Operation Research For Business Decisions

St. Joseph’s College of Commerce 2015 II Sem Operation Research For Business Decisions Question Paper PDF Download

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ST. JOSEPH’S COLLEGE OF COMMERCE (AUTONOMOUS)

End Semester Examinations –  march/April 2015
M.I.B. – ii semester
P211201: OPERATION RESEARCH FOR BUSINESS DECISIONS
Duration: 3 Hours                                                                                             Max. Marks: 100
SECTION – A
I) Answer any SEVEN questions.  Each carries 5 marks.                                    (7×5=35)
  1. Discuss briefly the Hungarian method for solving an assignment problem.
  2. Write five differences between  PERT and CPM
  3. Explain principle of dominance in Game Theory?
  4. A newly developed diary has started producing cheese, butter, and milk candy. There are three departments: one is manufacturing department, and the other two are pasteurization and packing departments respectively. The following table shows the labor hours spent by one unit (kg) in each department.

The minimum working capacity of each department is 100, 75, and 80 hours respectively. The Profit on the sale of one kg of cheese, butter and milk candy is Rs.12, Rs.10 and Rs.8 respectively. You have to plan to schedule that maximizes the total profit. Bring out a mathematical model
  5. Find the graphical solutions of the following LP problems.

            Maximize Z=5x+10y

               Subject to 2x+y≥8

                               3x+4y≤24

                                       y≥2

                                    x,y≥0
  6. A child care shop dealing with children’s requirement has one cashier who handles all customers’ payments. The cashier takes on an average 4 minutes per customer. Customers come to the cashier’s area in a random manner but on an average of 10 people per hour. The management received a large number of customer’s complaint and decided to investigate the following questions.

  1. What is the average length of the waiting line to be expected under the existing conditions?
  2. What portion of his time is the cashier expected to be idle?
  3. What is the average length of time that a customer would be expected to wait to pay for his purchase?
  4. If it was decided that a customer would not tolerate a wait of more than 12 minutes, what is the probability that a customer would have to wait at least that length of time?
  7. Using the following cost matrix, determine:

a)      Optimal job assignment                         b) The cost of assignmetns.

Machinist                             Job

                        1        2          3         4        5

A                 10       3          3         2        8

   B                   9        7          8         2        7

   C                  7         5          6         2        4

   D                  3         5          8        2        4

   E                   9       10         9         6       10
   

8.

  

It is known that currently nothing can be sent from warehouse 1 to market A and from warehouse 3 to market C. Find the initial solution to the problem using Vogel’s Approximation Method.

 
   

9.

  

Obtain the dual of the following

Maximize Z=8X1+10X2+5X3

Subject to constraints

X1-X3≤4

   2X1+4X2≤12

   X1+X2+X3≥2

   3X1+2X2-X3=8

   X1,X2,X3≥0
   

10.

  

You are given the information about the cost of performing different jobs by different persons. The job-person marking x indicate that the individual involved cannot perform the particular job,. Using this information state

i)                   The optimal assignment of jobs (ii) the cost of such assignment

 
 

SECTION – B
II) Answer any THREE questions.  Each carries 15 marks.                                (3×15=45)
  11. A cabinet manufacturing company is planning to introduce a new model of

Cabinets which requires the following task.

The wheels are mounted after they are prepared. The base cannot be attached until the sides are assembled and wheels mounted. The  top cannot be attached nor the brackets inserted until the sides are assembled. The shelves are inserted after the brackets are installed. The back panel is attached after the base and top are attached. The doors are attached after the selves are inserted and the top and bases are attached. The unit is painted after the back and the doors Are attached.

  1. Identify the immediate predecessors of each task and draw the network.
  2. Find the critical path(s) and list the critical activities.
  3. Obtain the earliest and the latest start and the completion times of all the a ctivities and their total floats, free float and independent floats.
   

12.

  

Find x1 and x2 so as to maximize Z=24×1 +18×2

Subject to constraints 4×1 +6×2≤24

6×1  + 3×2≤18

With                                x1,x2≥0
   

13.

  

Maximize  Z=2X1+4X2-X3

Subject to constraints

X1+X2+X3≥8

X1-X2≥1

3X1+4X2+X3≤40

Where X1, X2, X3≥0
   

14.

  

a) In a game of matching coins with two players suppose A wins one unit of value when there are two heads, wins nothing when there are two tails, and loses ½ unit of value when there are one head and one tail. Determine the payoff matrix A, the best strategies for each player and the value of the game to A.

b) The optimistic, most likely and pessimistic times of the activities of a project are given below. Activity 40-50 must not start before 22 days while activity   70-90 must end by 35 days. The scheduled completion time of the project is 46 days. Draw the network and determine the critical path. What is the probability of completing the project in scheduled time?

 

 

Activity  To-Tm-Tp Activity  To-Tm-Tp
10-20 4-8-12 50-70 3-6-9
20-30 1-4-7 50-80 4-6-8
20-40 8-12-16 60-100 4-6-8
30-50 3-5-7 70-90 4-8-12
40-50 0-0-0 80-90 2-5-8
40-60 3-6-9 90-100 4-10-16

 
  15. Solve the following game using graphical approach

      B’s  strategy  
A’s strategy B1 B2 B3 B4
 A1 8 5 -7 9
A2 -6 6 4 -2
 

SECTION – C
III) Case Study                                                                                                              (1×20=20)
  16. An investment company wants to study the investment projects based on market demand, profit and the investment required, which are independent of each other. Following probability distributions are estimated to each of these three factors.

Annual demand

(units in thousands)

 25 30 35 40 45 50 55
Probability 0.05 0.10 0.20 0.30 0.20 0.10 0.05

 

Profit per unit (Rs) 3 5 7 9 10
Probability 0.10 0.20 0.40 0.20 0.10

 

Investment

Required

(in thousand rupees)

 2750 3000 3500
Probability 0.25 0.50 0.25

 

Using Simulation process repeat the trial 10 times, compute the values for demand, profit and investment. What is the most likely return percent on investment?

Random numbers : (30, 12, 16)  (59, 09, 69) (63,94,26) (27, 08, 74) (64, 60, 61) (28, 28, 72)  (31, 23, 57) (54,85,20) (64, 68, 18)  (32, 31, 87)

In the bracket above use the first number for demand, second for profit and last for investment required.

 

*************************************

 

 

 

M.I.B. – II SEMESTER

OPERATIONS RESEARCH

 

Answer key 4.

 

 

Answer key 5.

 

 

 

 

 

 

 

 

 

 

Answer key:6.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer 7.

 

 

 

 

Answer 8.

 

It is known that currently nothing can be sent from warehouse 1 to market A and from warehouse 3 to market C. Solve the problem and determine the least

Cost of transportation schedule. Is the optimal solution obtained by you unique? If no, what are/is the other optimal solutions?

the above loop is done.

 

 

ANSWER 9.

 

ANSWER 10. Solution : Adding dummy row.

 

 

 

ANSWER 11.

 

 

 

 

 

 

 

 

 

 

 

 

 

SECTION – B

ANSWER 12.

 

ANSWER 13.    

 

 

 

 

 

 

 

 

 

 

ANSWER 14

 

B.

ANSWER 15.

 

 

Answer case study 16:      

 

 

 

 

 

St. Joseph’s College of Commerce 2016 Operation Research For Business Decisions Question Paper PDF Download

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REG NO:

ST. JOSEPH’S COLLEGE OF COMMERCE (AUTONOMOUS)
END SEMESTER EXAMINATION – MARCH/APRIL 2016
M.COM(I.B.) – II  SEMESTER
P4 15 AR 201: OPERATION RESEARCH FOR BUSINESS DECISIONS
Duration: 3 Hours                                                                                              Max. Marks: 100
SECTION – A
I. Answer any SEVEN questions.  Each carries 5 marks.                                    (7×5=35)
  1. Three products are processed through three different operations.  The times (in minutes) required per unit for each product, the daily capacity of the operations (in minutes per day) and the profit per unit sold for each product (in rupees) are as follows:

Operation Time per unit (Minutes) Operation capacity (Minutes/day)
Product I Product II Product III
1 3 4 3 143
2 5 0 4 146
3 3 6 2 142
Profit/unit (Rs.) 20 10 30  

The zero time indicates that the product does not require the given operation.  It is assumed that all units produced are sold.  Moreover, the given profits per unit are net values that result after all pertinent expenses are deducted.  The problem is to determine the optimum daily production for three products that maximizes the profit.

Formulate the above production planning problem in a linear programming problem.
  2. Find the dual of the following problem:

Maximize               Z = 6x1+8x2

Subject to              2x1 + 3x2  ≤ 16

4 x1  + 2x2  ≥ 16

2x1    +  x2     =  16

x1  ,  x2     ≥ 0

 
  3. A director in a Management Institute has the problem of assigning courses to teachers with a view to maximizing educational quality in his Institute.  He has available to him one professor, two associate professors, and one teaching assistance (TA).  Four courses must be cleared and, after appropriate evaluation, has arrived at the following relative ratings regarding the ability of each instructor to teach each of the four courses

  Course 1 Course 2 Course 3 Course 4
Prof.  1 60 40 60 70
Prof.  2 20 60 50 70
Prof.   3 20 30 40 60
TA 30 10 30 40

How should he assign his staff to the courses to realize his objective?

 
  4. At Dr. Prachi’s clinic patients arrive at an average of 6 patients per hour.  The clinic is attended to by Dr. Prachi herself.  The doctor takes 6 minute per patient to serve them.  It can be assumed that arrivals follow a Poisson distribution and the doctor’s inspection time follows an exponential distribution.  Determine:

i)                   The percent of time a patient can walk right inside the doctor’s cabin without having to wait.(Traffic Intensity)

ii)                 The average no. of patients in Dr. Prachi clinic.

iii)              The average no. of patients waiting for their turn.

iv)               The average time a patients spends in the clinic.

 
  5. Plot separately the following equations in a Graph. Also highlight the feasible area in each case.

i)       4x + 6y  ≤ 12

ii)     4x + 6y  ≥ 12

iii)   4x + 6y = 12

 
  6. A project consists of the following activities with the time estimates noted against each:

Activity Time Estimate

(weeks)

 Activity Time Estimate

(weeks)
1-2 2 3-7 5
1-3 2 4-6 3
1-4 1 5-8 1
2-5 4 6-9 5
3-6 8 7-8 4
    8-9 3

Required:

i)                   Draw a network diagram.

ii)                 Determine the critical path and its duration.

 
  7. Baroda lather shop has three manufacturing plants and four sales outlets.  Data of daily demand at various sales outlets and supply from various plants along with transportation cost per unit between plants and sales outlets are given in the following table.  Determine the Initial Basic Feasible Solution (IBFS) by using the Least Cost Method

Plants Sales outlet Supply
A B C D
1 10 2 20 11 15
2 12 7 9 20 25
3 4 14 16 18 10
Demand 5 15 15 15  
  8. A company’s management and labour union are negotiating a new three years settlement.  Each party has four strategies these are:-

I.                     Hard and aggressive bargaining approach.

II.                 Reasoning and logical Approach

 

III.               Legalistic approach

IV.              Conciliatory Approach

The cost to the company for every pair of strategy choices are given in the table below:

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

Which strategy should the management adopt and which strategy should the union adopt? Solve and find the value of the game through the Maximin Minimax principle.

 
  9. Sofitel Plumbing and Heating maintains a stock of 30 –gallon hot water heaters that is sells to homeowners and installs for them.  Owner Mr. Gupta likes the idea of having a large supply on hand to meet customer demand, but he also recognizes that it is expensive to do so.  He examines hot water heater sales over the past 50 weeks and notes the following.

Hot water heater sales per week Number of weeks this number was sold.
4 6
5 5
6 9
7 12
8 8
9 7
10 3
                                                                                               Total:    50

 

a)      Simulate the data and determine the average number of sales per week over a 6- week period?  Random Numbers: 10, 24, 03, 32, 23, 59.

 

b)     If Mr. Gupta maintains a constant supply of 8 hot water heaters in any given week, calculate the closing stock at the end of the 6-week period?

 
  10. “Operation Research is concerned with scientifically deciding how best to design and operate man machine systems usually under conditions requiring the allocation of scare resources.  In light of this statement explain the main features of Operation Research.

 
SECTION – B
II. Answer any THREE questions.  Each carries 15 marks.                                (3×15=45)
  11. Two caters, A & Co. and B & Co. are competing for an increased market share.  The payoff matrix, shown in the following table, describes the increase in market share for A & Co. and decrease in market share of B & Co.

 

B & Company

A & Company Give Coupons Decrease Prize Maintain Present Strategy Increase Advertising
Give Coupons 2 -2 4 1
Decrease Price 6 1 12 3
Maintain Present Strategy -3 2 0 6
Increase Advertising 2 -3 7 1

i)       Determine optimal strategies for both the manufactures and the value of game by using the Dominance method.

ii)     Determine the probability of company A and B choosing each strategy.
  12.        i) Obtain an IBFS to the following transportation problem by VAM.

ii) Is the solution degenerate?  Test the optimality of the solution thus

Obtained using the MODI method. If not, optimize the solution.

  D1 D2 D3 D4 Ai
O1 19 30 50 10 14
O2 70 30 40 60 18
O3 40 8 70 20 36
Bj 10 16 14 28 68
  13. A city corporation has decided to carry out road repairs on main four entries of the city.  The govt. has agreed to make a special grant of Rs.106 lakhs towards the cost with a condition that the repairs must be done at the lowest cost & quickest time.  If conditions warrant, then supplementary grants will also be considered favorable.  The corporation has floated tenders and 5 contractors have sent in their bids (in lakhs).  In order to expedite work, one road will be awarded to only one contractor:

Contractors/Road R1 R2 R3 R4
C1 18 28 38 30
C2 14 34 40 38
C3 18 36 42 36
C4 20 24 36 38
C5 20 30 42 32

i)                   Find the best way of assigning the repair work to the contractors to reduce the costs.

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

iii)              Which of the five contractors will be unsuccessful in his bid?
  14. A single counter Ticket Booking Centre employs one Booking Clerk.  A Passenger on arrival immediately goes to the Booking Counter for being served if the Counter is free.  If, on the other hand, the Counter is engaged, the Passenger will have to wait.  The Passengers are served on first come first served basis.  The time of arrival and the time of service varies from 1 minute to 6 minutes.  The distribution of arrival and services time is as under:

Arrival/Service time(Minutes) Arrival (Probability) Service (Probability)
1 0.05 0.10
2 0.20 0.20
3 0.35 0.40
4 0.25 0.20
5 0.10 0.10
6 0.05 –
  1. Simulate the arrival and service of 10 passengers starting from 9 A.M. by using the following random numbers in pairs respectively for arrival and services.  Random Numbers (60 09) (16 12) (08 18) (36 65) (38 25) (07 11) (08 79) (59 61) (53 77) (03 10).
  2. Determine the total duration of Idle time of Booking Clerk and Waiting time of passengers.

 
  15. Solve by Simplex method the model given below

Max Z = 22x1 + 18 x2

 

Subject to constraints

x1 +x2                     ≤ 20

360x1 + 240x2 ≤ 5760

Where x1  , x2  ≥ 0
SECTION – C
III. Case Study                                                                                                              (1×20=20)
  16. A small project consists of seven activities, whose time estimates are given in the following:

Activity      : 1-2 1-3 1-4 2-5 3-5 4-6 5-6
to                          1 1 2 1 2 2 3
tm                         1 4 2 1 5 5 6
tp                         7 7 8 1 14 8 15

a)       Determine the expected time and variance for each activity.

b)     Draw the network and determine the project duration and critical path.

c)      Determine the total float, free float and independent float for each activity.

d)     What is the probability that the project is completed 4 days earlier than expected?

e)      What is the probability that the project is completed 4 days later than expected?

 

 

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