LINEAR PROGRAMMINGLINEAR PROGRAMMING

ByAVINASHPRADEEP SINGH JOSHUA

What is Linear What is Linear Programming?Programming?Programming is a technique which is

applied for finding maximum or minimum values in problems confronting the decision making authorities subject to certain constraints.

Therefore it helps to find out maximum or minimum values when side conditions are inequalities and not equations. So, when side conditions are inequalities, the inequalities should be of first degree (i.e. linear), the method of solving problem is known as Linear programming method.

Definition of Linear Definition of Linear Programming:Programming:It is defined as a widely used

mathematical modeling technique to determine the optimum allocation of scarce resources among competing demands.

Resources – Machinery, Time, Money, Raw materials, manpower, etc.

Linear programming method is applied to many different types of real business problems in areas like Finance, Production, Sales and distribution, Personnel, Marketing and many more areas of management.

Properties of Linear Properties of Linear programming modelprogramming model

The following properties form the linear programming model:

1)Relationship among decision variables must be linear in nature.

2)A model must have an objective function.

3)Resource constraints are essential.4)A model must have a non-negativity

constraint.

Formulation of Linear Formulation of Linear Programming:Programming:

Linear Programming method in its standard form involves three parts.

1)Objective function.2)Constraints3)Non negativity constraint.

OBJECTIVE FUNCTION:

The objective function represents the aim or goals of the system.

The objective function should be expressed as a linear function of decision variables.

The objective of the problem is identified and converted into a suitable objective function.

The objective refers to the aim to optimize i.e. maximize the profits or minimize the costs.

For example, Assume that a furniture manufacturer

produces tables and chairs. If the manufacturer wants to maximize his profits, he has to determine the optimal quantity of tables and chairs to be produced.

Let x1 = Optimal production of tables. P1 = Profit from each table sold. x2 = Optimal production of chairs. P2 = Profit from each chair sold.Hence, Total profit from tables = P1 x1 Total profit from chairs = P2 x2

The objective function is formulated as below: Maximize Z or Zmax = P1 x1 + P2 x2

CONSTRAINTS: It states the side conditions on

the different activities of problem.i.e. When availability of resources are

in surplus, there will be no problem in making decisions.

But in real life, organizations normally have scarce resources within which the job has to be performed in the most effective way. Therefore, problem situations are within confined limits in which the optimal solution to the problem must be found.

Considering the previous example of furniture manufacturer,

Let w be the amount of wood available to produce tables and chairs. Each unit of table consumes w1 unit of wood and each unit of chair consumes w2 units of wood.

For the constraint of raw material availability, mathematical expression is,

w1 x1 + w2 x2 ≤ w

NON NEGATIVITY CONSTRAINT: Along with structural constraints, we

have non negativity constraints which assume that negative value of variables is not possible in the solution of Linear Programming problems.

i.e. Producing negative number of chairs and tables is not possible.

So, it is necessary to include the element of non negativity as a constraint.

i.e. x1, x2 ≥ 0.

General Linear Programming General Linear Programming ModelModel

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Let x1, x2 ….. xn represents decision variables.

Standard form of L.P.P Standard form of L.P.P After formulating the Linear programming problem,

the next step is to obtain its solution. In order to obtain the solution, a particular form is used before using any analytical method which is called standard form. The standard form is used to develop the general procedure for solving any linear programming problem.

The main characteristics of standard form are:1)All variables are non-negative.2)The right hand side of each constraint is non-

negative.3)All constraints are expressed as equations.4)Objective function may be of maximization or

minimization type.

Basic assumptions of Linear Basic assumptions of Linear Programming:Programming:

The application of Linear Programming is based on several assumptions,

1)A well defined objective function.2)Presence of constraints in activities.3)Non negativity of decision variables.4)Linear relations in problem.5)Constant prices. – Prices are assumed

to be given and constant. Prices are not considered as variables.

6)Divisibility. – Activity levels are permitted to take fractional or integer values.

7) Proportionality: A basic assumption of linear programming is

that proportionality exists in the objection function and the constraints. This assumption implies that if a product yields a profit of Rs.10, the profit earned from sale of 12 such products will be Rs. (10*12) = Rs 120. This may not always be true because of quantity discounts. Further, even if the sale price is constant, the manufacturing cost may vary with the number of units produced and so may vary the profit per unit.

8) Additivity: It means that if we use t1 hours on machine A

to make product 1 and t2 hours to make product 2,

the total time required to make products 1 and 2 on machine A is (t1 + t2)hours. This, however true only if the change-over time period from product 1 to product 2 is negligible. Some process may not behave in this way.

9) Continuity: Another assumption underlying the linear

programming model is that the decision variables are continuous i.e. they are permitted to take any non-negative values that satisfy the constraints. However, there are problems wherein variables are restricted to have integral values only.

10) Certainty: Another assumption underlying a linear

programming model is that the various parameters, namely, the objective function coefficients, R.H.S coefficients of the constraints and resource values in the constraints are certainly and precisely known and their values do not change with time. Thus the profit or cost per unit of the product, labour and materials required per unit, availability of labour and materials, market demand of the product produced etc. are assumed to be known with certainty. The LPP is therefore assumed to be deterministic in nature.

11) Finite Choices: A linear programming model also

assumes that a finite (limited) number of choices (alternatives) are available to the decision maker and that the decision variables are interrelated and non-negative. The non-negative condition shows that linear programming details with real life situations as it is not possible to produce negative quantities.

Besides those assumptions, there are some implicit assumptions of this technique.

1)The number of activities and constraints should be finite, otherwise solution will not exist.

2)It is assumed that each coefficient used in the relations is known with certainty.

This is often not a realistic assumption.

BASIC TERMS USED IN BASIC TERMS USED IN LPP:LPP:

Feasible solution: It is a solution which specifies such values

to all the variables involved in objective function of problem which would satisfy both types of constraints. i.e. Structural and Non negativity.

Basic solution: It is a solution wherein the number of

variables getting non zero value is equal to the number of structural constraints.

Optimal solution: Optimal solution of any L.P problem lies in

basic feasible solutions, and in such solutions number of non zero valued variables is exactly equal to number of structural constraints in the problem.

Optimal basic feasible solution: It is the basic feasible solution that also optimizes the objective function.

Degenerate basic feasible solution: It is a basic feasible solution in which one or more of ‘m’ basic variables are equal to zero.

Non-degenerate basic feasible solution: It is a basic feasible solution in which all the ‘m’ basic variables are positive (>0) and the remaining ‘n’ variables are zero each.

Unbounded solution: If the value of the objective function can be increased or decreased indefinitely, the solution is called unbounded solution.

Decision variables: Mathematical symbols representing levels of activity of an operation.

Feasible region: In a constrained optimization problem, the set of solutions satisfying all equalities or inequalities is called feasible region.

The region bounded by these straight lines is called the feasible region

Corner Point Theorem 1: Let R be the feasible region for a linear programming problem and let ‘Z = ax + by’ be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.

Corner Point Theorem 2: If the feasible region is bounded, then the objective

function has both a maximum and a minimum value and each occur at one or more corner points.

If the feasible region is unbounded ,then objective function may not have a maximum or a minimum. But if a maximum or minimum value exists, it will occur at one or more corner points.

Corner Point TheoremCorner Point Theorem

Advantages of Linear Advantages of Linear Programming:Programming:

1) It helps in attaining the optimum use of productive factors. Linear programming indicates how a manager can utilize his productive factors most effectively by a better selection and distribution of these elements.

E.g.:- More efficient use of man power and machines can be made using this method.

2) It improves the quality of decisions. The individual becomes more objective than subjective. The individual having a clear picture of the relationships within the basic equations, inequalities or constraints can have a better idea about the problem and its solution.

3) It also helps in providing better tools for adjustments to meet changing conditions. It can go a long way in improving the knowledge and skill of future executives.

4) Most business problems involve constraints like raw materials availability, market demand etc. which must be taken into consideration. Just because we can produce so many units of products does not mean that they can be sold. Linear programming can handle such situations also since it allows modification of its mathematical solutions.

5) It highlights the bottlenecks in the production processes. When bottlenecks occur, some machines cannot meet demand while others remain idle, at least part of the time. Highlighting of bottlenecks is one of the most significant advantages of linear programming.

Limitations Of Linear Programming Model:

1) For large problems having many limitations and constraints, the computational difficulties are enormous, even when assistance of large digital computers is available. The approximations required to reduce such problems to meaningful sizes may yield the final results far different from the exact ones.

2) Another limitation of linear programming is that it may yield fractional valued answers for the decision variables, whereas it may happen that only integer valued of the variables are logical.

3) It is applicable to only static situations since it does not take into account the effect of time. The O.R team must define the objective function and constraints which can change due to internal as well as external factors.

4) It assumes that the values of the coefficients of decision variables in the objective function as well as in all the constraints are known with certainty. Since in most of the business situations, the decision making variable coefficients are known only probabilistically, it cannot be applied to such situations.

5) In some situations it is not possible to express both the objective function, constraints in linear form.

6) Linear programming deals with the problems that deal with problems that have a single objective. Real life problems may involve multiple and even conflicting objectives. One has to apply goal programming under such situations.

Applications of Linear Applications of Linear Programming method:Programming method:

Linear programming techniques are widely used to solve a number of business, industrial, military, economic, marketing, distribution and advertising problems.

Three primary reasons for its wide use are:1. A large number of problems from different

fields can be represented or at least approximated to linear programming problems.

2. Powerful and efficient techniques for solving L.P problems are available.

3. L.P models can handle data variation (sensitivity analysis) easily

Areas of application:Areas of application: A few areas of its application are given below:1)Industrial applications such as Product mix

problems, Blending problems, Production scheduling problems, Trim loss problems, Assembly line balancing, Make or buy problems.

2)Management applications such as Media selection problems, Portfolio selection problems, Profit planning problems, Transportation problems, Assignment problems, Manpower scheduling problems.

3)Miscellaneous applications such as Agriculture problems, Diet problems, Flight scheduling problems, Environment protection problems, Facilities location problems.

How Linear Programming How Linear Programming helps in Managerial decision helps in Managerial decision making?making?

A large number of managerial decision making problems are concerned with the efficient use or allocation of limited resources to meet the desired objectives. These problems are characterized by the large number of solutions that satisfy the basic conditions of each problem. The selection of a particular solution as the best solution to a problem depends on the overall objective of the problem.

A solution that satisfies both the conditions of the problem and the given objective is known as optimal solution. Linear programming is an technique for finding an optimal solution to such business problems of complex nature. Thus LP helps in managerial decision making for finding an optimal solution.

Brief answer: Linear programming is a mathematical

technique that determines the best way to use available resources. Managers use the process to help make decisions about the most efficient use of limited resources - like money, time, materials, and machinery.

How Linear Programming How Linear Programming helps in Managerial decision helps in Managerial decision making?making?

Methods of solving Linear Methods of solving Linear Programming problemsProgramming problems

The methods used for solving LPP are:

1)Graphical method.2)Simplex method.3)Big M method or Penalty method.

GRAPHICAL METHODGRAPHICAL METHOD

Once a problem is formulated as mathematical model, the next step is to solve the problem to get the optimal solution. A linear programming problem with only two variables presents a simple case for which the solution can be derived using a graphical method.

This method consists of following steps:

Step1: Represent the given problem in mathematical form. i.e. formulate the mathematical model for the given problem.

Step 2: Draw x1 and x2 axis. The non negativity constraints imply that the values of variables x1 and x2 can lie only in first quadrant.

Step 3: Plot each of the constraint on the graph. Connect the two plotted points by straight line. Thus each constraint is plotted as line in the first quadrant.

Step 4: Identify the feasible region that satisfies all the constraints. The area common to all the constraints is called feasible region and is shown shaded, which represents a feasible solution to given problem.

Additional variables used in solving L.P.P:

Any L.P. problem can be put in standard form with the help of some elementary transformations. The inequality constraints are changed to equality constraints by adding or subtracting a non-negative variable from the left hand sides of such constraints. These new variables are:

1) Slack variables: The non negative variable which is added when the constraint type is less than or equal to (≤), is called slack variable. Slack variables refer to the amount of unused resources like raw materials, labour and money. The slack variables are introduced in the constraints in order to convert the problem into standard form.

2) Surplus variable. The variable which is subtracted when the

constraint type is greater than or equal to (≥), is called surplus variable. This variable represents the surplus of left hand side over right hand side. It refers to the amount of resources by which the left hand side of the equation exceeds the minimum limit.

3) Artificial variable. Artificial variables are temporary slack

variables which are used for purposes of calculation, and are removed later. The artificial variables are fictitious and have no physical meaning. The coefficient of artificial variables are represented by a high value (penalty). And so, it is used in Penalty method.

SIMPLEX METHOD:SIMPLEX METHOD: In case a problem involves more than two

variables in objective function and structural constraints, then it would not lend itself easily to graphical solution. Such linear programming problems involving more variables and more constraints are dealt by using Simplex method which is the most general and powerful method of solving any linear programming problem. The simplex method solves the linear programming problem in iterations to improve the value of the objective function.

By the simplex procedure, we can obtain a maximum feasible solution in a finite number of steps starting with basic feasible solution. The different steps consists in finding a new feasible solution whose corresponding value of the objective function is more than the value of the objective function for the preceding solution. The process is continued until an optimal solution is reached.

The different steps involved are given below:

Step 1 : Express the problem in standard form.

Step 2 : Find initial basic feasible solution.Step 3 : Perform Optimality test.Step 4 : Iterate towards an Optimal

solution.

The flowchart of simplex method is shown below:

BIG M Method:BIG M Method:Introducing the artificial variables

technique, i.e. introduce artificial variables which are fictitious and have no physical meaning. They are merely a device to get the starting basic feasible solution so that simplex algorithm be applied as usual to get optimal solution. The coefficient of artificial variables are represented by a very high value M and hence the method is known as BIG-M method. Also known as ‘method of penalties’.

The Big M method consists of the following steps:

Step 1 : Express the linear programming problem in standard form by introducing slack variable. These variables are added to left hand sides of constraints of (≤) type and subtracted from the constraints of (≥) type.

Step 2 : Add non negative variables to left hand sides of all the constraints of initially (≥) or (=) type. These are called artificial variables.

Step 3 : Solve the modified linear programming problem by the simplex method.

Unbounded solutions in L.P.P. : In a linear programming problem, when a

situation exists that the value objective function can be increased infinitely, the problem is said to have an ‘unbounded solution’. This can be identified when all the values of key column are negative and hence minimum ratio values cannot be found.

Multiple solutions in L.P.P : In the optimal iteration table if (Cj – Zj)

value of one or more non basic variable is equal to zero, then the problem is said to have multiple or alternative solutions.

Duality in L.P.P. : All linear programming problems have another

problem associated with them, which is known as its dual. In other words, every minimization problem is associated with a maximization problem and vice versa. The original linear programming problem is known as primal problem, and the derived problem is known as its dual problem. The optimal solutions for the primal and dual problems are equivalent.

In the context of LPP , Duality implies that each Linear Programming Problem can be analyzed in two ways but having equivalent solution.

Procedures of Duality:

Step 1: Convert the objective function if maximization in the primal into minimization in the dual and vice versa.

Step 2: The number of variables in the primal will be the number of constraints in the dual and vice versa.

Step 3 : The coefficient in the objective function of the primal will be the RHS constraints in the dual and vice versa.

Step 4 : In forming the constraints for the dual, consider the transpose of the body matrix of the primal problems.

Note: Constraint inequality signs are changed.

In Dual L.P problem the main focus is to find its best marginal value. i.e. Dual price or shadow price.

Shadow price is defined as the rate of change in optimal objective function value with respect to the unit change in the availability of resources.

Shadow price = Change in optimal objective function

Unit Change in the availability of resources

Sensitivity analysis: Sensitivity analysis deals with making

individual changes in the co-efficient of the objective function and the right hand sides of the constraints. It is the study of how changes in the co-efficient of a linear programming problem affect the optimal solution.

In real world, the situation is constantly changing like change in raw material prices, decrease in machinery availability, increase in profit on one product and so on. Sensitivity analysis can be used to provide information and to determine solution with these changes.

Sensitivity analysis involves ‘what if’ questions.

What If: Change in Coefficients of Variables in the

Objective Function. Change in Coefficients of decision

variables on the L.H.S. of the constraints (aij).

Change in R.H.S of constraints/ Resources.

Adding New Variables. Adding New Constraints.

Review questions (given in Review questions (given in AIMA)AIMA)

1) Define Linear programming?2) What are the essential characteristics

required for a linear programming model?3) What is meant by objective function in LP

model?4) Enumerate the steps involved in solving a

LPP by graphical approach?5) List out the various constraint types in

formulating a LP model?6) What is meant by an unbounded solution?7) How are multiple solutions interpreted in

graphical method?

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