LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.A., DEGREE EXAMINATION – ECONOMICS

## THIRD SEMESTER – NOVEMBER 2003

**ST-3100/STA100 – RESOURCE MANAGEMENT TECHNIQUES**

13.11.2003 Max:100 marks

9.00 – 12.00

__SECTION-A__

__Answer ALL questions.__ (10×2=20 marks)

- Define optimal feasible solution.
- Mention any two disadvantages of using graphical method to solve a LPP.
- When should an artificial variable be introduced in LPP?
- Convert the following LPP into standard form:

Max. Z= 3x_{1} + 5x_{2} + 8x_{3}

S.t. 2x_{1} + 5x_{2} ≥ 10;

3x_{1} + 4x_{3} ≤ 25;

x_{1,} x_{2,} x_{3} ≥ 0.

- How will you covert a maximization type transportation problem into a minimization type?
- Mention the difference between transportation and assignment problems.
- What are the conditions under which a sequencing problem involving 3 machines can be converted to a problem of 2 machines?
- Define critical path.
- Define Total float.
- Give the formula for the cost equation in a single item EOQ model.

__SECTION-B__

__Answer any FIVE questions.__ (5×8=40 marks)

- Solve graphically the following LPP.

Minimize. Z= 3x_{1} + 2x_{2}

s.t. 5x_{1} + x_{2} ≥ 10; 2x_{1} + 2x_{2} ≥ 12; x_{1} + 4x_{2} ≥ 12; x_{1,} x_{2,} ≥ 0.

- Using Big M method, show that the following LPP does not possess any feasible solution.

Max. Z= 3x_{1} + 2x_{2}

s.t. 2x_{1} + x_{2} ≤ 2; 3x_{1} + 4x_{2} ≥ 12; x_{1,} x_{2,} ≥ 0.

- Find an initial feasible solution for the transportation problem using (i) North-West Corner rule (ii) Least cost Method.

Destination Availability

Origin D_{1} D_{2} D_{3} D_{4 }

O1 1 2 1 4 30

O2 3 3 2 1 50

O3 4 2 5 9 20

Requirement 20 40 30 10

- The following table gives the cost of transporting materials from supply points A,B,C,D to demand points E,F,G,H and J.

E F G H J

A 8 10 12 17 15

B 15 13 18 11 9

C 14 20 6 10 13

D 13 19 7 5 12

The present allocation is as follows:

AE : 90 units; AF : 10 units

BF : 150 units ; CF : 10 units

CG : 50 units ; CJ : 120 units

DH : 210 units ; DJ : 70 units

Check whether the above allocation is optimum. If not, find an optimum schedule.

- The following matrix shows the profit (in Rs.) of assigning various jobs to different machines Assign the jobs the machines so as to maximize the total profit.

__Machines__ __Jobs__

I II III IV V

1 5 11 10 12 4

2 2 4 6 3 5

3 3 12 5 14 6

4 6 14 4 11 7

5 7 9 8 12 5

- Determine the optimal sequence of jobs which minimizes the total elapsed time based on the following information. Also find the total elapsed time.

Job Processing time (in mts) of Machines

A_{i} B_{i} C_{i}

1 3 3 5

2 8 4 8

3 7 2 10

4 5 1 7

5 2 5 6

- Calculate (i) total float for each activity (ii) Critical path and its duration for the following network.

- Define the following :

(i) Overstock (ii) Lead time (iii) Price Break (iv) Set up cost.

__SECTION-C__

__Answer any TWO questions.__ (2×20=40 marks)

- a) Three products are processed through three different operations. The time (in mts)

required per unit of each product, the daily capacity of the operations (in mts per day)

and the profit per unit sold for each product (in Rs.) are as follows:

Time per units (in mts) Operation capacity

Operation Product I Product II Product II (mts / dat)

1 3 4 3 43

2 5 0 4 46

3 3 6 2 42

Profit/unit(inRs.) 2 1 3

The problem is to determine the optimum daily production for the products that maximizes the profit. Formulate the above production planning problem as a LPP.

- b) Solve the following LPP using simplex method.

Minimize Z = x_{1} – x_{2} + x_{3} + x_{4} + x_{5} – x_{6}

s.t. x_{1} + x_{4} + 6x_{6} = 9;

3x_{1} + x_{2} – 4x_{3} + 2x_{6} = 2;

x_{1} + 2x_{3} + x_{5} + 2x_{6} = 6;

x_{i,} ≥ 0 for i = 1, 2, …, 6. (5+15)

- a) Explain the problem of transportation with an example.
- b) Solve the following transportation problem for minimizing the costs.

Destination Availability

Origin D_{1} D_{2} D_{3} D_{4 }

O1 2 3 11 7 6

O2 1 0 6 1 1

O3 5 8 15 9 10

Demand 7 5 3 2

Use Vogel’s method to get the initial feasible solution. (5+15)

- a) Average time taken by an operator on a specific machine is tabulated below. The

management is considering to replace one of the old machines by a new one and the

estimated time for operation by each operator on the new machine is also indicated.

Machines

Operators 1 2 3 4 5 6 New

A 10 12 8 10 8 12 11

B 9 10 8 7 8 9 10

C 8 7 8 8 8 6 8

D 12 13 14 14 15 14 11

E 9 9 9 8 8 10 9

F 7 8 9 9 9 8 8

- Find out an allocation of operators to the old machines to achieve a minimum operation time
- Reset the problem with the new machine and find out the allocation of operators to each machine and comment on whether it is advantageous to replace an old machine by the new one.

- b) Solve the following assignment problem with restrictions.

__Jobs__ __Machines__

I II III IV V

1 ¥ 4 7 3 4

2 4 ¥ 6 3 4

3 7 6 ¥ 7 5

4 3 3 7 ¥ 7

5 4 4 5 7 ¥ (14+6)

- a) The following table list the jobs of a network along with their time estimates (in days):

Job: (1,2) (1,6) (2,3) (2,4) (3,5) (4,5) (6,7) (5,8) (7,8)

Optimistic: 3 2 6 2 5 3 3 1 4

Most Likely: 6 5 12 5 11 6 9 4 19

Pessimistic: 15 14 30 8 17 15 27 7 28

- Draw the project network diagram.
- Calculate the expected task time and variances for each job.
- Find the critical path.

- b) A manufacturer has to supply his customer with 600 units of his product per year.

Shortages are not allowed and the storage cost amounts to Rs.0.60 per unit per year.

The set up cost per run is Rs.80/-. Find the optimum run size and minimum average

yearly cost. (14+6)

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