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TUTORIAL SHEET NO. 1

Q.1 SolveMax Z= 5A +8B

Subjected to, 3A+2B 3

A+4B 4

A+B 5

A,B 0 (Z= 40)

Q.2 In relation to linear programming explain the implications of the following assumption of the

model.

(i) Linearity of the objective function(ii) Continuous variables

(iii) Certainty

Q.3 Define the term

(i) Basic variable(ii) Basic Solution

(iii) Basic feasible solution(iv) DegeneracyQ.4 Write short notes on

(i) Inconsistency and redundancy of constraint

(ii) Cycling in linear programming problemsQ.5 Explain the use of artificial variable in linear programming

Q.6 Define a basic solution to a given system of m simultaneous linear equations in n unknowns

Q.7 In the course of simplex table calculations, describe how you will detect a degenerate, an

unbounded and a non existing feasible solutionQ.8 How would you resolve the following complications in LPP

(i) Minimization

(ii) Equalities in constraints(iii) Tie for the leaving basic variable

Q.9 Short notes on

(i) Alternative optima in LPP(ii) Generalized LPP

Q.10 Solve, Max Z = 3A+2B

Sub to, 2A+ B 2

3A +4B 12

A,B 0 (No solution)

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TUTORIAL SHEET NO 2

Q.1 Solve by Big M methodMinimize Z = 4A +B

Sub to, 3A+B =3

4A + 3B 6

A+2B 4

A,B 0 (Ans A= 2/5, BB= 9/5, Z= 17/5)

Q.2 Solve the prob1 by 2-Phase method.

Q3. Use 2 phase Simplex Method to :Maximise Z = 5 A + 2B

Subject to 2A+B 1A+ 4B 6

A, B 0 ( Ans . No Feasible Solution )

Q4. What is the function of Minimum ratio rule in Simplex Method.

Q5. What is sensitivity analysis and why do we perform it.

Q6. Solve the following LP problem.

Maximize Z = 3A+2B-5C

Subject to A+B 2

2A+B+6C 6

A-B+3C = 0A,B,C 0

( ANS. A = 1, B= 2 , C= 0 , Z= 5 )

Q7. Solve the above LP problem if the RHS of primal is changed from( 2,6,0 ) to (2,10,5) . Find the

new optimal solution. (Ans A=2,B=0, C=1, Z=1)

Q8. Solve the following LPP using Dual- simplex method :

Minimize Z = A+2B + 3 C

Subject 2A-B +C 4

A+B+2C 8

A-C 2A,B,C 0

(ANS. A=3, B=2, Z=7 )

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Q9. Solve the following LP problem

Maximize Z= 20 A + 80 B

Subject to4A + 6B 90

8A + 6B 100

A, B 0

( Ans A= 0, B= 15, Z= 1200 )

Q10. In the above problem if the following new constraint is added to the model, find the solution of the

problem.New Constraint is 5A + 4B 80 ( Ans. No Change )

Tutorial Sheet No 3

1. Find the initial basic feasible solution of the following transportation problem by Vogels approximationmethod.

WarehousesW1 W2 W3 W4 CAPACITY

F1 10 30 50 10 7

Factory F2 70 30 40 60 9

F3 40 8 70 20 18

Requirement 5 8 7 14 34

2. A company has four salesmen who are to be assigned to four different sales territories. The monthly salesincreases estimated for each sales man in different territories (in lac rupees) are shown in the followingtable

Sales territory1 2 3 4 5

A 75 80 85 70 90

Salesman B 91 71 82 75 85

C 78 90 85 80 80

D 65 75 88 85 90Suggest optimal assignment for the sales man who is to be assigned to four different sales territories.Which sales territories will remain un-assigned? What will be the maximum sales increase every month?

3. In above question, if for certain reasons D cant be assigned to territory 3,will the optimal assignmentschedule be different? If so , show the new assignment schedule.

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The penalty costs are not satisfying demand at the shops D, E and F are Rs. 5 , Rs 3 and Rs. 2 .respectively.Determine the optimal distribution.

(Ans (A,E) =10 units, (B,D) =60 units, (B,E) =10 units, (B,F) =10 units, (C,D) =15 units, Penalty Rs 80)10. Consider a transportation problem with m=3 and n=4 where

C11=2 C12=3 C13=11 C14=7C21=1 C22=0 C23=6 C24=1

C31=5 C32=8 C33=15 C34=9Suppose S1 =6 ,S2=1 and S3=10 whereas D1=7 ,D2=5 and D4=2.Apply the transportation simplex method to finan optimal solution.

(Ans minimum cost=100).

TUTORIAL SHEET N0. 4

Q1. State the Transportation problem in general terms.

Q2. Explain the problem of degeneracy . How does one over come it.

Q3. Differentiate between Transportation and Assignment Problems.

Q4. How transportation problem is solved when demand and supply are not equal.

Q5. Using VAM, solve the following transportation problem for maximum profit:

A B C D SUPPLY

12 18 06 25 200

8 7 10 18 500

14 03 11 20 300

DEMAND 180 320 100 400

( ANS 15300 )

Q6. Find the optimal transportation cost for the following :

A B C D SUPPLY

03 07 09 10 120

04 05 07 12 170

09 12 05 06 210

DEMAND 290 150 70 90

( ANS Rs. 2430/- )

Q7. A student has to take examination in three courses X, Y , Z. He has three days available for studies.

He feels that it would be best to devote the whole day to study the same course so that he may study thecourse for one day, two days or three days or not at all. His estimate of grades he may get by studying

are as follows :

Study Days/ courses X Y Z

0 1 2 1

1 2 2 2

2 2 4 4

3 4 5 4

How should he plan to study so that he maximizes the sum of his grades.

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( Ans : Maximum return = 5, X =1, Y= 0, Z=2)

Q8. A company has plants A,B,C which have capacities to produce 300 kg, 200 kg, 500 kg respectively

of a particular chemical per day. The production cost per kg in these plants are Rs. 70/- , Rs. 60/- and Rs66/- respectively. Four bulk consumers orders for the product on the following basis:

Consumer

Kg reqd /day Price offered(Rs/kg)

I 400 100

II 250 100III 350 102

IV 150 103

Shipping costs (Rs/kg)from plants to consumers are given in the table below

From

I II III IV

A 3 5 4 6

B 8 11 9 12

C 4 6 2 8

Work out an optimal schedule for the above situation. Under what conditions would you change the

schedule. (Ans Total profit Rs 30700)

Q.9 Consider the following transportation problem

Factory Godowns Stock

available

1 2 3 4 5 6

A 7 5 7 7 5 3 60

B 9 11 6 11 - 5 20

C 11 10 6 2 2 8 90

D 9 10 9 6 9 12 50DEMAND 60 20 40 20 40 40

It is not possible to transport any quantity from factory B to godown5. Determine

(1) Initial solution by VAM

(2) Optimal basic solution

(3) Is the optimum solution unique?If not , find the alternative optimum solution.(Ans degeneracy,1120Rs, no,min cost 1120/-)

Q10. Salesman has to reach city no 10 starting from city no 1 by a motor car. Though its starting anddestination points are fixed, he has considerable choice as to which city to travel through en route. The

cost Cij (in Rs) for the standard policy on the motor car run from city i to city j is given below

1 2 3 5 6 7 8 9 10

1 2 4 3 2 7 4 6 5 1 4 8 3

3 3 2 4 6 6 3 9 4

4 4 1 2 7 3 3

Find the safest route of traveling so that the total traveling cost is minimum.

(Ans min. cost= 11)( 1 4 5 8 10), ( 1 4 6 9 10)

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TUTORIAL SHEET NO- 5

Q.1 What are the steps in decision making analysis?

Q.2 Explain various quantitative methods which are useful for decision making under uncertainty.

Q3. With the given payoff tables , determine the optimal strategy under(i) Maximin criteria

(ii) Minimax criteria

(iii) Hurwicz criteria with = 0.6

(iv) Laplace criteria

(Ans E.A,)Events Strategies

A B C D

E1 3 4 4 5E2 6 2 1 2

E3 1 8 8 3

Q.4 Explain 2 person zero sum game by giving suitable example.

Q.5 A manager has a choice between

(i) a risky contact promising Rs 7 lakh with probability 0.6 to Rs 4 lakh with prob. 0.4(ii) Adiversified portfolio consisting of 2 contracts with independent outcomes each promising

Rs 3.5 lakh with pro. 0.6 and Rs 2 lakh with prob 0.4. Construct a decision tree for it using

EMV criterion. (Ans EMV Rs 5.8 lakh)

Q.6 Give an example of a good decision you made not resulted in bad outcome. Also give an example of

a good decision you made that had a good outcome. Why was each decisions good or bad.

Q.7 Game theory provide a systematical quantitative approach for analyzing competitive simulation in

which the competitors make use of logical processes to techniques in order to determine an optimal

strategy for winning. Comment.

Q.8 Consider the game for following payoff matrix

Player BB1 B2

Player B 2 b

-2

(i) Show that the game is strictly determinable whatever may be.(ii) Determine the value of game (Ans 2)

Q.9 Solve the following game using Dominance propertyPlayer B

I II III

I 1 7 2

Player A II 6 2 7III 5 2 6

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(Ans A(2/5,3/5,0), B (1/2,1/2,0) , V= 4)

Q.10 A producer of boats has estimated the following distribution of demand for a particular kind aboat.

No.

demanded

0 1 2 3 4 5 6

Probability 0.14 0.27 0.27 0.18 0.09 0.09 0.01

Each boatcosts him Rs 7000 and he sells them for Rs 10,000 each. Boats left unsold at the end of the

season must be disposed off for Rs 6000 each. How many boats should be in stock so as to minimize his

expected profit. ( Ans Max. EMV 4080stock 3 boats)

TUTORIAL SHEET NO. 6

Q.1 A flower merchant purchase roses at Rs 10/dozen and sell them for Rs 30. Unsold flowers aredonated to the temple. The daily demand for rose has the following probability distribution

Demand(dozen) 7 8 9 10Probability 0.1 0.2 0.3 0.4

How many dozen of roses should be purchased in order to minimize the profit.

(Ans 9 dozen)

Q.2 In prob.1 ,also apply EOL criterion.

Q.3 Explain minimax criterion as applied to theory of games.

Q.4 Let (Uij) be the payoff matrix for a 2 person zero sum game.If V denotes the maximin value and

V the minimax value of the game, then prove that V V.Q.5 A decision problem has been expressed in the following payoff table.

Action Events

I II III

A 10 20 26

B -30 30 60

C 40 30 20

Determine which action the executive has to choose if he adopts

(i) Maximin criterionAns (20,C),

Q.6 Solve the above problem by Minimax criterion

Ans (60,B)

Q.7 Solve the problem5 using Hurwicz criterion with = 0.7

Ans (34,C),

Q8 Solve the problem4 using Laplace criterion Ans (30,C),

Q9 Solve the problem4 using Minimax criterion ( Ans (34,A)

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Q.10 Find the value of game .Also indicate whether they are fair or strictly determinable.

B

A

1 9 6 0

2 3 8 -1

-5 -2 10 -3

7 4 -2 -5(Ans V= 0(A1,B4),fair)

TUTORIAL SHEET NO.7

Q.1 Define reorder point.

Q.2 How is reorder point related to lead time demand?

Q.3 Explain the difference between a periodic review system and a continuous review system .

Q.4 Out of periodic review and continuous review system, which one is better and why?

Q.5 What is the unit for holding cost parameter.

Q.6 Is it possible to have a negative inventory level? If so, explain how?

Q.7 Discuss the probability of inventory control when the stochastic demand is uniform, production of

commodity is instantaneous and lead time is negligible (discrete case).

Q.8 In a private canteen, the daily demand for packet meal follows uniform distribution as presented

below,

P(X) = 1/(450-230) 230 X 450The cost of production per packet of meals is Rs 8. The S.P. is Rs 16 /packet . The surplus packets oneach day are sold at Rs 6 /pkt in a nearby public place. Find the optimal no. of pkts of meals to be

prepared each day. (Ans 406)

Q.9 A fish stall is planning for its optimal purchases of a costly variety of fish. The daily demand of fish

follows normal; distribution with a mean of 800 kg and standard deviation of 75 kg. The purchase price

of the fish is 1.50/- per kg. The S.P. is 200/-/kg If the fish is not sold on the day of purchase, it is sold toa dry fish manufacturing firm at 110/- per kg. Find the optimal daily purchase quantity of fish

(Ans 810.5 kg)

Q.10 Describe the basic characteristic of an inventory system.

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TUTORIAL 8

Q.1. Explain the nature of the probabilistic model in inventory control.Q.2 Consider a shop which produces 3 items. The items are produced in lots. The demand rate for each

item inconstant and can be assumed to be deterministic. No back orders are to be allowed. The data for

the items are given below

Items 1 2 3

Holdind cost (Rs) 20 20 20

Set up cost(Rs) 50 40 60

Unit cost(Rs) 6 7 5

Demand rate/ year 10000 12000 7500

Determine approximately the economic order quantities when the total value of average inventory levels

of these items is Rs 1000 . (Ans 114,110.5,109 units)

Q.3 A baking company sells one of its types of cakes by weight. It makes a profit of Rs 3.20 per pound

on every pound of cake sold on the day it is baked. It disposses off all the cakes not sold on the day theyare baked at the rate of 50 paiseper pound. If demand is known to be triangular with probability density

function,

f(R) = 0.02- 0.0002R 0 X 100

Find the optimum amount of cake the company should bake daily. (Ans 63.3 pounds)

Q.4 A baking company sells one of its types of biscuits by weight. It makes a profit of Rs 9.50 per

pound on every kg of biscuits sold on the day it is baked. It disposes off all the biscuits not sold on the

day they are baked at the rate of 1.50 Rs per kg. If demand is known to be rectangular between 300kgand 400kg, find the optimum amount of biscuits the company should bake daily.

(Ans386.4 kg)

Q 5 Derive the expression for optimum order level for the case; demand, set up cost zero, stock levelsdiscrete and lead time zero.Q.6 A person want to decide the constituents of a diet which will fulfill his daily requirement of proteins

fats & carbohydrates at the minimum cost. The choice is to be made from 4 different type of foods .The

yield per unit of these foods are given as Yield per unitFood type Protein Fat Carbohydrate Cost per unit

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 .

Q7- Solve the following game by reducing them to 2 x 2 game by graphical method

A

3 0 6 -1 7

-1 5 -2 2 1

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Q8 Solve the following 3 x3 game by linear programming

Player B

Player A

Q9- A company gives a lease on a certain property . It may sell for Rs 75,000 or may dull the said property for oil. Various

possible results are as under along with the probabilities of happening and rupees consequence

Possible result Probabilities Rupees Consequences

Dry well .10 -100000

Gas well only .40 45000

Oil & gas combination .30 98000

Oil well .20 199000

(a) Draw a discussion tree for the problem & determine whether the company should drill or sell.Q10- The number of machines arriving per day at a factory repair bench has been noted over a range period of time and

joined to have the following distribution

Arrivals: 0 1 2 3 4 5 6 7 8 9Percentage 2 7 15 20 20 16 10 6 31

Repair times depend on the type of fault. The number of repairs completed per day. They have been recorded & found

to be represented by.

Arrival : 1 2 3 4 5 6 7 8 9 10

Percentage: 4 8 14 18 18 15 10 7 4 2

Simulate 25 days to find the maximum queue length use the following series of random numbers.

55 86 87 70 05 71 69 29 74 73 37 17 42 16 89 98 35 81

07 16 94 42 27 18 13 70 05 35 86 85 02 79 13 35 17 38

57 19 97 38 45 82 40 32 63 29 10 67 41 99

Tutorial Sheet No. 9

Q.1 List five disadvantages in using simulation analysis.

Q.2 Explain why a simulation experiment must usually be executed over long periods of simulated time.

Q.3 What is a random number generator ?Q.4 Why is this term(random no. generator) a misnomer?

Q.5What are the desirable properties of random number generators?

Q6 What is the difference between model verification and model validation .

Q.7 Describe the general probability of M/M/K queuing system and also deduce an explicit expression for the steady state

probability of the length of the queue in an M/M/1 system.

1 -1 -1

-1 -1 3

-1 2 -1

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Q.8 Four counters are being run on the frontier of a country to check the passport and necessary papers of the tourists. The

tourists choose the counter at random. If the arrival at the frontier is poisson at the rate and the service time is exponential

with parameter/2. What is the steady state average queue at each counter.

(Ans 4/23)

Q9In a railway yard , goods train arrive at a rate of 30 trains per day. Assuming that the inter-arrival time follows an

exponential distribution and the service time is also exponential with an average 36min. Calculate

(a) The mean queue size (line length)

(b) The probability that the queue size exceeds 10.If the input of trains increases to an average 33 per day , what will be the change in (a) and (b).

(Ans 3 trains, 0.06, 4.8 or 5 trains, 0.2 approx)

Q.10 List 5 reasons why one would want to use a micro computer based simulation analysis.

TUTORIAL 10

Q.1 List 5 advantages of using a simulation language.

Q.2 Problems arriving at the computer center in a poisson function at an average rate of 5/day. The rules of the computing

centre are that any man waiting to get his problem solved must aid the man whose problem is solved.If the time for solving

one problem has an exponential distribution with mean time of 1/3rd

day if the average solving time is inversely proportionalto the no. of people working on the problem, approximate the expected time in the center for a person entering .

(Ans. 1/3 day or 8 days)

Q.3 In the production shop of a company, the breakdown of the machines is found to be poisson with an average rate of 3

machine/hour. Breakdown time at one machine costs Rs 40/hr to the company. There are two choices before the

company to hiring the repairman. One of the repairman is slow but cheap and other fast but expensive. The slow cheap

repairman demand Rs 20/hr and will repair the broken down machines exponentially at the rate of 4/hr. The fast

expensive repairman demands Rs. 30/hr will repair machines exponentially at an average rate of 6/hr. Which repairmanshould be hired.

(Ans cost- 180, cost 70, fast-expensive repairman )

Q.4 Show that for a single service station, Poisson arrivals and exponential service times, the probability that exactly n

calling units are in queuing system is (1-) n ,n 0. ( = traffic intensity).Also find the expected line length.

Q.5 The material manager of the firm wishes to determine the expected mean demand for a particular item in stock during

the re order lead time. The information is needed to determine how far in advance to reorder, before the stock level is reducedto zero. However both the lead time and the demand/day for the item are random variables described by the probability

distribution,

Lead time (in days) Probability Demand/day(units) Probability

1 0.4 1 0.15

2 0.3 5 2 0.25

3 0.25 3 0.40

4 0.20

Manually simulate the probability for 30 re-orders to estimate the demand during lead time.

Q.6 In a railway yard ,goods train arrive at a rate of 30 trains per day. Assuming that the interarrival time follows an

exponential distribution and the service time distribution is also exponential with an average 36 min., calculate the following,(i) Average no. of customers in the system

(ii) The probability that the queue size exceeds 10.

(Ans 3,0.06)

Q.7 For the above problem, if the inputs of the trains increases to an average33 /day, what will be the change in (i) and (ii).

(Ans 0.2)

Q.8 A mechanics repairs 4 machines. The mean time between service requirements is 5 hrs for each machine and forms an

exponential distribution. The mean repair time is 1 hr and also follows the same distribution pattern. Machine downtime costs

Rs 25 per hr and the machine costs Rs 55 per day. Find the expected number of operating machines.( Ans 3)

Q.9 Determine the expected downtime lost per day in the above problem.

(Ans Rs 200/day)

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Q.10 In prob1 ,would it be economical to engage 2 mechanics, each repairing only 2 machines

(Ans 0-4 hr/day)

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TUTORIAL SHEET NO.11

Q.1 Most of the businessman view Inventory is a necessary evil . Do you agree with this.

Q.2 Formulate and solve a discrete stochastic model for a single product with lead time zero. The shortage and shortage

costs are independent of time. Set up cost is constant.Q.3 Discuss the probabilistic inventory models with instantaneous demand and no set up cost.

Q.4 Consider an inventory model in which the holding cost of one unit in an inventory for a specified period is C and the cost

of shortage per unit is B .Suppose the demand follows a known continuous probability distribution. Determine the optimuminventory level in the beginning of the period.

Q.5 Write a note on Newspaper boy problem.

Q.6Define inventory and its importance.

Q.7 What are the advantages and disadvantages of having inventories.

Q.8 Formulate and solve continuous probabilistic reorder point for lot size model to determine optimal reorder point for apresented lot size. Lead time is finite. Shortages are allowed and fully backlogged.

Q.9 A newspaper boy buys paper for 60 paise each.and sales them for Rs 1.40 each. He cannot return unsold papers. Daily

demand form the following distribution.

NO. OF

CUSTOMERS

23 24 25 26 27 28 29 30 31 32

PROBABILITY .01 0.03 0.06 0.10 0.2 0.25 0.15 0.10 0.05 0.05

If each days demand is independent of the previous day demand, how many papers should be ordered each day.(Ans 28)

Q.10Some of the spare parts of a ship costs Rs100000 each. These spare parts can only be ordered with the ship. If not

ordered at the time when shipnwas construted, these parts cant be available on need. Suppose that a loss of Rs 1,00,00,000

is suffered for each spare that is needed as replacement during the life term of the class of the ship discussed are,

Spare parts

reqd

0 1 2 3 4 5

Prob. 0.9488 0.0400 0.0100 0.0010 0.0002 0.0

How many spare parts should be procured. (Ans 3 parts)

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