MEE 437 OPERATIONS RESEARCH

Prof K KARUNAMURTHY

SMBS/VIT Chennai

Lecture #2 Formulation of LPP

Prof K Karunamurthy / VIT

CONTENTS

Recap

Formulation of LPP

Prof K Karunamurthy / VIT

NON-NEGATIVITY CONSTRAINTS

STRUCTURED CONSTRAINTS

General form of a LPP

Optimize (Maximize or Minimize) Z=c1x1+c2x2cnxn Subject to a11x1+a12x2+.a1nxn()b1

a21x1+a22x2+.a2nxn()b2

. . . . . .

am1x1+am2x2+.amnxn()bm

and

x1,x2,x3.xn >0

Objective function

RECAP

Prof K Karunamurthy / VIT

Tons of raw material per ton of

Exterior paint Interior paint Max

availability (ton)

Raw Material C1

6 4 24

Raw material C2

1 2 6

Profit per ton (Rs)

5000 4000

Problem 1. Berger Paint manufactures both exterior and interior paints from two raw materials C1 and C2. The following data provides the details:

A survey indicates that the daily demand for interior paint cannot exceed that for exterior paint by more than 1 ton. Also, the maximum demand for interior paint is 2 tons. Berger paint wants to determine the optimum product mix of interior and exterior paints that maximizes the total daily profit.

Formulation of LPP

NOTE: Product Mix Problem Prof K Karunamurthy / VIT

Formulation of LPP

Max.Z = 5000x1 + 4000 x2 Subject to

Where

1 26 4 24x x

1 22 6x x

2 1 1x x

2 2x

1 2, 0x x

Prof K Karunamurthy / VIT

Problem 2. Three products are made from Lakshmi industries. These products are processed on three different machines. The time required to manufacture one unit of each of the three products and the daily capacity of the three machines are

It is required to determine the daily number of units to be manufactured for each product. The profit per unit of product 1, 2, and 3 is Rs.4, Rs.3 and Rs 6 respectively. It is assumed that all the units produced are consumed in the market. Formulate the LPP model to maximize the profit.

Formulation of LPP

Machine

Time per unit (min) Machine capacity (minutes/day)

Product 1 Product 2 Product 3

M1 2 3 2 440

M2 4 - 3 470

M3 2 5 - 430

Note: Production Scheduling Problem Prof K Karunamurthy / VIT

Formulation of LPP

1 2 3

1 2 3

1 3

1 2

1 2 3

. 4 3 6

to,

2 3 2 440

4 +3 470

2 5 430

, , 0.

Max Z x x x

subject

x x x

x x

x x

where x x x

Prof K Karunamurthy / VIT

Problem 3. A company desires to utilize the excess capacity of lathe, shaper and milling machine to make the products A, B, and C. The time availability/month of these machines are

The profit per product is Rs 20, Rs 16, and Rs 12 respectively on product A, B and C. Formulate a LPP to maximize the profit.

Formulation of LPP

Product

Time required (hrs)

Lathe Shaper Milling

Machine

A 6 2 4

B 2 2 -

C 3 - 3

Available hours/month

200 100 180

Note: Production Scheduling Problem Prof K Karunamurthy / VIT

Formulation of LPP

1 2 3

1 2 3

1 2

1 3

1 2 3

. 20 16 12

to,

6 2 3 200

2 +2 100

4 3 180

, , 0.

Max Z x x x

subject

x x x

x x

x x

where x x x

Prof K Karunamurthy / VIT

Formulation of LPP Problem 4. The manager of an oil refinery must decide on the optimal mix

of two possible blending processes of which the inputs and outputs per production run are as follows.

The availability of the two varieties of crude is limited to the extent of 400 units and 450 units per day. The market demand indicates that at least 200 units and 240 units of superior and ordinary quality of petrol is required per day. The profitability analysis indicate that process A contributes Rs 18,000/day while the process B contributes Rs24,000/day. Determine the number of production runs of process A and process B to maximize the companys profit. Formulate the LPP.

Inputs (Units) Outputs (Units)

Process Arabian Crude

Iraqi Crude

Petrol (Superior)

Petrol ( Ordinary)

A 10 6 10 16

B 12 15 12 12

Note: Blending Problem Prof K Karunamurthy / VIT

Max Z = 18000 X1 + 24000 X4 Subject to

10X1 + 12X2 400

6X1 + 15X2 450

10X1 + 12X2 200

16X1 + 12X2 240

Where X1, X2 0

Prof K Karunamurthy / VIT

Formulation of LPP

Formulation of LPP

Problem 5. A person wants to fulfill his daily requirements of protein, fat and carbohydrates at the minimum cost. The choice is to be made from four different types of foods. The yields per unit of these foods are given below,

Formulate the LPP

Food Type Protein Fat Carbohydrates Cost / Unit (Rs)

1 3 2 6 45

2 4 2 4 40

3 8 7 7 85

4 6 5 4 65

Minimum Requirement

800 200 700

Note: Diet Problem Prof K Karunamurthy / VIT

= 451 + 402 + 853 + 654 Subject to, 31 + 42 + 83 + 64 800 21 + 22 + 73 + 54 200 61 + 42 + 73 + 44 700 Where 1, 2 , 3, 4 0

Prof K Karunamurthy / VIT

Formulation of LPP