Linear Programming

Linear Programming

Dr. T. T. Kachwala

Introduction to Linear Programming

Linear programming is a method of mathematical

programming that involves optimization of a certain function

called objective function subject to defined constraints and

restrictions.

Although it has very wide applications, the common type of

problems handled in linear programming call for

determining a product-mix that would maximize the total

profit, given the profit rates of the products involved and the

resource requirements for each of them, along with the

amount of resources available.

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Introduction to Linear Programming

Thus, linear programming method is a technique for choosing the best alternative from a set of feasible alternatives, in situations where the objectives function, as well as the constraints, are expressed as linear mathematical functions.

Generally, the constraints in maximization problems are of the `<’ type, and in the minimization problems of the `>’ type. But a given problem may involve a mix of constraints involving the signs of ‘<’, ‘>’, and / or ‘=’. Usually the decision variables are non-negative.

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Example of Product Mix Problem

M/s. P.M.S. Industries makes two kinds of leather purses for ladies. Purse type ‘A’ is of high quality and purse type ‘B’ is of lower quality. Contributions per unit were Rs.4 and Rs.3 for purse type ‘A’ and type ‘B’ respectively. Each purse of type ‘A’ requires two hours of machine time per unit & that of type ‘B’ requires one hour per unit. The company has 1000 hours per week of maximum available machine time. Supply of leather is sufficient for 800 purses of both types combined per week. Each type requires the same amount of leather. Purse type ‘A’ requires a fancy zip and 400 such zips are available per week, there are 700 zips for purse type ‘B’ available per week. Assuming no market or finance constraints recommend an optimum product mix.

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Example of Product Mix problem

1. Three types of data are defined in Product Mix Problem:

a. Requirement of resources per unit of product

b. Availability of resources, i.e., constraints of linear

programming problem

c. Objective function

2. The product mix problem is defined as follows; “How many units of

each type of product should be manufactured and sold, given the

requirement of resources within the constraints of maximum

availability such that the profit contribution is maximized.”

Example of Marketing Mix problemSuppose a media specialist has to decide on allocation of advertisement in three

media vehicles. The unit costs of a message in the three media are Rs.1000,

Rs.750 & Rs. 500. The available budget is Rs.20,000 for a campaign period of one

year. The first medium is a monthly magazine and it is desired to advertise not more

than one insertion in one issue. Also at least six messages should appear in the

second medium. The number of messages in the third medium should strictly lie

between 4 & 8. The expected effective audience for unit message in the media

vehicle is shown below :------------------------------------------------------------------------------Vehicle : 1 2 3------------------------------------------------------------------------------Expected effective Audience : 80000 60000 45000------------------------------------------------------------------------------

Draft this as an LP problem to find the optimal allocation that would maximize total

effective audience.

Example of Investment Mix problem

The Boffin Investment Trust wishes to invest Rs.10,00,000. There are five different

investment choices. The current returns on investments are as follows:------------------------------------------------------------------------------------Investment choice : L M N O P------------------------------------------------------------------------------------Annual yield % : 10 8 6 5 9------------------------------------------------------------------------------------

Because of risk element involved, management restricts the investment in L to not

more than the combined total investment in N, O and P. Total investment in M & P

combined must be at least as large as that in N. Also management wishes to restrict

its investment in M to a level not exceeding that of O.

Construct an LP model to determine the optimum allocation of investment funds

amongst these 5 choices.

Formulation of Linear Programming Problem

Formulation of LPP means translating the problem

statements in to mathematical equations.

All solutions to linear programming problem starts with

formulation. This involves:

1. Defining the decision variables

2. Imposing the restrictions on the decision variables

as defined by the constraints

3. Defining the objective function in terms of the

decision variables

Significance of the title Linear Programming

The word Linear signifies that all equations are linear,

i.e., variables are raised to power of index 1(alternately

this means that Linear Programming can only be

applied to Business applications that can be expressed

as a linear equation – Limitation of Linear

Programming)

Programming is interpreted as a set of logical

instructions (iteration – step by step improvement)

which are repeatedly applied to the problem till we

generate an optimum solution.

Graphical Solution to the LPP

Solution of LPP graphically requires plotting all the constraints of the problem. After this, the feasible region for the constraints is identified. The feasible region for the problem is the area that is common to all the constraints. It thus represents the region in which any point would satisfy all the constraints.

The optimal solution may be found by determining extreme points of the feasible region (the variable values at each point, and the corresponding objective function value). The pair of values of decision variables that optimizes, that is, maximizes for a maximization problem and minimizes for a minimization problem, is taken to be the optimal solution.

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Graphical Solution to Linear Programming Working Rules and Guidelines

1. Formulation of Linear programming problem: This involves:

a. defining the decision variables

b. imposing the restrictions on the decision variables as defined by the constraints

c. defining the objective function in terms of the decision variables.

2. Draw the graph corresponding to each equation of LPP.

3. Identify feasible solution area and the corresponding corner points

4. Calculate the value of the objective function (Z or C) corresponding to each corner

point and identify the corner point corresponding to the optimum value of the

objective function. Such a corner point represents optimum solution for the LPP.

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Simplex (Mathematical) Solution to Linear Programming Working Rules and Guidelines

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Formulate the LP problem: (same as Graphical Solution)1. identification of decision variables.2. imposing the restrictions on the decision variables as defined

by the constraints of LPP.3. defining the objective function in terms of the decision

variables.

Convert the inequalities into “=” sign: If the inequalities is of “≤” type, then add a slack variable

‘S’. If the inequalities is of “≥” type, then add an artificial

variable ‘A’ and subtract a surplus variable ‘S’. If it is “=” sign then add artificial variable ‘A’. This is done to form a unit matrix and thereby help to get an

initial basis.


Define the objective function (in terms of all the variables):

If the problem is of the minimization type, then the objective

function is given by C = c1 x1 + c2 x2 + ----, where c1, c2 --- are the

cost coefficients for the decision variables x1, x2 ------.

However, if the problem is of the maximization type, then the

objective function is given by Z = p1 x1 + p2 x2 + --, where p1, p2

----- are the profit coefficients for the decision variables x1,x2 ---.

Convert the objective function to minimization by multiplying the

profit function by ‘-1’. We obtain the equivalent cost function as

follows. C = -p1x1 – p2x2 ----.

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“-pi / ci” are referred as associate cost coefficients. On similar lines the associate cost coefficient for Slack variable “S” is zero & Artificial variable “A” is M. The associated cost coefficient is indicated in the table below:

Variable Profit Cost

Xi -pi ci

Si 0 0

Ai M M


Simplex table (Iterations):

1. Layout the initial Simplex table as per the standard format

2. Calculate the net cell evaluation for all the variables listed in

the Simplex table. It will be observed that the net cell

evaluation for some of the variables is greater than or equal to

zero. This condition implies that the initial Simplex table is not

an optimum solution

3. Initial Simplex table is only a reference table

4. Modify the initial Simplex table through a process of iteration (step

by step improvement)

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Simplex table (Iterations):

1. Modification of the initial Simplex table through a process of

iteration involves some mathematical calculations

(Replacement ratio, Gauss-Jordan rules for new coefficients

etc.).

2. The process continues till we reach the Optimum Simplex

Table (such that the net cell evaluation of all the variables in the

simplex table are ≤ 0).

Calculate the minimum cost / maximum profit from the objective

function for the values of the decision variables corresponding to

optimum solution.

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Unique, Multiple, Infeasibility and Unbounded Solutions

A problem may have a unique optimal solution, multiple optimal solutions, an unbounded solution,

or infeasible solution:

1. Unique optimal solutions means one optimum solution is available for LPP.

2. Multiple optimal solutions means more than one optimum solution is available for

LPP.

3. Unbounded solution is present when the feasible region is unbounded from above and the

objective function is of maximization type, so that it is possible to increase the objective

function value indefinitely. Optimum solution is not available. A minimization problem with

non-negative variables will not have unbounded solution.

4. Infeasibility (no feasible solution) exists when there is no common point in the feasible areas

for the constraints of a problem. The feasible region is empty in such a case. Optimum

solution is not available for such a LPP.

Slide 17

Linear Programming

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