Decision Science

Part A

Give the role & significance of O.R. in business & industry for scientific decisions.
Answer:

“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.

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.

What do you mean by Simulation? Explain Monte Carlo Simulation in present business decision making.

Explain the terms: 
(a) Basic feasible solution; 

 (b) Non-degenerate basic feasible solution;  

(c) Optimal solution and 

(d) Pivot column

Part B

Explain in brief with examples: (i) North West Corner rule :

(ii) Vogel’s Approximation Method.

Show that assignment problems are particular cases of transportation problem. Can an assignment problem ever be a non degenerate transportation problem? Explain.

“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     

Determine the optimal tie-up 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

G.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 jobs. 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

4. 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

5. 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

6. 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

7. 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.

8. 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

9. 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

10. 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

11. The most difficult aspect of performing a formal economic analysis of queueing systems is

Estimating the service cost
Estimating the waiting cost
Estimating use

12. 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)

13. The scientific method in O.R. study generally involves
Judgement phase
Research phase
Action phase
All of the above

14. The operations Research models can be classified according to
degree of abstraction
structure
purpose
nature of environment
all of the above

15. 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 accurate.
Simulation is applicable in cases where there is an element of randomness in a system.
All of the above.

16. 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.

17. 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

18. 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

19. In business & management decision making, the O.R. study helps to have
Better control
better system
better decisions
all of the above

20. Simulation is not possible if the complete knowledge of the system is not known.
True
False
Cannot say
None of the above

21. Linear programming is
a constrained optimization model
a constrained decision making model
a mathematical programming model
all of the above

22. 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

23. Which of the following assertions is true of an optimal solution to an LP?
Every LP has an optimal solution.
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 corner.

24. 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

25. 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

26. 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

27. 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 evaluated.
The path may skip over unused squares or stones.

28. 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 Rs. 28,000; 7 tonnes of product B & none of A or C
Maximum Rs. 22,000; 5 tonnes of product A & none of B or C
Maximum Rs. 20,000; 5 tonnes of product C & none of A or B
Maximum Rs. 28,000; 7 tonnes of product A & none of B or C

29. The North-west 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

30. 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.
190
200
290
390

31. 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= 2000.
Maximum Value of Z= 10,000; X1= 2000 & X2= 4000. 
(d)    Maximum Value of Z= 10,000; X1= 2000 & X2= 2000.

32. 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.

A-II, B-IV, C-V, D-III, E-II;  Minimum cost = 60

A- III, B-II, 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, D-V, E- III; Minimum cost = 120

33. 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

34. 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     

300
324
225
356

35. 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

36. 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)

37. 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

38. A queue is formed when

Customers wait for services
Service facilities stand idle & wait for customers
Either (a) or (b)
Both

39. 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 indexes.
None of the above

40. 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