Optimization Techniques

(MA- 051)

Overall Marking Scheme

Minor Test 1 20Minor Test 2 20

Major Test 50 Internal Assessment 10

Total 100 marks

References

Taha,H.A.(1992)Taha,H.A.(1992) Operations Research- An Introduction, New Operations Research- An Introduction, New York : Macmillan,York : Macmillan,

Hadley, G.(1962)Hadley, G.(1962) Linear Programming, Massachusetts : Linear Programming, Massachusetts : Addison-Wesley,1962. Addison-Wesley,1962.

Hiller, F.S. and G.J.Lieberman(1995) Introduction to Operations Hiller, F.S. and G.J.Lieberman(1995) Introduction to Operations Research, San Francisco : Holden-Day.Research, San Francisco : Holden-Day.

Harvey M. Wagner (1975)Harvey M. Wagner (1975) Principles of Operations Research with Principles of Operations Research with Applications to Managerial Decisions, Prentice Hall of India Pvt. Ltd.Applications to Managerial Decisions, Prentice Hall of India Pvt. Ltd.

Quantitative Techniques in Management by ND Vohra(Second Edition). Schaum Series of Operation Research.

Applications

Personnel Administrator• Forecasting the manpower requirement, recruitment policies,

job assignments. Marketing

Product selection,timing,competitive actions. Financial Controller

in finding out a profit plan for the company. Scheduling : of aircrews and the fleet of airlines,of the operating

theatres in a hospital. Indian Railways, Media Planning.

Objective : OR attempts to locate the best or optimal solution to the problem under consideration.

Methodology• Formulate the problem.

» decision variables that define the problem» constraints that limit the decision choices» objectives of the decision maker

• Mathematical representation of the problem.» Optimize z = f(x1,x2,….,xn)

subject to gi(x1,x2,….,xn) bi , i = 1,2,…..,m

• Solution of the model.• Validation of the model.• Implementation of the solution.

A solution of the model is feasible if it satisfies all the constraints. Along with it if it yields the best(max or min) value of the objective

function then it is a optimal solution. In OR, we do not have a single general technique that solves all

mathematical models that arise in practice. Few of them are linear programming, integer programming, nonlinear programming.

Mathematical Formulation

A firm is engaged in producing two products, A and B. Each unit of product A requires 2 kg of raw material and 4 labour hours for processing, whereas each unit of product B requires 3 kg of raw material and 3 hours of labour, of the same type. Every week, the firm has an availability of 60 kg of raw material and 96 labour hours.One unit of product A sold yields Rs 40 and one unit of product B sold gives Rs 35 as profit. Formulate this problem as a linear programming problem to determine as to how many units of each of the products should be produced per week so that the firm can earn the maximum profit. Assume that there is no marketing constraint so that all that is produced can be sold.

Aim: To determine the no. of units of each of the products to be produced per week so that the firm can earn the maximum profit.

Decision variables : Let x1 and x2 represent the number of units produced per week of the products A and B respectively.

Return Function : Profit function Z = c1x1 + c2x2

Constraints : Raw material constraint, Labour hours constraint.

Trivial Constraints: xj 0.

A B Availability

Raw Material 2 kg 3 kg 60

Labour hours 4 3 96

Profit Rs 40 Rs 35

Maximise Z = 40x1 + 35x2 ProfitSubject to

2x1 + 3x2 60 Raw material constraint 4x1 + 3x2 96 Labour hours constraint

x1, x2 0 Non-negativity restriction

Example

Evening shift resident doctors in a government hospital work five consecutive days and have two consecutive days off. Their five days of work can start on any day of the week and the schedule rotates indefinitely. The hospital requires the following minimum number of doctors working : Sun Mon Tues Wed Thurs Fri Sat 35 55 60 50 60 50 45No more than 40 doctors can start their five working days on the same day. Formulate this problem as an LP model to minimize the number of doctors employed by the hospital.

Let xj be the number of doctors who start their duty on day j( j = 1,2,….,7 ) of the week.

Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7

Subject to x1 + x4 + x5 + x6 + x7 35

x2 + x5 + x6 + x7 + x1 55

x3 + x6 + x7 + x1 + x2 60

x4 + x7 + x1 + x2 + x3 50

x5 + x1 + x2 + x3 + x4 60

x6 + x2 + x3 + x4 + x5 50

x7 + x3 + x4 + x5 + x6 45

xj 40

xj 0 for all j

A multinational company has two factories that ship to three regional warehouses.The costs of transportation per unit are :

Transportation Costs(Rs)

Factory F2 is old and has a variable manufacturing cost of Rs 20 per unit. Factory F1 is modern and produces for Rs 10 per unit. Factory F2 has a monthly capacity of 250 units, Factory F1 has a monthly capacity of 400 units. The requirements at the warehouses are :

W1 W2 W3

F1 2 2 5

F2 4 2 3

How should each factory ship to each warehouse in order to minimize the total cost? Formulate this problem as a linear programming model.

Warehouse RequirementW1 200

W2 100

W3 250

Let xij be the quantity shipped from ith factory to jth warehouse.

FactoryWarehouse

AvailabilityW1 W2 W3

F1 12 12 15 400

F2 24 22 23 250

Requirement 200 100 250 550/650

The total cost(manufacturing plus transportation ) matrix is given below:

x11 + x21 = 200

x12 + x22 = 100

x13 + x23 = 250

xij 0 for i = 1,2 and j = 1,2,3

Minimise Z = 12x11 + 12x12 + 15x13 + 24x21 + 22x22 + 23x23

Subject to x11 + x12 + x13 400

x21 + x22 + x23 250

A 400-meter medley relay involves four different swimmers, who successively swim 100 meters of the backstroke,breaststroke,butterfly, and freestyle. A coach has six very fast swimmers whose expected times(in seconds) in the individual events are given below:

Event 1(backstroke)

Event 2(breaststroke)

Event 3(butterfly)

Event 4(freestyle)

Swimmer 1 65 73 63 57

Swimmer 2 67 70 65 58

Swimmer 3 68 72 69 55

Swimmer 4 67 75 70 59

Swimmer 5 71 69 75 57

Swimmer 6 69 71 66 59

How should the coach assign swimmers to the relay so as to minimize the sum of their times?

Let xij designate the number of times swimmer i will be assigned to event j(where i = 1,2,…….,6 ; j = 1,2,3,4)

Minimise Z = 65x11 + 73x12 + 63x13 + 57x14 +………….. + 66x63 + 59x64

Since no swimmer can be assigned to more than one event.

x11 + x12 + x13 + x14 1

x21 + x22 + x23 + x24 1

………………………

x61 + x62 + x63 + x64 1

Since each event must have one swimmer assigned to it.x11 + x21 + x31 + x41 + x51 + x61 = 1

…………………………………… x14 + x24 + x34 + x44 + x54 + x64 = 1.

xij 0 for i = 1,2,…..,6 and j = 1,2,3,4.

Graphical Solutions of LPP

LPP involving two decision variables can be easily solved by graphical methods.

To solve using graphical methods following steps are involved(a) Identify the problem-the decision variable,the objective

function,and the constraints.(b) Plot a graph representing all the constraints of the problem and

identify the feasible region.(c) Obtain the point on the feasible region that optimises the objective

function-the optimal solution.(d) Interpret the results.

Maximization Case

Maximise Z = 40x1 + 35x2

Subject to 2x1 + 3x2 60

4x1 + 3x2 96

x1, x2 0

(18,8)

(0,20)

(24,0)

40(18) + 35(8)=1000 Q

P

O R

J

T

Minimisation Case

Use graphical method to solve the following LPP:

Minimise Z = 3x1 + 2x2

Subject to 5x1 + x2 10

x1 + x2 6

x1 + 4x2 12

x1, x2 0

2 128

2

12

8

(1,5)

(0,10)

(4,2)3(1) + 2(5) = 13

Multiple Optimal Solutions

Maximise Z = 8x1 + 16x2

Subject to x1 + x2 200

x2 125

3x1 + 6x2 900

x1, x2 0

50 200 300

50

200

(50,125)

(100,100)

Z = 8(50) + 16(125) = 2400

Z = 8(100) + 16(100) = 2400

Inferences:• The objective function is parallel to a constraint that forms the

boundary of the feasible solutions region.

• The constraint should form a boundary on the feasible region in the direction of optimal movement of the objective function.

Unbounded Solution

Maximise Z = 6x1 + x2

Subject to 2x1 + x2 3

x2 - x1 0

x1, x2 0

Inference:

When the values of the decision variables can be increased indefinitely w/o violating any of the constraints, then the solution is stb unbounded.

There is a difference b/w feasible region being unbounded and an LP problem being unbounded.

Infeasible Solution

Maximise Z = x1 + x2/2

Subject to 3x1 + 2x2 12

5x1 = 10

x1 + x2 8

-x1 + x2 4

x1, x2 0 1

2

3

4

Inference:

Infeasibility is a condition that arises when no value of the variables satisfy all of the constraints simultaneously.

Redundancy: A redundant constraint does not affect the feasible solution region.

Exercise

Solve graphically the LPP:

Minimise Z = 6x1 + 14x2

Subject to 5x1 + 4x2 60

3x1 + 7x2 84

x1 + 2x2 18

x1, x2 0

Non-Linear Programming Problem

Solve graphically the following NLP problem:

Maximize Z = 2x1 + 3x2

Subject to

0x,x

8.xx

20xx

21

21

22

21

(2,4)

(4,2)

Z = 2(2) + 3(4) = 16