Questions:
Decision Science
Part A
1.Give the role & significance of O.R. in business & industry for scientific decisions.
2. “The primary contribution of the game theory has been its concept rather than its formal application to solving real life problems.” Do you agree? Discuss.
3. What is queuing theory? Describe the different types of costs involved in a queuing system. In what areas of management can queuing theory be applied successfully? Give examples.
4. What do you mean by Simulation? Explain Monte Carlo Simulation in present business decision making.
5. Explain the terms:
(a) Basic feasible solution;
(b) Nondegenerate basic feasible solution;
(c) Optimal solution and
(d) Pivot column
PART B
1.Explain in brief with examples:
(i) North West Corner rule :
(ii) Vogel’s Approximation Method.
2. Show that assignment problems are particular cases of transportation pr Can an assignment problem ever be a non degenerate transportation problem? Explain.
3.“Basic Problem in queuing theory is to strike an economic balance between the service cost and the waiting cost.” Elucidate this statement by taking an example.
Case Study
M/s Gupta chemicals Ltd. Markets its product through five area distributors. The company has three plants the particulars of which are given below:
Plant

Monthly production
capacity (kgs)

Fixed cost of
production (Rs/month)

Variable cost of
production) (Rs./unit)

P1

6,000

2,40,000

120

P2

15,000

5,00,000

110

P3

22,500

6,00,000

90

The selling price excluding freight is Rs. 250 per kg and the company has commitments to supple the following quantities to the distributors:
Distributors

Quantity to be supplied (kg.)

I

3,750

II

3,750

III

7,500

IV

15,000

V

6,000

The transportation cost, rupees per unit (borne by the manufacturer), for supply for plants to distributor are as given below:
Plant/Distributors

I

II

III

IV

V

P1

1.2

1.5

1.0

1.5

1.0

P2

1.5

1.8

1.2

1.2

1.5

P3

1.6

1.7

1.0

0.9

0.5

1.Determine the optimal tieup between the plants and the distributors and the maximum profit company can make.
PART C
 Every corner of the feasible region is defined by
 the intersection of 2 constraints lines
 Some subset of constraint lines and non negativity condition
 Neither of the above
 J. Breveries Ltd. Have two bottling plants, one located at ‘G’ and the other at ‘J’. Each plant produces three drinks – whisky, beer and brandy names A, B and C respectively. The number of bottles produced per day are as follows:
Drinks

Plant

G

J

Whisky

1,500

1,500

Beer

3,000

1,000

Brandy

2,000

5,000

A market survey indicated that during the month of July, there will be a demand of 20,000 bottles of whisky, 40,000 bottles of beer and 44,000 bottles of brandy. The operating costs per day of plants at G & J are 600 and 400 monetary units. For how many days each plant be run in July so as to minimize the production cost, while still meeting the market demand?
(a) x1= 10, x2= 4, Max Z= 8,800
(b) x1= 12, x2= 4, Max Z= 8,800
(c) x1= 10, x2= 4, Max Z= 4,400
(d) x1= 12, x2= 2, Max Z= 2,200
 Five machines are available to do five different From past records, the time (in hrs.) that each machine takes to do each job is known & given in the following table:
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

Find the assignment of machines to jobs that will minimize the total time taken
 10 hours
 12 hours
 13 hours
 22 hours
 In performing a simulation it is advisable to
 Use the results of earlier decisions to suggest the next decision to try
 Use the same number of trials for each decisions
 Simulate all possible decisions
 None of the above
 The assignment problem consists of the following elements
 A set of n jobs
 A set of n facilities
 A set of cost, one for each pair of job facility
 All of the above
 Find the optimal strategies for two stores from the following payoff matrix showing gain or loss of customers for store 1.
Action of StoreY

Action of
Store X


A

B

C

I

0

20

60


II

30

10

20

III

70

80

30

 Optiamal strategy (II, A), Value of game= 20
 Optiamal strategy (I, D), Value of game= 40
 Optiamal strategy (II, C), Value of game= 20
 Optiamal strategy (II, B), Value of game= 40
 The phrase ‘unbounded LP’ means that
 at least one decision variable can be made arbitrarily large without leaving the feasible region
 The objectives contours can be moved as far as desired, in the optimizing direction, and still touch at least one point in the constraint set.
 Two firms A and B (manufacturing of detergent powder) are planning to make fund allocation for advertising their products. The matrix given below shows the percentage of market share of firm A for its various advertising policies. Determine the optimal strategy for firm A.
Firm A

Firm B

Strategies

No
advertising

Medium
advertising

Large
advertising

No
advertising

60

50

40

Medium
advertising

70

55

45

Large
advertising

80

60

50

 Large advertising, 60
 Medium advertising, 55
 No advertising, 50
 Large advertising, 50
 An advantage of simulation, as opposed to optimization, is that
 Often multiple measures of goodness can be examined
 Some appreciation for the variability of outcomes of interest can be obtained
 More complex scenarios can be studied
 All of the above
 Following is payoff matirx in terms of increase in votes to X(loss to Y) using three defferent strategies available to each player for advertising. Find optimal strategy to be adopted by X for the campaign and the number of votes X will gain with this strategy.
Candidate Y

Candidate X

Strategy

I

II

III

A

300

200

100

B

600

500

400

C

600

400

600

 (A, I), (C, II); Value of game = 400
 (B, II), (A, III); Value of game = 500
 (C, III), (B, II); Value of game = 300
 (C, I), (C, III); Vlaue of game = 600
 The most difficult aspect of performing a formal economic analysis of queueing systems is
 Estimating the service cost
 Estimating the waiting cost
 Estimating use
 In a typical simulation model input provided by the analyst includes
 Value for the parameters
 Value for the decision variables
 Value for the measure of effectiveness
 Both (a) and (b)
 The scientific method in O.R. study generally involves
 Judgement phase
 Research phase
 Action phase
 All of the above
 The operations Research models can be classified according to
 degree of abstraction
 structure
 purpose
 nature of environment
 all of the above
 One of the disadvantages of simulation is that:
 it is very expensive & requires to develop large repetitions of data
 Simulation solution can be cent percent accura
 Simulation is applicable in cases where there is an element of randomness in a sy
 All of the above.
 An Operations Research model is good as
 It provides some logical & systematic approach to the problem
 It incorporates useful tools which help in eliminating duplication of methods applied to solve specific problem
 It helps in finding avenues for new research & improvements ina system.
 It indicates the nature of measurable quantities in a problem.
 All of the above.
 With small sample sizes the results of a simulation can be very sensitive to the initial conditions.
 True
 False
 Can not say
 None of the above
 Which of the following does not apply to the basic queuing model?
 Exponentially distributed arrivals
 Exponentially distributed service times
 Finite time horizon
 Unlimited queue size
 In business & management decision making, the O.R. study helps to have
 Better control
 better system
 better decisions
 all of the above
 Simulation is not possible if the complete knowledge of the system is not known.
 True
 False
 Cannot say
 None of the above
 Linear programming is
 a constrained optimization model
 a constrained decision making model
 a mathematical programming model
 all of the above
 The non negativity requirement is included in an LP because
 it makes the model easier to solve
 it makes the model correspond more closely to the real world problem
 Both (a) and (b)
 Neither of the above
 Which of the following assertions is true of an optimal solution to an LP?
 Every LP has an optimal
 The optimal solution always occurs at an extreme point
 The optimal solution uses up all resources
 If an optimal solution exists, there will always be at least one at a corne
 In Vogel’s approximation method, the opportunity cost associated with a row is determined by
 the difference between the smallest cost & the next smallest cost in that row
 the difference between the smallest unused cost & the next smallest unused cost in that row
 the difference between the smallest cost & the next smallest unused cost in that row
 None of the above
 Once a queue model has been constructed, analysis of the model can be performed in
 Through analytical solution
 Through simulation
 Either (a) or (b)
 Both
 An unbalanced transportation problem is the one in which
 the number off jobs are not equal to number of facilities
 the total supply is not equal to total requirement
 the total supply is same as total requirement
 None of the above
 A closed path has all the following characteristics except:
 It links an unused square with itself.
 Movements on the path may occur horizontally, vertically, or diagonally.
 The corners of the path must all be stones, except for the corner at the unused square being evaluate
 The path may skip over unused squares or stone
 A company produces three products A, B & C. These products require three ores O1, O2 and O3. The maximum quantities of the ores O1, O2 and O3 available are 22 tones, 14 tones and 14 tones respectively. For one tonne of each of these products , the ore requirements are:
Product

O1

O2

O3

Profit per tonne
(in Rs.)

A

3

1

3

1

B



2

2

4

C

3

3

0

5

How many tonnes of each product A, B & C should company produce to maximize the profits?
 Maximum 28,000; 7 tonnes of product B & none of A or C
 Maximum 22,000; 5 tonnes of product A & none of B or C
 Maximum 20,000; 5 tonnes of product C & none of A or B
 Maximum 28,000; 7 tonnes of product A & none of B or C
 The Northwest corner rule
 Is used to find an initial feasible solution.
 Is used to find an initial optimal solution.
 Is based on the concept of minimizing opportunity cost
 None of the above
 A cement factory manager is considering the best way to transport cement from his three manufacturing centers P, Q, R to depots A, B, C, D and E. The weekly production and demands alongwith transortation costs per tonne are given below:
Manufacturing
centre Depot

A

B

C

D

E

Supply

P

4

1

3

4

4

60

Q

2

3

2

2

3

35

R

3

5

2

4

4

40

Demand

22

45

20

18

30

135

Calculate the minimum total transportation cost.
 Roma pharmaceutical company products two popular drugs A & B which are sold at the rate of Rs. 9.60& Rs. 7.80 respectively. The main ingredients are X, Y & Z & they are required in the following properties:
Drugs

X

Y

Z

A

50%

30%

20%

B

30%

30%

40%

The total available quantities (gms.) of different ingredients are 1,600 in X, 1400 in Y & 1200 in Z. The costs of X, Y & Z per gm are Rs. 8, Rs. 6 & Rs. 4 respectively.
Estimate the most profitable quantities of A & B to be produced, using the simplex method.
(a) Maximum Value of Z= 20,000; X1= 2000 & X2= 2000.
 Maximum Value of Z= 40,000; X1= 4000 & X2=
 Maximum Value of Z= 10,000; X1= 2000 & X2=
(d) Maximum Value of Z= 10,000; X1= 2000 & X2= 2000.
 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 given in the following table:
Jobs/Machines

I

II

III

IV

V

A

11

17

8

16

20

B

9

7

12

6

15

C

13

16

15

12

16

D

21

24

17

28

26

E

14

10

12

11

15

Find out minimum cost possible through optimal assignment of machines to jobs.
 AII, BIV, CV, DIII, EII; Minimum cost = 60
 A III, BII, C I, D IV, E V; Minimum cost = 120
 A I, B III, C IV, D II, E V; Minimum cost = 60
 A IV, B II, C I, DV, E III; Minimum cost = 120
 The maximum number of items that can be allocated to an unused route with the stepping stone algorithm is
 the maximum number in any cell
 the minimum number in any cell
 the minimum number in an increasing cell
 the minimum number in a decreasing cell on the stepping stone path for that route
 Obtain an initial basic feasible solution to the following transportation problem:
Warehouse/
stores

I

II

III

IV

Supply

A

7

3

5

5

34

B

5

5

7

6

15

C

8

6

6

5

12

D

6

1

6

4

19

Demand

21

25

17

17

80

 A major goal of queuing is to
 Minimizing the cost of providing service
 Provide models which help the manager to trade off the cost of service
 Maximize expected return
 Optimize system characteristics
 Determine the optimal strategies for both Firm A and Firm B and the value of the game (using maximin minimax principle):
Firm B

Firm A

3

1

4

6

7

1

8

2

4

12

16

8

6

14

12

1

11

4

2

1

 Optimal strategy (2, 2)
 Optimal strategy (3,4)
 Optimal strategy (3, 5)
 Optimal strategy (3, 3)
 Solve the value of game whose payoff matrix is given by:
Strategy

B1

B2

B3

B4

A1

16

60

56

58

A2

20

28

18

24

A3

24

8

0

24

 For player A= 42 &for player B= 42
 For player A= 20 &for player B= 20
 For player A= 24 &for player B= 24
 For player A= 28 &for player B= 28
 A queue is formed when
 Customers wait for services
 Service facilities stand idle & wait for customers
 Either (a) or (b)
 Both
 The MODI method uses the stepping stone path
 to calculate the marginal cost of unused cells
 to determine how many items to allocate to the selected unused cell
 To determine the values of the row and column indexe
 None of the above
 Characteristics of queues such as ‘expected number’ in the system:
 Are relevant after the queue has reached a steady state
 Are probabilistic statements
 Depend on the specific model
 All of the above