ASSIGNMENT : DECISION SCIENCE -204

Submission: 5th Nov,2016

___________________________________________________________________________

LINEAR PROGRAMMING

1. Production of a certain chemical mixture should contain minimum 80 mg chlorides, 28 mg

nitrates and 36 mg of sulphate per kg. The company can use two substances. Substance X

contains 8 mg chlorides, 4 mg nitrates and 6 mg sulphates per gram. Substance Y contains 10

mg chloride, 2 mg Sulphates per gram. Both substances cost Rs. 20 per gram. It is required to

produce the mixture using substances X and Y so that the total cost is minimum. Formulate the

problem as LPP.

2. A firm uses lathes, milling and grinding machines to produce two machine parts. Table given

below represents times available on different machines and profits on each machine part.

Type of machine

Machining time required

for machine part in

minutes

Maximum time available in

minutes

I II

Lathes 12 6 3000

Milling 4 10 2000

Grinding 2 3 900

Profit per unit Rs. 40 100

Find the number of parts I and II to be manufactured per week to maximize the profit.

3. Consider a small plant which makes two types of automobile parts, say A & B. it buys castings

that are machined, bored & polished. The capacity of machining is 25 per hour for A & 24 per

hour for B, capacity for boring is 28 per hour for A and 35 per hour and the capacity of polishing

is 35 per hour for A and 25 per hour for B. castings for part A cost Rs. 2 and sell for Rs. 5 each

and those for part B cost Rs. 3 and sell for Rs. 6 each. The three have running costs of Rs. 20,

Rs. 14 &Rs. 17.50 per hour. Assuming that any combination of parts A & B can be sold.

Formulate & solve this problem as an LPP to determine the optimum product mix & maximum

profit.

4. A refinery makes 3 grades of petrol A, B and C from crude oils D, E and F. Crude oil F can be

used in any grade but the others must satisfy the following specifications.

Grade

Selling Price

per Litre

(Rs)

Specifications

A Rs. 48 Not less than 50% crude D Not

more than 25% crude E

B Rs. 50 Not less than 25% crude D Not

more than 50% crude E

C Rs. 49 No Specifications

There are capacity limitations on the amounts of 3 crude elements than can be used

Crude Capacity (KL) Price per Litre (Rs.)

D 500 49.5

E 500 47.5

F 360 48.5

Formulate LPP to maximize Profit.

5. A manufacturing company is engaged in producing three types of products: A, B & C. The

production department produces, each day, components sufficient to make 50 units of A, 25

units of B & 30 units of C. The management is confronted with the problem of optimizing the

daily production of the products in the assembly department, where only 100 man-hours are

available daily for assembling the products. The following additional information is available:

Type of Product Profit Contribution per Unit

of Product (Rs.)

Assembly Time per

Produ (Hrs.)

A 12 0 .8

B 20 1.7

C 45 2.5

The company has a daily order commitment for 20 units of products A and a total of 15 units of

products B & C. Formulate this problem as an LP model so as to maximise the total profit.

6. A company has two plants, each of which produces & supplies two products: A & B. The plants

can each work up to 16 hours a day. In plant 1, it takes 3 hrs to prepare & pack 1,000 gallons

of A & 1 hr to prepare & pack one quintal of B. In plant 2, it takes two hours to prepare & pack

1,000 gallons of A & 1.5 hrs to prepare & pack a quintal of B. In plant 1, it costs Rs. 15,000 to

prepare & pack 1,000 gallons of A and Rs. 28,000 to prepare & pack a quintal of B, whereas in

plant 2 these costs are Rs. 18,000 &Rs. 26,000 respectively. The company is obliged to produce

daily at least 10,000 gallons of A & 8 quintals of B. Formulate this problem as an LP model to

find out as to how the company should organize its production so that the required amounts of

the two products be obtained at the minimum cost.

7. A businessman is opening a new restaurant and has budgeted Rs. 8,00,000 for advertisement,

for the coming month. He is considering four types of advertising: i). 30 second television

commercials ii). 30 second radio commercials iii). Half-page advertisement in a newspaper

iv). Full-page advertisement in a weekly magazine which will appear four times during the

coming month.

The owner wishes to reach families (a) with income over Rs. 50,000 and (b) with income under

Rs. 50,000. The amount of exposure of each media to families of type (a) & (b) and the cost of

each media is shown below:

Media Cost of

Advertisement (Rs.)

xposure to families w i income over

Rs. 50,000

xposure to families

wi

income

under

Rs. 50,000

Television 40,000 2,00,000 3,00,000

Radio 20,000 5,00,000 7,00,000

Newspaper 15,000 3,00,000 1,50,000

Magazine 5,000 1,00,000 1,00,000

To have a balanced campaign, the owner has determined the following four restrictions: i). There

should be no more than 4 television advertisements ii). There should be no more than 4

advertisements in the magazine iii). There should not be more than 60 per cent of all

advertisements in newspaper & magazine put together. iv). There must be at least 45,00,000

exposures to families with annual income of over Rs. 50,000.

Formulate this problem as an LP model to determine the number of each type of advertisement to

be given so as to maximise the total number of exposures.

8. A tape recorder company manufactures models A, B and C, which have profit contributions per

unit Rs15, Rs.40 and Rs.60, respectively. The weekly minimum production requirements are

25 units for model A, 130 units for model B and 55 units for model C. Each type of recorder

requires a certain amount of time for the manufacturing of the component parts, for assembling

and for packing. Specifically, a dozen units of a model A require 4 hours for manufacturing, 3

hours for assembling and 1 hour for packaging. The corresponding figures for a dozen unit of

model B are 2.5, 4 and 2 and for a dozen units of model C are 6, 9 and 4. During the forthcoming

week, the company has available 130 hours of manufacturing, 170 hours of assembling and 52

hours of packaging time. Formulate this problem as an LP model so as to maximize the total

profit to the company.

9. A company manufacturing animal feed must produce 500 kgs of a mixture daily. The mixture consists

of two ingredients F1 & F2. Ingredient F1 costs Rs. 5 per kg and ingredient F2 costs Rs. 8 per kg.

Nutrient constitutions dictate that the feed contains not more than 400 kgs of F1 and a minimum of

200kgs of F2. Formulate the LPP and find the quantity of each ingredient used to minimise cost.

10. Solve the following LPP by using Graphical Method

Max. Z = 300X1 +400X2

Subject to the constraints

5X1 + 4X2≤200

3X1 + 5X2≤ 150

5X1 +4X2≤ 100

8X1 + 4X2 ≥ 80 X1≥

0, X2≥ 0

11. Solve by graphical method.

Minimize z= 6x1+ 14x1

Subject to 5x1 + 4x2≥ 60

3x1 + 7x2≥ 84 x1+ 2x2≥ 18

x1, x2≥ 0

12. Find the quantity of each type of chair to be produced to maximize profit X- Quantity of chairs

of type A

Y- Quantity of chairs of type B

Using following constraints

6x +4y ≤ 3600, 2x +4y ≤ 2000, 4.6x + 4y ≤ 500, y ≤ 400, x, y ≥ 0

TRANSPORTATION PROBLEMS

1. Determine the optimum shipping cost for the following transportation problem.

Distribution

centre

Retails outle t Availability

A B C D E

Agra 55 30 40 50 40 8

Allahabad 35 30 100 45 60 4

Calcutta 40 60 95 35 30 8

Requirement 5 2 4 6 3

2. The wholesale company has three warehouses from which supplies are drawn for four

retail customers. The company deals in a single product. The supplies of which at each

warehouse are.

Warehouse no. Supply( units) Customer No. Demand units

1 20 1 15

2 28 2 19

3 17 3 13

4 18

Conveniently total supply at the ware houses is equal to total demand from the customer.

The following table gives the transportation costs per unit shipment from each ware

house to each customer.

Ware house Customer

1 2 3 4

1 3 6 8 5

2 6 1 2 5

3 7 8 3 9

Determine the allocation to minimize overall transportation cost.

3. Find the initial basic feasible solution of the following transportation problem using

i) North west corner method. ii)

Matrix minimum method.

iii) Vogel’s approximation method.

Also find corresponding costs.

Factory Warehouse

A B C

D

Capacity units

X

Y

Z

11 31 51

11

71 31 41 61

41 09 71 21

7

9

18

Requirement 05 08 07

Unit 15

4. Solve the following T. P.

Unit Transportation Cost (Rs.)

Factory Warehouses

Capacity D E F

A 5 1 7 10

B 6 4 6 80

C 3 2 5 15

Requirement 75 20 50

5. Goods have to be transported from sources S1, S2 & S3 to destinations D1, D2 & D3. The

transportation cost per unit, capacities of the sources, and the requirements of the

destinations are given in the following table. Determine a transportation schedule so that

cost is minimised.

Demand

Supply

D1 D2 D3

S1 8 5 6 120

S2 15 10 12 80

S3 3 9 10 80

150 80 50

6. Solve the following T. P.

D1 D2 D3 D4 Supply

S1 19 30 50 10 7

S2 70 30 40 60 9

S3 40 8 70 20 18

Demand 5 8 7 14 34

7. A company has factories at F1, F2 and F3 that supply products to warehouses at W1, W2

and W3. The weekly capacities of the factories are 200, 160 and 90 units respectively. The

weekly warehouse requirements are 180, 120 and 150 units respectively. The unit shipping

costs (in rupees) are as follows:

Factory

Warehouse

W1 W2 W3

F1 3 6 8

F2 6 1 2

F3 7 8 3

8. The following table provides all the necessary information on the availability of supply to

each warehouse, the requirement of each market, and the unit transportation cost (in Rs.)

from each warehouse to each market.

Market Supply

P Q R S

Warehouse

A 6 3 5 4 22

B 5 9 2 7 15

C 5 7 8 6 8

Demand 7 12 17 9 45

The shipping clerk of the shipping agency has worked out the following schedule, based on

his own experience: 12 units from A to Q, 1 unit from A to R, 8 units from A to S, 15 units

from B to R, 7 units from C to P and 1 unit from C to R.

a). Check and see if the clerk has the optimal schedule.

b). Find the optimal schedule and minimum total transport cost.

c). If the clerk is approached by a carrier of route C to Q, who offers to reduce his rate in the

hope of getting some business, by how much should the rate be reduced before the clerk

would offer him the business.

9. A manufacturer wants to ship 22 loads of his product as shown below. The matrix gives

the kilometers from sources of supply to the destinations.

Destination

Source D1 D2 D3 D4 D5 Supply

S1 5 8 6 6 3 8

S2 4 7 7 6 5 5

S3 8 4 6 6 4 9

Demand 4 4 5 4 8

10. Four hospitals have decided to send their patients needing linear accelerator therapy to

three locations in their region. They have contract with an ambulance company which

charges Rs. 2/km. Each location can handle only a certain number of outside patients per

day. Find the optimal allocations of patients to locations of therapy given the following

details.

Hospitals Dis tance from locations (kms) Patients

1 17 31 45 5

2 12 14 23 8

3 46 32 13 7

4 38 16 19 5

Place 5 11 5

ASSIGNMENT PROBLEMS

1. A department of a company has five employees with five jobs to be performed. The time

(in hours) that each man takes to perform each job is given in the effectiveness matrix.

Employees

I II III IV V

Jobs

A 10 5 13 15 16

B 3 9 18 13 6

C 10 7 2 2 2

D 7 11 9 7 12

E 7 9 10 4 12

How should the jobs be allocated, one per employee, so as to minimize the total man-hours.

2. A company operates in four territories, and four salesman available for an assignment.

The territories are not equally rich in their sales potential. It is estimated that a typical

salesman operating in each territory would bring in the following annual sales:

Territory : I II III IV

Annual Sales (Rs): 126000 105000 84000 63000

The four salesmen also differ in their ability. It is estimated that, working under the same

conditions, their yearly sales would be proportionately as follows:

Salesmen : A B C D

Proportion: 7 5 5 4

If the criterion is maximum expected sales, the intuitive answer is to assign the best salesman to

the richest territory, the next best salesman to the second richest, and so on; verify this answer

by the assignment technique.

3. Five men are available to do five different jobs. From past records, the time (in hours)

that each man takes to do each job is known and is given in the following table:

Jobs

I II III IV V

Men 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

Find out how men should be assigned the jobs in way that will minimize the total time taken.

4. A company has 4 machines on which to do 3 jobs. Each job can be assigned to one and

only one machine. The cost of each job assignments for minimum cost?

Job Machine

W X Y Z

A

B

C

7 23 27 31

12 16 18

14 18 21

5. A pharmaceutical company producing a single product sold it through five agencies

situated in different cities. All of a sudden, there rouse a demand for the product in

another five cities that didn’t have any agency of the company. The company is now

facing the problem of deciding on how to assign the existing agencies in order to

despatch the product to needy cities in such a way that the travelling distance is

minimized. The distance between the surplus and deficit cities (in km) is given in the

following table.

Deficit cities

a b c d e

Surplus

Cities

A 160 130 115 190 200

B 135 120 130 160 175

C 140 110 125 170 185

D 50 50 80 80 110

E 55 35 80 80 105

6. A national truck rental service has a surplus of one truck in each of the cities, 1, 2, 3, 4,

5 & 6 and a deficit of one truck in each of the cities 7, 8, 9, 10, 11 & 12. The distances

(in km) between the cities with a surplus and cities with deficit are displayed in the table

below:

To

7 8 9 10 11 12

From

1 31 62 29 42 15 41

2 12 19 39 55 71 40

3 17 29 50 41 22 22

4 35 40 38 42 27 33

5 19 30 29 16 20 23

6 72 30 30 50 41 20

7. In the modification of a plant layout of a factory, four new machines M1, M2, M3 & M4

are to be installed in a machine shop. There are five vacant places A, B, C, D& E

available. Because of limited space, machine M2 cannot be placed at C and M3 cannot

be placed at A.

The cost of locating a machine at a place (in hundred rupees) is as follows.

Location

A B C D E

Machine

M1 9 11 15 10 11

M2 12 9 - 10 9

M3 - 11 14 11 7

M4 14 8 12 7 8

Find the optimal assignment schedule.

8. A city corporation has decided to carry out road repairs on four main arteries of the city.

The government has agreed to make a special grant of Rs. 50 lakh towards the cost with

a condition that the repairs be done at the lowest cost and quickest time. If the conditions

warrant, a supplementary token grant will also be considered favourably. The

corporation has floated tenders and five contractors have sent in their bids. In order to

expedite work, on road will be awarded to only one contractor.

Cost of Repairs (Rs. In lakh)

R1 R2 R3 R4

ontractors / Road

C1 9 14 19 15

C2 7 17 20 19

C3 9 18 21 18

C4 10 12 18 19

C5 10 15 21 16

a). Find the best way of assigning the repair work to the contractors and the costs.

b). If it is necessary to seek supplementary grants, what should be the amount sought?

c). Which of the five contractors will be unsuccessful in his bid?

9. 5 salesmen are to be assigned to 5 territories. Based on the past performance, the

following table shows the annual sales (in Rs. Lakhs) that can be generated by each

salesman in each territory. Find the optimum assignment.

Salesman Te rritory

T1 T2 T3 T4 T5

S1 26 14 10 12 9

S2 31 27 30 14 16

S3 15 18 16 25 30

S4 17 12 21 30 25

S5 20 19 25 16 10

10. Solve the following assignment problem. (Sales in Rs. Thousands)

Salesman Distri cts

D1 D2 D3 D4

S1 20 36 33 18

S2 25 24 19 21

S3 18 20 22 20

S4 25 20 17 22

11. The Head of the department have five jobs A, B, C, D, E and five subordinates V, W, X,

Y, Z. The number of hours each man would take to perform each job is as follows.

V W X Y Z

A 3 5 10 15 8

B 4 7 15 18 8

C 8 12 20 20 12

D 5 5 8 10 6

E 10 10 15 25 10

Find the optimum allocation of jobs to the subordinates.

12. 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 earlier understanding only

one job was given to one typist and the typist was paid for a full hour, even if he worked for a

fraction of an hour. Find the least cost allocation for the following data:

Typist Rate per hour (Rs) No of Pages Typed/Hour

A 5 12

B 6 14

QUEUING THEORY

1. A television repairman finds that the time spent on his jobs has an exponential distribution with a

mean of 30 minutes. If he repairs the sets in the order in which they came in, and if the arrival of

sets follows a Poisson distribution with an approximate average rate of 10 per 8-hour day, what is

the repairman’s expected idle time each day? How many jobs are ahead of the average set just

brought in?

C 3 8

D 4 10

E 4 11

Job No. of Pages

P 199

Q 175

R 145

S 298

T 178

2. In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day. Assuming that the

inter arrival time follows an exponential distribution and the service time (the time taken to hump

a train) distribution is also exponential with an average of 36 minutes. Calculate: a). Expected

queue size (line length).

b). Probability that the queue size exceeds 10.

If the input of trains increases to an average of 33 per day, what will be the change in a & b.

3. Arrivals at telephone booth are considered to be Poisson with an average time of 10 minutes

between one arrival and the next. The length of phone calls is assumed to be distributed

exponentially, with a mean of 3 minutes.

a). What is the probability that a person arriving at the booth will have to wait?

b). The telephone department will install a second booth when convinced that an arrival would expect

waiting for at least 3 minutes for a phone call. By how much should the flow of arrivals increase

in order to justify a second booth?

c). What is the average length of the queue that forms from time to time?

d). What is the probability that it will take a customer more than 10 minutes altogether to wait for the

phone and complete his call?

4. A warehouse has only one loading dock manned by a three person crew. Trucks arrive at the

loading dock at an average rate of 4 trucks per hour and the arrival rate is Poisson distributed. The

loading of a truck takes 10 minutes on an average and can be assumed to be exponentially

distributed. The operating cost of a truck is Rs 20 per hour and the members of the loading crew

are paid Rs 6 each per hour. Would you advise the truck owner to add another crew of three

persons?

5. A road transport company has reservation clerk on duty at a time. He handles information of bus

schedules and makes reservations. Customers arrive at a rate of 8 per hour and the clerk can, on

an average, service 12 customers per hour. After stating your assumptions, answer the following:

a). What is the average number of customers waiting for the service of the clerk?

b). What is the average time a customer has to wait before being served?

c). The management is contemplating to install a computer system for handling information and

reservations. This is expected to reduce the service time from 5 to 3 minutes. The additional cost

of having the new system works out to Rs. 50 per day. If the cost of goodwill of having to wait is

estimated to be 12 paise, per minute spent waiting, before being served, should the company install

the computer system?

6. The tool room company’s quality control department is manned by a single clerk who takes an

average of 5 minutes in checking parts of each of machine coming for inspection. The machine

arrive once in every 8 minutes on the average. One hour of the machine is valued at Rs. 15 and a

clerk’s time is valued at Rs. 4 per hour. What are the average hourly queuing system costs

associated with the quality control department.

7. A bank plans to open a single server drive in banking facility at a particular centre. It is estimated

that 20 customers will arrive each hour on an average. If on an average, it requires 2 minutes to

process a customer’s transaction, determine:

i). The proportion of time that the system will be idle.

ii). On the average, how long the customer will have to wait before reaching the server.

DECISION THEORY

1. Following table gives profits matrix for different events & actions. Calculate EVPI

Events

States of

Nature

Probability

Actions

A1 A2 A3

E1 0.20 40 52 45

E2 0.35 70 28 40

E3 0.35 30 70 -50

E4 0.10 30 -50 -70

2. A certain output is manufactured at Rs. 8 and sold at Rs. 14 per unit. The product is such that if it

is produced but not sold during a day time it becomes worthless. The daily sales records in the

past are as follows.

emand per day: 30 40 50 60 70

o. of days each sales level was recorded: 24 24 36 24 12

i. Calculate the average expected sales of a day.

ii. Find the expected pay-offs and the optimum policy.

iii. Also find the value of perfect information.

3. The Probability distribution of demand for cakes is given below:

o. of Cakes demanded (

arbitrary units) 0 1 2 3 4 5

robability 0.05 0.10 0.25 0.30 0.20 0.10

If the cost per cake is Rs. 3 per unit & selling price is Rs. 4 per unit, how many cakes should the baker

make to maximise his profit. Assume that if cake is not sold at the end of the day its value is zero.

4. A farmer wants to decide which of the 3 crops he should plant. The farmer has categorised the

amount of rainfall is high, medium & low.

Estimated profit is given below.

Estimated Profit (

inRs

.)

Rainfall Crop A Crop B Crop C

High 8,000 3,500 5,000

Medium 4,500 4,500 4,900

Low 2,000 5,000 4,900

Farmer wishes to plant one crop, decide the best crop using: i).

Hurwicz criteria (take degree 0.6)

ii). Laplace criteria iii).

Minimax regret criteria.

5. The past experience shows that the number of copies of a book in demand are between 25 & 30

copies. Some agency purchases such unsold copies for Rs. 35. The vendor purchases the copies

at Rs. 83 each & sales them at Rs. 110 each.

Find the number of copies to be kept in stock using EMV criteria if probability of demand are

known as:

Demand 25 26 27 28 29 30

Probability 0.05 0.10 0.30 0.32 0.16 0.07

6. The manager of a flower shop promises its customers delivery within four hours on all flower

orders. All flowers are purchased on the previous day and delivered to Parker by 8.00 am the next

morning. The daily demand for roses is as follows.

Dozens of roses: 70 80 90 100

Probability : 0.1 0.2 0.4 0.3

The manager purchases roses for Rs.10 per dozen and sells them for Rs.30. All unsold roses are

donated to a local hospital. How many dozens of roses should Parker order each evening to

maximize its profit? What is the optimum expected profit?

GAME THEORY

1. Find the optimal strategies for A and B in the following game. Also obtain the value of the game.

A’s

Strategies

B’s Strategies

B1B2 B3

A1 9 8 -7

A2 3 -6 4

A3 6 7 -7

2. Determine the optimal strategies for each firm and value of the game for the following payoff

matrix

Firm B

Firm A B1 B2 B3 B4

A1 35 35 25 5

A2 30 20 15 0

A3 40 50 0 10

A4 55 60 10 15

3. Consider the game with the following pay off table:

Player B

B1 B2

Player

A

A1 2 6

A2 -2 ʎ

4. For the following 2*2 Game check whether saddle point exists or not? If not, find the probabilities

for each action of player A and each action for player B. Also find value of the Game.

Player B

B1 B2

Player

A

A1 11 7

A2 9 10

5. Solve the following game

Player B

B1 B2 B3 B4 B5

Player A

A1 3 5 4 9 6

A2 5 6 3 7 8

A3 8 7 9 8 7

A4 4 4 8 5 3

6. Solve the following game.

B

A 1 2 3 4 5 6

1 0 0 0 0 0 0

2 4 2 0 2 1 1

3 4 3 1 2 2 2

4 4 3 7 -5 1 2

5 4 3 4 -1 2 2

6 4 3 3 -2 2 2

7. A company is currently involved in negotiations with its union on the upcoming wage contract.

Positive signs in table represent wage increase while negative sign represents wage reduction.

What are the optimal strategies for the company as well as the union? What is the game value?

Conditional costs to the company (Rs. in lakhs)

Union Strategies

U1 U2 U3 U4

Company Strategies

MARKOV CHAIN

1. It was found in the survey that the mobility of the population in a state to the village, town & city

is in following percentage.

To

From Village Town City

Village 50 % 30 % 20 %

Town 10 % 70 % 20 %

City 10 % 40 % 50 %

What will be the proportion of population in village, town & city after two years given that present

population proportion is 0.7, 0.2 & 0.1 resp.

2. Market survey is made on two brands of breakfast foods A & B. Every time a customer purchases,

he may buy the same brand or switch to another brand. The transition matrix is given below.

From To

A B

A 0.8 0.2

B 0.6 0.4

At present 60% of people buy brand A and 40% people buy brand B. Determine market shares of

brands A & B in the steady state.

3. On January 1 (this year), Bakery A had 40 cent of its local market share while the other two

bakeries B and C had 40 per cent and 20 per cent, respectively, of the market share. Based upon

a study by a marketing research firm, the following facts were complied. Bakery A retains 90 per

cent of its own customers, while gaining 5 per cent of B’s customers and 10 per cent of C’s

customers. Bakery B retains 85 per cent of its customers, while gaining 5 per cent of A’s

customers and 7 per cent of C’s customers. Bakery C retains 83 per cent of its customers and

gain 5 per cent of A’s customers and 10 per cent of B’s customers. What will each firm’s share

be on January 1 next year and what will each firm’s market share be at equilibrium?

C1 0.25 0.27 0.35 -

0.02

C2 0.20 0.06 0.08 0.08

C3 0.14 0.12 0.05 0.03

C4 0.30 0.14 0.19 0.00

CPM/PERT

1. The activities of a project with details are as given bellow Activity Predecessor

Activity to tm tp

A - 6 7 8

B A 5 7 9

C B 1 2 3

D B 6 7 8

E B 3 5 7

F C 2 4 6

G D 4 6 14

H F 5 7 9

I E,G,H 1 2 9

a. Draw Network Diagram. b. Find project duration & identify critical path c. Calculate EST, EFT, LST, and LFT & Float for each activity.

2. A company has two grades of inspectors 1 & 2, the members of which are to be assigned

for a quality control inspection. It is required that at least 2000 pieces be inspected per 8-

hour day. Grade 1 inspectors can check pieces at the rate of 40 per hour, with an accuracy

of 97 percent. Grade 2 inspectors check at the rate of 30 pieces per hour with an accuracy

of 95 percent. The wage rate of a Grade 1 inspector is Rs. 5 per hour while that of Grade 2

inspector is Rs. 4 per hour. An error made by an inspector costs Rs. 3 to the company.

There are only 9 Grade 1 inspectors and 11 Grade 2 inspectors available to the company.

The company wishes to assign work to the available inspectors so as to minimize the total

cost of the inspection. Formulate this problem as an LP model so as to minimize the daily

inspection cost.

3. Use the graphical method to solve the following LP problem.

Maximize Z = 7x1 + 3x2

Subject to the constraints: x1 + 2x2 >= 3

x1 + x2 <= 4

x1 <= 2.5 x2

<= 1.5 x1 , x2

>= 0