unit of product C requires 2,4,5 hours respectively in three departments. Available time in each department is 60 hours, 72 hours and 100 hours respectively.

• If the unit contribution of product A is Rs. 5, of B is Rs. 10 that of C is Rs. 8, determine the product mix to be produced every day to maximize the total contribution. Also determine if capacity of any department remains unutilized? (Use simplex method)

Cont..

• If the optimal solution obtained does not require the production of some product, explain, as to why such product would not be produced. In this context, indicate the quantity of other product/s that would be foregone for producing such product.

• What would be the effect on the solution of each of the following– Obtaining an order for 6 units of A, which has to be

met

Optimal table

Basis Coeff (Cj)

Value (bi)

x1 x2 x3 S1 S2 S3

X2 10 8 1/3 1 0 1/3 -1/6 0

X3 8 10 2/3 0 1 -1/3 5/12 0

S3 0 18 -8/3 0 0 1/3 -17/12 1

Deltacj-zj

-11/3 0 0 -2/3 -5/3 0

Also find• If the profit of product B was increased to 18/- would

the optimal solution change?• If the capacity of finishing department was

increased by 20%. Would the optimal solution change?

• An increase of 20% capacity in fabrication. What would be the effect?

• . Suppose that the firm is contemplating introduction of a new product with a likely profit margin of 8/-. Production department’s study shows that this product shall consume 3hrs, in each of the fabrication & finishing dept. and 2hrs in packaging dept. should the firm decide for the introduction of this product?

Example 2

• Quality air conditioning manufactures three home air conditioners: an economy model, a standard model and a deluxe model. The profits per unit are $63, $95 and $135 respectively. The production requirements per unit are as follows.

Data

Number of fans

Number of cooling coils

Manufacturing time (hours)

Economy 1 1 8

Standard 1 2 12

Deluxe 1 4 14

• For the coming production period, the company has 200 fan motors, 320 cooling coils and 2400 hours of manufacturing time available. How may economy models, standard models, and deluxe models should the company produce in order to maximize the profit?

• Formulate the problem as an LPP• The simplex solution table is given below. What

is the optimal solution?• Which constraints show extra capacity? How

much?• If the profit of the deluxe model were increased

to $150 per unit, would the optimal solution change?

Basic variables

Cost of the basic variables

Value of the basic variables

X1 X2 X3 S1 S2 S3

X1 63 80 1 0 -10 2 -1 0

X2 95 120 0 1 11 -1 1 0

S3 0 320 0 0 -38 -4 -4 1

H .W

• 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.

Data

Resource Unit resource requirement Maximum availability

Model 1 Model 2

Resistor 2 3 1200

Capacitor 2 1 1000

Chips 0 4 800

Unit Profit(Rs) 3 4

– After formulating the above problem as a LPP the following optimal Simplex Solution table is obtained.

– Determine the value of each resource– In terms of optimal profit, determine the

worth of one Resistor, one Capacitor and Chip.

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

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

CB Basis Solution Values

X1 X2 S1 S2 S3

3 X1 450 1 0 -1/4 ¾ 0

0 S3 400 0 0 -2 2 1

4 X2 100 0 1 1/3 -1/2 0

C-Z 0 0 -5/4 -1/4 0