Download Linear Programming Problems and Solutions and more Exams Management Fundamentals in PDF only on Docsity!
ARHOLIADAU EXAMINATIONS
Ionawr 2011 January 2011
AC31210 ANALYTICAL TECHNIQUES FOR MANAGEMENT CONTROL
Time allowed: TWO hours
Answer THREE questions only. All questions carry equal marks.
Graph paper will be provided
Only Casio FX83ES/GT or FX85ES/GT calculators may be used
QUESTION 1
A small manufacturer produces two products P and Q. Each product has to be processed in three departments: mechanical, electrical and assembly. Each unit of P requires 12 minutes in the mechanical department, 6 minutes in the electrical department and 3 minutes in assembly. The corresponding times for each unit of Q are 6 minutes, 9 minutes and 3 minutes respectively. Each month there are 120 hours available in the mechanical department, 90 hours available in electrical and 35 hours available in assembly. Assume that no other resources are needed. Each unit of P contributes £5 to the monthly profit and each unit of Q contributes £3. The manager wishes to know how many units of P and Q the company should produce in order to maximise the monthly profit.
Required:
a) Formulate this scenario as a linear programming problem. (5 marks)
b) Write down the initial tableau for a solution to the problem by the simplex algorithm. (5 marks) c) Find the pivot element and compute the second tableau. (5 marks)
d) The final tableau is given below (units are minutes and pounds):
Variable p q S 1 S 2 S 3 Solution p 1 0 1/6 0 -1/3 500 S 2 0 0 1/2 1 -4 600 q 0 1 -1/6 0 2/3 200 Profit 0 0 1/3 0 1/3 3100
Interpret the final tableau and advise the company of its best strategy. Include a discussion of the shadow prices and any unused processing time. (10 marks) (Total 25 marks)
Turn Over
QUESTION 2
Consider the following primal linear programming problem:
Maximise: P = 2.5 x 1 +2 x 2 Subject to: x 1 (^) 2 x 2 8000
3 x 1 (^) 2 x 2 9000 x 1 (^) , x 2 0
Required:
a) Write down the dual problem and explain the relationship between the solutions of the dual and primal problems. (5 marks)
b) Solve the dual problem graphically – showing the feasible region clearly on your graph. (8 marks)
c) Determine algebraically the shadow prices for the dual problem. (5 marks)
d) Use the solution of the dual problem to find the solution to the primal problem. (2 marks)
e) Compare and contrast alternative methods that can be used to solve optimization problems. (5 marks) (Total 25 marks)
QUESTION 3 – Answer both parts (A) and (B) (A) A company needs to deliver 125 units of goods to 4 supermarkets from 3 storage depots. The number of units to be delivered to the 4 supermarkets (S1, S2, S3 and S4) is 45, 20, 30 and 30 respectively. The number of units of goods available from each of the 3 storage depots (D1, D2 and D3) is 35, 50 and 40 respectively. The transportation costs (in £ per unit ) from each depot to each supermarket are given below. x ij denotes the unknown number of units.
Storage Depots Supermarkets S1 S2 S3 S4 Supply D 1 1
8 x (^) 12
x (^) 13
x (^) 14
x^35 D 21
x (^) 22
x (^) 23
x (^) 24
x^50 D 31
x (^) 32
x (^) 33
x (^) 34
x^40 Demand 45 20 30 30 125
Question Continues Overleaf Turn Over
QUESTION 4
In a game two players, A and B, choose a number 1, 2 or 3. If both choose the same number, A pays to B the amount of the chosen number in pounds. Otherwise, B pays to A the amount A chooses in pounds.
Required :
a) Write down the payoff matrix for player A. (2 marks)
b) Formulate A’s problem in selecting a mixed strategy as a linear programming problem (6 marks)
c) Write down the linear programme representing B’s problem. (4 marks)
d) Determine the best mixed strategies for A and B and the value of the game. (8 marks)
e) Describe an example of the game referred to as “The Prisoner’s Dilemma” and discuss how this type of game can be interpreted in economic situations. (5 marks) (Total 25 marks)
QUESTION 5 A small company produces five different types of computer chip. Each type must pass through two manufacturing processes and then be assembled by hand. The following table shows the number of minutes that each type of chip takes in each of the processes, the number of minutes of labour required to assemble each chip and the amount of profit each chip brings.
Chip A B C D E Process 1 12 20 0 25 15 Process 2 10 8 16 0 0 Labour 20 19 21 18 22 Profit(pence) 55 60 35 40 20
The company wishes to know how many chips are needed to be manufactured in any week to maximise profit. 440 man-hours of labour are available each week together with 408 hours of Process 1 and 272 hours of Process2.
Required: a) Assuming that unlimited supplies of raw materials are available, formulate this as a linear programming problem. (5 marks)
Using the attached Excel Solver output answer the following questions:
b) State the optimal numbers of each chip which should be made per week and the maximum profit for the company. (4 marks)
c) Explain the terms binding , non-binding and slack in relation to this problem. (6 marks)
d) Interpret the sensitivity report_._ (10 marks) (Total 25 marks)
Turn Over
Solver Output Answer Report
Target Call (Max) ___ _ Cell Name Original Value Final Value_ $n$6 0 803.
Adjustable Cells Cell Name Original Value Final Value $I$6 A 0 366 $J$6 B 0 1004 $K$6 C 0 0 $L$6 D 0 0 $M$6 E 0 0
Constraints Cell Name Cell Value Formula Status Slack $J$9 Process 2 11693.02326 $J$9<=16320 Not Binding 4626. $J$8 Process 1 24480 $J$8<=24480 Binding 0 $J$10 Labour 26400 $J$10<=26400 Binding 0
Sensitivity Report
Adjustable Cells ________ Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $I$6 A 366 0 0.55 0.081578947 0. $J$6 B 1004 0 0.6 0.077777778 0. $K$6 C 0 -0.113953488 0.35 0.113953488 1E+ $L$6 D 0 -0.222965116 0.4 0.222965116 1E+ $M$6 E 0 -0.42122093 0.2 0.42122093 1E+
Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H Side Increase Decrease $J$9 Process 2 11693.02326 0 16320 1E+30 4626. $J$8 Process 1 24480 0.2234 24480 3309.473684 8640 $J$10 Labour 26400 0.3213 26400 7651.307692 3144
END OF PAPER
