- JOSEPH’S COLLEGE OF COMMERCE (Autonomous)
End Semester Examination – APRIL 2011
M.COM – II SMester
OPERATION RESEARCH FOR BUSINESS DECESIONS
Duration: 3 Hours Max. Marks:100
SECTION – A
I)Answer ALL the questions. (10×2=20)
- How do the graphical and simplex methods of LP differ? How are these same?
- What is simulation?
- How do you determine that there are multiple alternative solutions to an assignment problem? How do you find the alternative solutions?
- For the following equation -2x + y< 1, show the feasible region with the help of a diagram.
- Define the term operation research. Mention any two characteristics of operation research.
- How can degeneracy be resolved at the initial solution and at the any iteration?
- What do you understand by the term feasible solution?
- Explain the term jockeying.
- Explain the term artificial variable.
- Differentiate between Static and Dynamic Models.
SECTION – B
- II) Answer any FOUR questions (4×5=20)
- The simplex tableau for a maximisation problem of linear programming is given here
| Product mix Cj | Basic variables | X1 | X2 | S1 | S2 | Quantity |
| 5 | X2 | 1 | 1 | 1 | 0 | 10 |
| 0 | S2 | 1 | 0 | -1 | 1 | 3 |
| Cj | 4 | 5 | 0 | 0 | ||
| Zj | 5 | 5 | 5 | 0 | 50 | |
| Cj-Zj | -1 | 0 | -5 | 0 | ||
Answer the following and give reasons in brief-
- Is the solution optimal?
- Are there more than one optimal solution?
- Is the solution feasible?
- How many units of the two products X1 and X2 are being produced according to this solution and what is the total profit?
- A student has to select one and only one elective in each semester and the same elective should not be selected in the different semesters. Due to various reasons the expected grades in each subject, if selected in different semesters vary and they are given below
Semester Analysis Statistics Graph theory Algebra
I F E D C
II E E C C
III C D C A
IV B A H H
The grade points are : H:10, A=9, B=8, C=7, D=6, E=5, F=4. How will the students elect the electives in order to maximise the total expected points and what will be his maximum expected total points.
- What is dynamic programming? What are the essential characteristics of a dynamic programming problem?
- Let us assume that you have inherited Rs 1, 00,000 from your uncle that can be invested in a combination of only two stock portfolios, with the maximum investment allowed in either portfolio set at Rs 75,000. The first portfolio has an average rate of return of 10%, whereas the second has 20%. In terms of risk factors associated with these portfolios, the first has a risk rating of 4(on a scale from 0 to 10) and the second has 9. Since you wish to maximise your returns, you will not accept an average rate of return below 12% or a risk factor above 6(both taken together). Hence you then face an important question. How much should you invest in each portfolio? Formulate this as a linear programming problem and solve it by graphic method.
- A) Differentiate between seasonal and cyclical trends.
- b) Use the method of least squares to fit a trend line for the income from the following data. Estimate the income for the year 2003.
Year 1995 1996 1997 1998 1999 2000 2001 2002
Income 85 90 110 100 95 120 125 115
- Find the initial basic feasible solution to the following transportation problem using VAM given the cost matrix.
D1 D2 D3 D4 SS
S1 20 25 28 31 200
S2 32 28 32 41 180
S3 18 35 24 32 110
DD 150 40 180 170
SECTION – C
III) Answer any THREE questions. (3×15=45)
- The manager of a book store has to decide the number of copies of a particular tax law book to order. A book costs Rs 60 and is sold for Rs 80. Since some of the tax laws change year after year, any copies unsold while the edition is current must be sold for Rs 30. From past records, the distribution for this book has been obtained as follows:
| Demand No of copies | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
| Proportion | 0.05 | 0.08 | 0.20 | 0.45 | 0.10 | 0.07 | 0.03 | 0.02 |
Using the following sequence of random numbers, generate data on demand for 10 time period (years). Calculate the average profit obtainable under each of the courses of action to the manager. If producer for production of a) 15 books b) 20 books c) 18 books. What is the optimal policy?
| 14 | 02 | 93 | 99 | 18 | 71 | 37 | 30 | 12 | 10 |
| 88 | 12 | 00 | 57 | 69 | 32 | 18 | 08 | 92 | 72 |
- A self-service store employs one cashier at its counter. 9 customers arrive on an average every 5 minutes while the cashier can serve 10 customers in 5 minutes. Assuming Poisson distribution for arrival rate and exponential distribution for service rate. Find –
- Average number of customers in the system
- Average number of customers in the queue
- Average time a customer spends in the system
- Average time a customer waits before being served.
- The following table shows all the necessary information on the availability of supply to each warehouse, the requirement of each market and unit transportation cost (in Rs) from each warehouse to each market
market
|
warehouse |
P | Q | R | S | Supply | |
| 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 has worked out the following schedule from experience: 12 units from A to Q, 1 unit from A to R, 9 units from A to S, 15 units from B to R, 7 units from C to P, and 1 unit from C to R.
- Check and see if the clerk has the optimal schedule.
- Find the optimal schedule and minimum total transport cost.
- If the clerk is approached by a carrier of route C to Q who offers to reduce his rate in the hope of getting business, by how much the rate should be reduced before the clerk will offer him the business.
- The ABC Company combines factors X and Y to form a product which must weigh 50 kgs. At least 20 kgs and no more than 40 kgs of Y can be used. X costs Rs 25 and Y Rs 10 per kg. Find out the amount of factor X and Y to be used manufacturing the product. Use Simplex Method.
- “Executives at all the levels in business and industry come across the problem of making decisions at every stage in their day to day activities. Operation research provides them with various quantitative techniques for decision making and enhances their ability to make long range plans and solve everyday problems of running a business and industry with greater efficiency, competence and confidence. “Elaborate the statement with examples.
SECTION – D
- IV) Compulsory question (1×15=15)
- As the Captain of the Indian Cricket team you have to allot five middle batting positions to five batsmen. The average runs scored by each batsman at these positions are as follows:
| BATSMEN BATTING POSITIONS | |||||
|
|
I | II | III | IV | V |
| Sachin | 40 | 40 | 35 | 25 | 50 |
| Dhoni | 42 | 30 | 16 | 25 | 27 |
| Gambhir | 50 | 48 | 40 | 60 | 50 |
| Sehwag | 20 | 19 | 20 | 18 | 25 |
| Yuvraj | 58 | 60 | 59 | 55 | 53 |
- Find the assignment of batsman to positions which would give the maximum number of runs.
- If another batsman Zaheer with the following average runs in the batting positions are given below:
| Batting positions | I | II | III | IV | V |
| Average runs | 45 | 52 | 38 | 50 | 49 |
Is added to the team, should be included to play in the team, if so who should be replaced by whom?