Chapter 2. LINEAR PROGRAMMING · 2019. 11. 19. · Darshan Institute of Engineering and Technology, Rajkot 3 7. Trim Loss Problem A paper mill produces rolls of paper used in making

PROBLEM FORMULATION

Operation Research (2171901)

Department of Mechanical Engineering

Darshan Institute of Engineering and Technology, Rajkot 1

Chapter 2. LINEAR PROGRAMMING

1. Production Allocation Problem

A firm produces three products. 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 given in the table below:

Machine Time per unit (minutes) Machine capacity

(minutes/day) Product 1 Product 2 Product 3

M1 2 3 2 440

M2 4 - 3 470

M3 2 5 - 430

It is required to determine the daily number of units to be manufactured for each product.

The profit per unit for product 1, 2 and 3 is Rs. 4, Rs. 3 and Rs. 6 respectively. It is assumed

that all the amounts produced are consumed in the market. Formulate the mathematical

(L.P.) model that will maximize the daily profit.

2. Diet Problem

A person wants to decide the constituents of a diet which will fulfil his daily requirements

of proteins, fats 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 in the table below:

Food Type Yield per unit

Cost per unit (Rs.) Proteins Fats Carbohydrates

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

Formulate linear programming model for the problem.

3. Blending Problem

A firm produces an alloy having the following specifications:

I. Specific gravity ≤ 0.98

II. Chromium ≥ 8 %

III. Melting point ≥ 450°C

Raw materials A, B and C having the properties shown in the table can be used to make

the alloy.

Property Properties of raw material

A B C

Specific gravity 0.92 0.97 1.04

Chromium 7 % 13 % 16 %

Melting point 440°C 490°C 480°C

PROBLEM FORMULATION




Costs of the various raw materials per ton are: Rs. 90 for A, Rs. 280 for B and Rs. 40 for

C. Formulate the L.P. model to find the proportions in which A, B and C be used to obtain

an alloy of desired properties while the cost of raw materials is minimum.

4. Advertising Media Selection Problem

An advertising company wishes to plan its advertising strategy in three different media –

television, radio and magazines. The purpose of advertising is to reach as large a number

of potential customers as possible. Following data have been obtained from market survey:

Television Radio Magazine I Magazine II

Cost of an advertising unit Rs. 30,000 Rs. 20,000 Rs.15,000 Rs.10,000

No. of potential customers

reached per unit 2,00,000 6,00,000 1,50,000 1,00,000

No. of female customers

reached per unit 1,50,000 4,00,000 70,000 50,000

The company wants to spend not more than Rs. 4,50,000 on advertising. Following are the

further requirements that must be met:

a. At least 1million exposures take place among female customers,

b. Advertising on magazines be limited to Rs.1,50,000,

c. At least 3 advertising units be bought on magazine I and 2 units on magazine II,

d. The number of advertising units on television and radio should each be between 5

and 10.

Formulate linear programming model for the problem.

5. Inspection Problem

A company has two grades of inspectors, I and II to undertake quality control inspection.

At least 1500 pieces must be inspected in an 8-hour day. Grade I inspector can check 20

pieces in an hour with an accuracy of 96%. Grade II I inspector checks 14 pieces in an hour

with an accuracy of 92%.

Wages of grade I inspector are Rs.5 per hour while those of grade II inspector are Rs. 4 per

hour. Any error made by an inspector costs Rs. 3 to the company. If there, are in all, 10

grade I inspectors and 15 grade II inspectors in the company, find the optimal assignment

of inspectors that minimizes the daily inspection cost.

6. Product Mix Problem

A chemical company produces two products, X and Y. Each unit of product X requires 3

hours on operation I and 4 hours on operation II, while each unit of product Y requires 4

hours on operation I and 5 hours on operation II. Total available time for operations I and

II is 20 hours and 26 hours respectively. The production of each unit of product Y also

results in two units of a by-product Z at no extra cost.

Product X sells at profit of Rs. 10/unit, while Y sells at profit of Rs. 20/unit. By-product Z

brings a unit profit of Rs. 6 if sold; in case it cannot be sold, the destruction cost is Rs.

4/unit. Forecasts indicate that not more than 5 units of Z can be sold. Formulate the L.P.

model to determine the quantities of X and Y to be produced, keeping Z in mind, so that

the profit earned is maximum.

PROBLEM FORMULATION




7. Trim Loss Problem

A paper mill produces rolls of paper used in making cash resisters. Each roll of paper is

100m in length and can be use in widths of 3, 4, 6 and 10cm. The company’s production

process results in rolls that are 24cm in width. Thus the company must cut its 24cm roll to

the desired widths. It has six basic cutting alternatives as follows:

Cutting alternatives Width of rolls (cm)

Waste (cm) 3 4 6 10

1 4 3 - - -

2 - 3 2 - -

3 1 1 1 1 1

4 - - 2 1 2

5 - 4 1 - 2

6 3 2 1 - 1

The minimum demand for the four rolls is as follows:

Roll width (cm) Demand

2 2000 4 3600

6 1600 10 500

The paper mill wishes to minimize the waste resulting from trimming to size. Formulate

the L.P. model.

CLASS TUTORIAL – GRAPHICAL METHOD

Operations Research (2171901)



1. A firm manufactures two products A & B on which the profits earned per unit are Rs. 3 and

Rs. 4 respectively. Each product is processed on two machines M1 and M2. Product A

requires one minute of processing time on M1 and two minute on M2, while B requires one

minute on M1 and one minute on M2. Machine M1 is available for not more than 7 hrs. 30

minutes, while machine M2 is available for 10 hrs. during any working day. Find the number

of units of products A and B to be manufactured to get maximum profit.

2. Mohan-Meakins Breveries Ltd. has two bottling plants, one located at Solan and the other at

Mohan Nagar. Each plant produces three drinks, whiskey, beer and fruit juices named A, B

and C respectively. The number of bottles produced per day are as follows:

Plant at

Solan (S) Mohan Nagar (M)

Whiskey, A 1500 1500

Beer, B 3000 1000

Fruit juices, C 2000 5000

A market survey indicates that during the month of April, there will be a demand of 20,000

bottles of whiskey, 40,000 bottles of beer and 44,000 bottles of fruit juices. The operating

costs per day for plants at Solan and Mohan Nagar are 600 and 400 monetary units. For how

many days each plant be run in April so as to minimize the production cost, while still

meeting the market demand?

3. Find the minimum value of

𝑍 = −𝑥1 + 2𝑥2,

Subject to,

−𝑥1 + 3𝑥2 ≤ 10,

𝑥1 + 𝑥2 ≤ 6,

𝑥1 − 𝑥2 ≤ 2,

𝑥1, 𝑥2 ≥ 0.

4. The standard weight of a special purpose brick is 5 kg and it contains two basic ingredients

B1 and B2. B1 costs Rs. 5/kg and B2 costs Rs. 8/kg. Strength considerations dictate that the

brick contains not more than 4 kg of B1 and a minimum of 2 kg of B2. Since the demand for

the product is likely to be related to the price of the brick, find graphically the minimum cost

of the brick satisfying the above conditions.

5. A firm uses lathes, milling machines and grinding machines to produce two machine parts.

Following table represents the machining times required for each part, the machining times

available on different machines and the profit on each machine part.

Type of machine

Machining time required for the

machine part (minutes)

Maximum time

available per week

(minutes) I II

Lathes 12 6 3000

Milling machines 4 10 2000

CLASS TUTORIAL – GRAPHICAL METHOD




Grinding machines 2 3 900

Profit per unit Rs. 40 Rs. 100

Find the no. of parts I and II to be manufactured per week to maximize the profit.

6. 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍 = 5𝑥1 + 4𝑥2,

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜, 𝑥1 − 2𝑥2 ≤ 1,

𝑥1 + 2𝑥2 ≥ 3,

𝑥1, 𝑥2 ≥ 0.

7. 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍 = 3𝑥 + 2𝑦,

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜, −2𝑥 + 3𝑦 ≤ 9,

3𝑥 − 2𝑦 ≤ −20,

𝑥, 𝑦 ≥ 0.


𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜, 𝑥1 − 𝑥2 ≥ 0,

2.5𝑥1 − 𝑥2 ≤ −3,

𝑥1, 𝑥2 ≥ 0.


𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜, 3𝑥1 + 5𝑥2 = 18,

5𝑥1 + 3𝑥2 = 14,

𝑥1, 𝑥2 ≥ 0.

CLASS TUTORIAL – LINEAR PROGRAMMING




1. Solve the first example of graphical method by the Simplex method.

2. Use Simplex method to solve the following linear programming problem:

Maximize 𝑍 = 2𝑥1 + 5𝑥2, Subject to,

𝑥1 + 4𝑥2 ≤ 24, 3𝑥1 + 𝑥2 ≤ 21,

𝑥1 + 𝑥2 ≤ 9, 𝑥1, 𝑥2 ≥ 0.

3. A firm produces three products. 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 given in the table below:

Machine Time per unit (minutes) Machine capacity

(minutes/day) Product 1 Product 2 Product 3

M1 2 3 2 440

M2 4 - 3 470

M3 2 5 - 430

It is required to determine the daily number of units to be manufactured for each product.

The profit per unit for product 1, 2 and 3 is Rs. 4, Rs. 3 and Rs. 6 respectively. It is assumed

that all the amounts produced are consumed in the market. Formulate the mathematical (L.P.)

model that will maximize the daily profit and solve example by Simplex method.

4. Use Simplex method to solve the following linear programming problem:

Minimize 𝑍 = 𝑥1 − 3𝑥2 + 3𝑥3, Subject to,

3𝑥1 − 𝑥2 + 2𝑥3 ≤ 7, 2𝑥1 + 4𝑥2 ≥ −12,

−4𝑥1 + 3𝑥2 + 8𝑥3 ≤ 10, 𝑥1, 𝑥2, 𝑥3 ≥ 0.

5. A food processing company produces three canned fruit products mixed fruit fruit

cocktail and fruit delight The main ingredients in each product are pears and peaches Each

product is produced in lots and must go through three processes, mixing, canning

and packaging The resource requirement for each product and each process are shown in the

following L.P. formulation

Maximize Z= 10x1 + 6x2 + 8x3, (profit., Rs)

subject to 20x1+10x2+16x3 ≤ 320 (pears, kg)

10x1+ 20x2+ 16x3 ≤ 400, (peaches, kg)

x1 + 2x2 +2x3 ≤ 43, (mixing, hr)

x1 + x2 + x3 ≤ 60, (canning., hr)





2x1 + x2 + x3 ≤ 40, (packaging, hr)

x1, x2, x3 ≥ 0

Cj 10 6 8 0 0 0 0 0

Basic

variables

Solution

values x1 x2 x3 s1 s2 s3 s4 s5

Cb B b (=Xb)

10 x1 8 1 0 8/15 1/15 1/30 0 0 0

6 x2 16 0 1 8/15 -1/30 1/15 0 0 0

0 s3 3 0 0 2/15 0 -1/10 1 0 0

0 s4 36 0 0 -1/15 1/30 1/30 0 1 0

0 s5 8 0 0 -8/15 1/10 0 0 0 1

Zj 10 6 128/15 7/15 1/15 0 0 0

Cj - Zj 0 0 -8/15 -7/15 -1/15 0 0 0

On the basis of above information answer the following questions

(i) Is the above solution feasible?

(ii) Is the above solution optimal? If yes, what is it?

(iii) Is the above solution unbounded?

(iv) Is the above solution degenerate?

(v) Doesn’t have multiple solutions?

(vi) Determine the amount of used and unused resources.

6. Food X contains 6 units of vitamin A per gram & 7 units vitamin B per gram and costs 12

paise per gram. Food Y contains 8 units of vitamin A per gram & 12 units of vitamin B per

gram and costs 20 paise per gram. The daily minimum requirement of vitamin A and vitamin

B is 100 units and 120 units respectively. Find the minimum cost of product mix by the

simplex method.

7. Maximize Z = 3x1 - x2,

subject to 2x1 + x2 ≤ 2,

x1 + 3x2 ≥ 3,

x2 ≤ 4,

x1, x2 ≥ 0.

8. Use the two-phase simplex method to

Maximize Z = 5x1 + 3x2,

Subject to the constraints 2x1 + x2 ≤ 1,

x1 + 4x2 ≥ 6,

x1, x2, ≥ 0.





9. Use the two-phase simplex method to

Maximize Z = 5x1 - 4x2 + 3x3,

Subject to the constraints 2x1 + x2 – 6x3 = 20,

6x1 + 5x2 + 10x3 ≤ 76,

8x1 – 3x2 + 6x3 ≤ 50,

x1, x2, x3 ≥ 0.

10. Construct the dual of the problem

Minimize Z = 3x1 + 5x2

Subject to the constraints 2x1 + 6x2 ≤ 50,

3x1 + 2x2 ≤ 35,

5x1 - 3x2 ≤ 10

X2 ≤ 20

x1, x2 ≥ 0.


Minimize Z = 3x1 + 17x2 + 9x3

Subject to the constraints x1 - x2 + x3 ≥ 3,

-3x1 + 2x3 ≤ 1,

x1, x2, x3 ≥ 0.


Maximize Z = 3x1 – 2x2 + 4x3,

Subject to the constraints 3x1 + 5x2 + 4x3 ≥ 7,

6x1 + x2 + 3x3 ≥ 4,

7x1 – 2x2 – x3 ≤ 10,

x1 – 2x2 + 5x3 ≥ 3,

4x1 + 7x2 – 2x3 ≥ 2,

x1, x2, x3 ≥ 0.

CLASS TUTORIAL – TRANSPORTATION PROBLEM




Unit 3(a). TRANSPORTATION MODEL

1. A firm owns facilities at seven places. It has manufacturing plants at places A, B and C with

daily output of 500, 300 and 200 units of an item respectively. It has warehouses at places P,

Q, R and S with daily requirements of 180, 150, 350 and 320 units respectively. Per unit

shipping charges on different routes are given below:

To: P Q R S

From A: 12 10 12 13

From B: 7 11 8 14

From C: 6 16 11 7

The firm wants to send the output from various plants to warehouses involving minimum

transportation cost. Obtain Initial Feasible Solution by:

a. North-West Corner Method (NWC Method)

b. Least Cost Method (LCM)

c. Vogel’s Approximation Method (VAM)

Also obtain Optimal Solution by:

a. Stepping-stone Method

b. Modified Distribution Method (MODI)

2. Unbalanced Transportation Problem

A company has received a contract to supply gravel for three new construction projects

located in towns A, B and C. Construction engineers have estimated the required amounts of

gravel which will be needed at these construction projects.

The company has three gravel pits located in towns

W, X and Y respectively. The gravel required by

the construction projects can be supplied by these

three plants. The amount of gravel which can be

supplied by each pit is as follows:

Pit W X Y

Amount available (Truck loads) 152 164 154

Project

Location

Weekly requirement

(Truck loads)

A 144

B 204

C 82





The company has computed the

deliver cost from each pit to each

project site. These costs (in rupees) are

shown in the adjoining table:

Schedule the shipment from each plant to each project in such a manner so as to minimize

the total transportation cost within the constraints imposed by plant capacities and project

requirements. Find the minimum cost.

3. Degeneracy occurs at the initial solution

A manufacturing company has three plants X, Y and Z, which supply to the distributors

located at A, B, C, D and E. Monthly plant capacities are 80, 50 and 90 units respectively.

Monthly requirements of distributors are 40, 40, 50, 40 and 80 units respectively. Unit

transportation costs are given below in rupees.

From To

A B C D E

X 5 8 6 6 3

Y 4 7 7 6 6

Z 8 4 6 6 3

Determine an optimal distribution for the company in order to minimize the total

transportation cost.

4. Degeneracy occurs during the testing of the optimal solution

Ashoka transport company has trucks available at 4 different sites in the following number:

Site A – 5 trucks, Site B – 10 trucks, Site C – 7 trucks, Site D – 3 trucks

Customers W, X and Y require trucks as shown:

Customer W – 5 trucks, Customer X – 8 trucks, Customer Y – 10 trucks

Variable costs getting trucks to the customers are:

From A to W – Rs. 7, to X – Rs. 3, to Y – Rs. 6

From B to W – Rs. 4, to X – Rs. 6, to Y – Rs. 8

From C to W – Rs. 5, to X – Rs. 8, to Y – Rs. 4

From D to W – Rs. 8, to X – Rs. 4, to Y – Rs. 3

Solve the above transportation problem.

Pit A B C

W 8 16 16

X 32 48 32

Y 16 32 48





5. Prohibited Route Problem

Find out optimal solution of given problem having restricted route.

From To

Supply 1 2 3 4 5 6

1 70 50 70 50 50 30 600

2 90 110 60 110 - 50 200

3 110 100 60 20 20 80 900

4 90 100 90 60 90 120 500

Demand 600 200 400 200 400 400

6. Maximization Transportation Problem

A company has four factories F1, F2, F3 and F4 manufacturing the same product. Production

and raw material costs differ from factory to factory and are given in the followning table in

the first two rows. The transportation costs from the factories to the sales depots, S1, S2, S3

are also given. The last two columns in the table give the sales price and the total requirement

at each depot. The production capacity of each factory is given in the last row.

F1 F2 F3 F4

Sales

price/unit Requirement

Production

cost/unit 15 18 14 13

Raw material

cost/unit 10 9 12 9

Transportation

cost/unit

S1 3 9 5 4 34 80

S2 1 7 4 5 32 120

S3 5 8 3 6 31 150

Supply 10 150 50 100

Determine the most profitable production and distribution schedule and the corresponding

profit. The deficit production should be taken to yield zero profit.

LAB TUTORIAL – TRANSPORTATION PROBLEM




1. Agro-product collection centers A, B, C and D stock wheat. The stock levels at A, B, C and

D are 100, 150, 200, 150 tonnes respectively. The requirement of wheat at three distribution

center X, Y and Z is 150, 250 and 200 tonnes respectively. The costs of transportation from

collection center to distribution center in Rs. per tonne of wheat are:

CAX = 10, CAY = 15, CAZ = 15, CBX = 15, CBY = 15, CBZ = 10

CCX = 12, CCY = 10, CCZ = 13, CDX = 10, CDY = 15, CDZ = 12

Obtain Initial Feasible Solution by:

a. North-West Corner Method (NWC Method)

b. Least Cost Method (LCM)

c. Vogel’s Approximation Method (VAM)

Also obtain Optimal Solution by:

a. Stepping-stone Method

b. Modified Distribution Method (MODI)

2. Find the optimum solution to the following transportation problem in which the cells contain

the transportation cost in rupees.

W1 W2 W3 W4 W5 Available

F1 7 6 4 5 9 40

F2 8 5 6 7 8 30

F3 6 8 9 6 5 20

F4 5 7 7 8 6 10

Required 30 30 15 20 5

3. A product is produced by four factories A, B, C and D. The unit production costs in them are

Rs. 2, Rs. 3, Rs. 1 and Rs. 5 respectively. Their production capacities are: factory A – 50

units, B – 70 units, C – 30 units and D – 50 units. These factories supply the product to four

stores, demands of which are 25, 35, 105 and 20 units respectively. Unit transport cost in

rupees from each factory to each store is given in the table below:

Stores

Factories

1 2 3 4

A 2 4 6 11

B 10 8 7 5

C 13 3 9 12

D 4 6 8 3

Determine the extent of deliveries from each of the factories to each of the stores so that the

total production and transportation cost is minimum.

LAB TUTORIAL – TRANSPORTATION PROBLEM




4. A company has four plants producing same product and five distribution centers. Production

costs as well as costs of raw material differ from plant to plant. Given the data in the table

below, determine the best production and distribution schedule.

Plant

1 2 3 4

Raw material

cost Rs./unit 17 20 16 15

Production cost

Rs./unit 11 10 13 9

Plant Required

Units 1 2 3 4

Distribution

Center

1 2 8 4 3 115

2 0 6 3 4 110

3 4 7 2 9 150

4 6 2 7 11 100

5 3 4 5 6 150

Plant Capacity unit 150 200 175 100

5. A company has 3 factories manufacturing the same product and 5 sale agencies in different

parts of the country. Production costs differ from factory to factory and the sales prices from

agency to agency. The shipping cost per unit product from each factory to each agency is

known. Given the following data, find the production and distribution schedules most

profitable to the company.

Production cost/unit (Rs.) Maximum Capacity (No. of units)

Factory i

1 18 140

2 20 190

3 16 115

Factory i

1 2 2 6 10 5 Shipping

cost (Rs.) 2 10 8 9 4 7

3 5 6 4 3 8

Ajency j 1 2 3 4 5

Demand 74 94 69 39 119

Sales price (Rs.) 35 37 36 39 34

CLASS TUTORIAL – ASSIGNMENT PROBLEM




Unit 3(b). ASSIGNMENT MODEL

1. Five machines are available to do five different jobs. From past records, the time (in hrs.) that

each machine takes to do each job is known and given in the Table 1:

Find the assignment of machines to jobs that will minimize the total time taken.

2. Solve the assignment problem given in Table 2.

3. Optimal Alternate Solutions

Solve the assignment problem given in Table 3. Also suggest all the possible alternate

optimal solutions.

Table 3

A B C D

1 22 23 24 25

2 24 25 26 27

3 27 28 29 28

4 23 25 28 24

4. Unbalanced Assignment Problem

On a given day district head quarter has the information that one ambulance van is stationed

at each of the five locations A, B, C, D and E. A dispatch order is to be issued for the

ambulance van to reach 6 locations, viz I, II, III, IV, V and VI one each. The distance (km)

between present locations of ambulance vans and destination are given in the Table 4.

Table 2

I II III IV V

1 11 17 8 16 20

2 9 7 12 6 15

3 13 16 15 12 16

4 21 24 17 28 26

5 14 10 12 11 13

Table 1

Machine Job

I II III IV V

A 2 9 2 7 1

B 6 8 7 6 1

C 4 6 5 3 1

D 4 2 7 3 1

E 5 3 9 5 1

Table 4

I II III IV V VI

A 18 21 31 17 26 29

B 16 20 18 16 21 31

C 30 25 27 26 18 19

D 25 33 45 16 32 20

E 36 30 18 15 31 30





5. Maximization Problem

A firm produces four products. There are four operators who are capable of producing any of

these four products. The processing time varies from operator to operator. The firm records 8

hours a day and allows 30 minutes for lunch. The processing time in minutes and the profit

for each of the product are given in the Table 5. Find the optimal assignment of product to

operators.

Table 5

Operators Products

A B C D

1 15 9 10 6

2 10 6 9 6

3 25 15 15 9

4 15 9 10 10

Profit (Rs. per unit) 8 6 5 4

6. Restrictions on Assignments

Four new machines M1, M2, M3 and M4 are to be installed in a forging shop. There are five

empty spaces A, B, C, D and E. Because of limited space problem M2 cannot be placed at C

and M3 cannot be placed at A. Installation costs of machines to various places in rupees are

given in the Table 6.

7. Travelling Salesman Problem

A travelling salesman has to visit five cities. He wishes to start

from a particular city, visits each city once and then returns to

his starting point. The travelling time (in hrs) for each city from

a particular city is given in Table 7. What is the sequence of

visits of the salesman, so that total travel time is minimum?

Table 6

A B C D E

M1 4 6 10 5 6

M2 7 4 - 5 4

M3 - 6 9 6 2

M4 9 3 7 2 3

Table 7

From To

A B C D E

A ∞ 4 7 3 4

B 4 ∞ 6 3 4

C 7 6 ∞ 7 5

D 3 3 7 ∞ 7

E 4 4 5 7 ∞






For the matrix shown in Table 8 find the least cost route for the travelling salesman problem.

9. Airline Crew Assignment

An airline that operates 7 days a week has the time table shown below. Crew must have a

minimum layover of 6 hours between flights. Obtain the pairing of flights that minimizes

layover time away from home. For any given pairing the crew will be based on the city that

results in the smaller layover.

Delhi - Kolkata Kolkata - Delhi

Flight No. Depart Arrive Flight No. Depart Arrive

1 7:00 AM 9:00 AM 101 9:00 AM 11:00 AM

2 9:00 AM 11:00 AM 102 10:00 AM 12 Noon

3 1:30 PM 3:30 PM 103 3:30 PM 5:30 PM

4 7:30 PM 9:30 PM 104 8:00 PM 10:00 PM

For each pair also mention the town where the crew should be based.

Table 8

To City

1 2 3 4 5 6

From City

1 ∞ 12 7 6 5 5

2 9 ∞ 13 5 13 10

3 6 13 ∞ 7 10 8

4 4 9 10 ∞ 6 9

5 5 13 7 6 ∞ 4

6 5 11 9 6 5 ∞

LAB TUTORIAL – ASSIGNMENT PROBLEM




1. A company is producing a single product and is selling it through five agencies situated in

different cities. All of a sudden, there is a demand for the product in another five cities not

having any agency of the company. The company is faced with the problem of deciding on

how to assign the existing agencies to dispatch the product to needy cities in such a way that

the travelling distance is minimized. The distances (in kms.) between the surplus and deficit

cities are given in the following distance matrix:

Table 1

Deficit Cities I II III IV V

Surplus Cities

A 160 130 175 190 200

B 135 120 130 160 175

C 140 110 155 170 185

D 50 50 80 80 110

E 55 35 70 80 105

Determine the optimal assignment schedule.

2. Five wagons are available at stations 1, 2, 3, 4 and 5. These are required at five stations I, II,

III, IV and V. The mileages between various stations are given by the Table 2. How should

the wagons be transported so as to minimize the total mileage covered?

3. Alternate Optimal Solutions

Four different jobs are to be done on four machines, one job on each machine, as set-up costs

and times are too high to permit a job being worked on more than one machine. The matrix

given in Table 3 gives the time of producing jobs on different machines. Assign the jobs on

machines so that total time for completing all the jobs is minimum.

Table 2

I II III IV V

1 10 5 9 18 11

2 13 9 6 12 14

3 3 2 4 4 5

4 18 9 12 17 15

5 11 6 14 19 10

Table 3

Jobs Machines

M1 M2 M3 M4

J1 10 14 22 12

J2 16 10 18 12

J3 8 14 20 14

J4 20 8 16 6






On a given day district head quarter has the information that one ambulance van is stationed

at each of the five location A, B, C, D and E. A dispatch order is to be issued for the

ambulance van to reach 6 location, viz I, II, III, IV, V and VI one each. The distance (km)

between present locations of ambulance vans and destination are given in the Table 4.


A fast-food chain wants to build four stores. In the past the chain has used six different

construction companies, and having been satisfied with each, has invited them to bid for each

job. The final bids (in thousands of rupees) are shown in the Table 5. Since the fast-food

chain wants to have each of the new stores ready as quickly as possible, it will allot at most

one job to a construction company. What assignment will result in the minimum total cost?

6. Maximization Problem

The captain of cricket team has to allot five middle batting positions to five batsmen. The

average runs scored by each batsman at these positions are as follows:

Batsmen Batting positions

I II III IV V

P 40 40 35 25 50

Q 42 30 16 25 27

R 50 48 40 60 50

S 20 19 20 18 25

T 58 60 59 55 53

Table 4

I II III IV V VI

A 54 63 93 51 78 87

B 48 60 54 48 63 93

C 90 75 81 78 54 57

D 75 99 135 48 96 60

E 108 90 54 45 93 90

Table 5

Stores Construction Companies

1 2 3 4 5 6

1 85.3 90.0 87.5 82.4 89.1 91.3

2 78.9 84.5 99.4 80.4 89.3 88.4

3 82.0 31.3 28.5 66.5 80.4 109.7

4 84.3 34.6 86.2 83.3 85.0 85.5





Find the assignments of batsmen to positions which would give the maximum number of

runs.

7. Restrictions on Assignments

An airline has drawn up a new flight schedule involving five flights. To assist in allocating

five pilots to the flights, it has asked them to state their preference scores by giving each

flight a number out of 10. The higher the number, the greater the preference is. Certain of

these flights are unsuitable to some pilots owing to domestic reasons. These have been

marked with X.

Pilot Flight number

1 2 3 4 5

A 8 2 X 5 4

B 10 9 2 8 4

C 5 4 9 6 X

D 3 6 2 8 7

E 5 6 10 4 3

What should be the allocation of the pilots to flights in order to meet as many preferences as

possible?


Given the matrix of set-up costs in Table 6, show how to sequence production so as to

minimize set-up per cycle.

Table 6

A B C D E

A - 2 5 7 1

B 6 - 3 8 2

C 8 7 - 4 7

D 12 4 6 - 5

E 1 3 2 8 -


Products 1, 2, 3, 4 and 5 are to be processed on a machine. The set-up costs in rupees per

change depend upon the product presently on the machine and the set-up to be made and are

given by the Table 7.

Find the optimum sequence of products in order to minimize the total set-up cost.

Table 7

1 2 3 4 5

1 - 16 4 12 -

2 16 - 6 - 8

3 4 6 - 5 6

4 12 - 5 - 20

5 - 8 6 20 -





10. Airline Crew Assignment

An airline that operates 7 days a week has the time table shown below. Crew must have a

minimum layover of 5 hours between flights. Obtain the pairing of flights that minimizes

layover time away from home. For any given pairing the crew will be based on the city that

results in the smaller layover.

Delhi - Kolkata Kolkata - Delhi

Flight No. Depart Arrive Flight No. Depart Arrive

1 6:00 AM 8:00 AM 101 8:00 AM 10:00 AM

2 8:00 AM 10:00 AM 102 9:00 AM 11:00 AM

3 2:00 PM 4:00 PM 103 2:00 PM 4:00 PM

4 8:00 PM 10:00 PM 104 7:00 PM 9:00 PM

For each pair also mention the town where the crew should be based.

7. GAMES THEORY




Chapter 7. GAMES THEORY

Find the saddle point, optimum strategies and value of the game in the following pay-off

matrix:

For the following matrix of pay-offs find saddle point. If there is no saddle point then find

optimal strategies, their frequencies and value of the game.

For the following matrix of pay-offs find saddle point. If there is no saddle point then find

optimal strategies, their frequencies and value of the game by algebraic method.

Use the concept of dominance to solve the following games:

1) Y 2) Y 3) Y

A B A B C A B C D

X

I 4 -3

X

I -2 4 2

X

I 3 4 2 9

II 3 2 II 3 2 4 II 7 8 6 10

III 2 3 4 III 6 2 4 -1

4) B 5) B 6) B

I II I II I II

A

I -4 7 A

I -2 3 A

I 200 80

II 8 -5 II 4 -1 II 110 70

7) Y

1 2

X 1 2 -1

2 -1 1

8) Y

1 2

X I 1 7

II 6 2

10) Y

A B C

X

I 60 50 40

II 70 70 50

III 80 60 75

9) Y

J K L

X

P 9 8 -7

Q 3 -6 4

R 6 7 -7

11) Y

1 2 3

X

I 40 50 -70

II 10 25 -10

III 100 30 60

12) Y

A B C D

X

I 3 2 4 0

II 3 4 2 4

III 4 2 4 0

IV 0 4 0 8

7. GAMES THEORY




Find the optimum strategies and the value of the game for the pay-off matrix given below.

Solve the following games by graphical Method

16) Y

A B C D E

X I -1 3 4 -2 6

II 4 2 6 3 2

18) A company management and the labour union are negotiating a new three year

settlement. Each of these has 4 strategies:

I: Hard and aggressive bargaining

II: Reasoning and logical approach

III: Legalistic strategy

IV: Conciliatory approach

The cost to the company are given in the following table for every pair of strategy

choice. What strategy will the two side adopt? Also determine the value of the game.

Union

Strategies Company Strategies

I II III IV

I 20 15 12 35

II 25 14 8 10

III 40 2 10 5

IV -5 4 11 0

13) Y

A B C

X I -1 7 6

II 5 -3 3

14) B

1 2 3 4 5

A 1 15 6 -6 -9 -13

2 -5 -4 -1 1 7

15) Y

1 2

X

1 4 6

2 3 7

3 5 -4

4 8 -5

17) Y

A B

X

I -6 7

II 4 -5

III -1 -2

IV -2 2

V 7 -6

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Chapter 2. LINEAR PROGRAMMING · 2019. 11. 19. · Darshan Institute of Engineering and Technology, Rajkot 3 7. Trim Loss Problem A paper mill produces rolls of paper used in making