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Linear Programming:LINEAR PROGRAMMING PROBLEM

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Operations Research (MTH601)
96
L
LINEAR PROGRAMMING PROBLEM
Let
xi = decision variable for ith variable.
ci = profit or cost co-efficient of ith variable.
Z = function to be maximized or minimized.
Thus for n decision variables, the objective function is to maximize or minimize
Z = c1 x1 + c2 x2 + ... + cn xn
aij = co-efficient of the jth constraint and ith variable
Let
bi = resource limitation for ith constraint
Thus the restrictions may be expressed in the general form
a11x1 + a12x2 + ... + a1nxn < b1
a21x1 + a22x2 + ... +a2nxn < b2
:
:
:
:
am1x1 + am2x2 + ...+ amnxn < bm
and xi > 0 for all values of i from 1 to n
The same linear programming problem can be expressed in a more condensed form using summation
notation or matrix equation.
n
Z = ∑ ci xi
Maximize
i =1
n
∑ aij x  j
for all i = 1, 2, ..., m
Subject to
j =i
and xj > 0 for all j = 1, 2, ..., n
or
Maximize Z = CX , subject to AX < B
where C is a row vector and X and B are column vectors. A is a co-efficient matrix of the order m x n.
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Operations Research (MTH601)
97
The linear programming problem may have an objective function to minimize cost also.
The inequalities may be "greater than or equal" instead of "less than or equal". Further in some
cases, the restrictions involve "equalities".
The successful application of linear programming is the ability to recognize that the problem can be
formulated as a linear programming model.
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Operations Research (MTH601)
98
REVIEW QUESTIONS
1.
What do you understand by Linear Programming problem?
2.
Explain how linear programming can be applied to management problems.
3.
Explain the terms: objective function and restrictions in relation to linear programming problem.
4.
Give a mathematical format in which a linear programming problem is expressed.
5.
Enumerate the limitations of linear programming problem.
6.
In relation to linear programming explain the implications of the following assumptions of the
model.
·
Linearity for the objective function and constraints.
·
Continuous variables.
·
Certainty.
7.
Discuss in brief linear programming as a technique for resource utilization.
8.
A company makes products A, B, C and D which flow through four departments: Drilling, Milling,
Lathe and Assembly. The variable time per unit of different products are given below in hours:
Product
Drilling
Milling
Lathe
Assembly
A
3
0
3
4
B
7
2
4
6
C
4
4
0
5
D
0
6
5
3
The unit contribution of the four products and hours of availability in the four departments are:
Product
Contribution/Unit
Department
Hours
Rs.
Available
A
9
Drilling
70
B
18
Milling
80
C
14
Lathe
90
D
11
Assembly
100
Formulate a linear programme for maximizing the contribution.
9.
A pension fund manager is considering investing in two shares A and B. It is estimated that:
(i) Share A will earn a dividend of 12% per annum and share B 4 % per annum.
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Operations Research (MTH601)
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(ii) Growth in the market value in one year of share will be 10 paise per Re. 1 invested and in B 40
paise per Re. 1 invested.
He requires investing the minimum total sum, which will give
·
dividend income of at least Rs. 600 per annum and
·
growth in one year of atleast Rs. 1000 on the initial investment.
you are required to state the mathematical formulation of the problem.
10.
A manufacturer uses three raw products a, b, c priced at 30, 50, 120 rupees per kg respectively. He
can make three different products A, B and C, which can be sold at 90, 100, 120 rupees per kg
respectively. The raw products can be obtained only in limited quantities, namely 20, 15 and 10 kg
per day. Given: 2 kg of a plus 1 kg of b plus 1 kg of c will yield 4 kg of A; 3 kg of a plus 2 kg of b
plus 2kg of c will yield 7 kg of B; 2kg of b plus 1 kg of c will yield 3 kg of C.
Make a production plan, assuming that the other costs are not influenced by the
choice among the alternatives. Formulate the model of the problem.
11.
A marketing manager wishes to allocate his annual advertising budget of Rs. 20,000 in two media
vehicles A and B. The unit of a message in media A is Rs. 1000 and that of B is Rs. 1500, Media A
is a monthly magazine and not more than one insertion is desired in one issue. At least 5 messages
should appear in media B. The expected effective audience for unit messages in the media A is 40,
000 and media B is 55, 000.
·
Develop a mathematical model.
·
Solve for maximizing the total effective audience.
12.
A company produces four products 1 to 4. Raw material requirements, storage space needed,
production rates and profits are given in the table below. The total amount of raw material
available per day for all four products is 180 kg, total space available for storage is 230 sq. mtr.
and 7 hours/day is used for production.
1
2
3
4
Raw materials (Kg/piece)
2
2
1.5
4
Space (sq. mtr./piece)
2
2.5
2
1.5
Production rate (pieces/hr)
15
30
10
15
Profit (Rs./piece)
55
56.6
55
55.5
How many units of each product should be produced to maximize total profit?
13.
A ship has three cargo holds, forward, aft and centre.
The capacity limits are:
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Operations Research (MTH601)
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Forward
2000 tons
1000 cubic meter
Centre
3000 tons
1350 cubic meter
Aft
1500 tons
300 cubic meter
The following cargoes are offered; the ship owners may accept all or any part of each commodity.
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Table of Contents:
  1. Introduction:OR APPROACH TO PROBLEM SOLVING, Observation
  2. Introduction:Model Solution, Implementation of Results
  3. Introduction:USES OF OPERATIONS RESEARCH, Marketing, Personnel
  4. PERT / CPM:CONCEPT OF NETWORK, RULES FOR CONSTRUCTION OF NETWORK
  5. PERT / CPM:DUMMY ACTIVITIES, TO FIND THE CRITICAL PATH
  6. PERT / CPM:ALGORITHM FOR CRITICAL PATH, Free Slack
  7. PERT / CPM:Expected length of a critical path, Expected time and Critical path
  8. PERT / CPM:Expected time and Critical path
  9. PERT / CPM:RESOURCE SCHEDULING IN NETWORK
  10. PERT / CPM:Exercises
  11. Inventory Control:INVENTORY COSTS, INVENTORY MODELS (E.O.Q. MODELS)
  12. Inventory Control:Purchasing model with shortages
  13. Inventory Control:Manufacturing model with no shortages
  14. Inventory Control:Manufacturing model with shortages
  15. Inventory Control:ORDER QUANTITY WITH PRICE-BREAK
  16. Inventory Control:SOME DEFINITIONS, Computation of Safety Stock
  17. Linear Programming:Formulation of the Linear Programming Problem
  18. Linear Programming:Formulation of the Linear Programming Problem, Decision Variables
  19. Linear Programming:Model Constraints, Ingredients Mixing
  20. Linear Programming:VITAMIN CONTRIBUTION, Decision Variables
  21. Linear Programming:LINEAR PROGRAMMING PROBLEM
  22. Linear Programming:LIMITATIONS OF LINEAR PROGRAMMING
  23. Linear Programming:SOLUTION TO LINEAR PROGRAMMING PROBLEMS
  24. Linear Programming:SIMPLEX METHOD, Simplex Procedure
  25. Linear Programming:PRESENTATION IN TABULAR FORM - (SIMPLEX TABLE)
  26. Linear Programming:ARTIFICIAL VARIABLE TECHNIQUE
  27. Linear Programming:The Two Phase Method, First Iteration
  28. Linear Programming:VARIANTS OF THE SIMPLEX METHOD
  29. Linear Programming:Tie for the Leaving Basic Variable (Degeneracy)
  30. Linear Programming:Multiple or Alternative optimal Solutions
  31. Transportation Problems:TRANSPORTATION MODEL, Distribution centers
  32. Transportation Problems:FINDING AN INITIAL BASIC FEASIBLE SOLUTION
  33. Transportation Problems:MOVING TOWARDS OPTIMALITY
  34. Transportation Problems:DEGENERACY, Destination
  35. Transportation Problems:REVIEW QUESTIONS
  36. Assignment Problems:MATHEMATICAL FORMULATION OF THE PROBLEM
  37. Assignment Problems:SOLUTION OF AN ASSIGNMENT PROBLEM
  38. Queuing Theory:DEFINITION OF TERMS IN QUEUEING MODEL
  39. Queuing Theory:SINGLE-CHANNEL INFINITE-POPULATION MODEL
  40. Replacement Models:REPLACEMENT OF ITEMS WITH GRADUAL DETERIORATION
  41. Replacement Models:ITEMS DETERIORATING WITH TIME VALUE OF MONEY
  42. Dynamic Programming:FEATURES CHARECTERIZING DYNAMIC PROGRAMMING PROBLEMS
  43. Dynamic Programming:Analysis of the Result, One Stage Problem
  44. Miscellaneous:SEQUENCING, PROCESSING n JOBS THROUGH TWO MACHINES
  45. Miscellaneous:METHODS OF INTEGER PROGRAMMING SOLUTION