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Linear Programming Problem

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Linear Programming Problem

- Linear Programming was developed by George B Dantzing in 1947 for solving military logistic operations.

- Meaning of Linear Programming
– The word Linear refers to linear relationship among variables. i.e. a given change in one variable will always cause a resulting proportional change in another variable. For example, doubling the investment on a certain project will exactly double the rate of return.

- The word programming refers to modeling & solving a problem mathematically that involves the economic allocation of limited resources by choosing a strategy among various alternative strategies to achieve the desired objective.

Introduction

- Linear Programming (LP) is a mathematical modeling technique useful for the allocation of limited resources, such as labour, material, machine, time, warehouse space, capital, energy etc. to several competing activities, such as products, services, jobs, new projects etc.

- Also, the general LPP calls for optimizing a linear function of variables called the objective function subject to a set of linear equations and /or inequalities called the constraints or restrictions.

- Decision Variables
- The Objective Function
- The Constraints

- Decision Variables: The activities that are competing one another for sharing the resources available. These variables are usually interrelated in terms of utilization of resources and need simultaneous solutions. All these variables are considered as continuous, controllable and non-negative.

- The Objective Function: A LPP must have an objective which should be clearly identifiable and measurable in quantitative terms. It could be of maximization of profit (sales), minimization of cost etc. The relationship among variables representing objective must be linear.

- The Constraints:There are always certain limitations or restrictions or constraints on the use of resources, such as labour, space, raw material, money etc. that limit the degree to which an objective can be achieved. Such constraints must be expressed as linear inequalities or equations in terms of decision variables.

- Certainty
- Additivity
- Linearity (Proportionality)
- Divisibility (continuity)

- Certainty: In all LLP’s, it is assumed that all the parameters; such as availability of resources, profit contribution of a unit or cost contribution of a unit of decision variable and computation of resources by a unit decision variable must be known and fixed. Or we can say that, all the coefficients in this objective function as well as in the constraints are completely known with certainty and do not change during the period

of study. Thus, the profit per unit of the product, requirements of material and labour per unit, availability of material etc. are given and known in the problem. The LP is obviously deterministic in nature.

- Additivity: The value of the obj. function for the given values of decision variables and the total sum of resources used, must be equal to the sum of the contributions (profit or loss) earned from each decision variable and the sum of the resources used by each decision variable respectively. For example, the total profit earned by the sale of two products A & B must

be equal to the sum of the profits earned separately from A & B. Similarly, the amount of a resource consumed by A & B must be equal to the sum of resources used for A & B individually.

- Linearity or Proportionality: This assumption requires the contribution of each decision variable in both the obj function and the constraints to be directly proportional to the value of the decision variable. Or we can say that, the amount of each resource used ( or supplied) and its contribution to the profit (or cost) in obj. fun must be proportional to the value of each decision variable. For eg., if

production of a one unit of a product uses 5 hrs of a particular resource, then making 3 units of that product uses 3*5=15 hrs of that resource.

- Divisibility or Continuity: This implies that solution values of the decision variables and resources can take on any non-negative values, including fractional values of the decision variables. For eg., it is possible to produce 8.35 quintals of wheat or 7.453 thousand gallons of a solvent or 43.45 thousand kiloliters of milk. Such variables are not divisible and hence are to be assigned

integer values. When it is necessary to have integer variables, the integer programming problem is considered to attain the desired values.

Introduction

The term formulation referred to the process of converting the verbal description and numerical data into mathematical expressions which represents the relationship among relevant decision variable (factors), objective & restrictions on the use of resources.

Introduction

- The term formulation referred to the process of converting the verbal description and numerical data into mathematical expressions which represents the relationship among relevant decision variable (factors), objective & restrictions on the use of resources.

The XYZ garment company manufactures men's shirts and women’s t-shirts for ABC Discount stores. ABC will accept all the production supplied by the company. The production process includes cutting, sewing and packaging. XYZ employs 25 workers in the cutting department, 35 in the sewing department and 5 in the packaging department. The factory works one 8-hour shift, 5 days a week.

The following table gives the time requirements and the profits per unit for the two garments:

Minutes per unit

Determine the optimal weekly production schedule for XYZ.

Assume that XYZ produces x1 shirts and x2 t-shirts per week.

Profit got =

8 x1 + 12 x2

Time spent on cutting =

20 x1 + 60 x2 mts

Time spent on sewing =

70 x1 + 60 x2 mts

Time spent on packaging =

12 x1 + 4 x2 mts

The objective is to find x1, x2 so as to

maximize the profit z = 8 x1 + 12 x2

satisfying the constraints:

20 x1 + 60 x2≤ 25 40 60

70 x1 + 60 x2 ≤ 35 40 60

12 x1 + 4 x2 ≤ 5 40 60

x1, x2≥ 0, integers

This is a typical optimization problem.

Any values of x1, x2 that satisfy all the constraints of the model is called a feasible solution. We are interested in finding the optimumfeasible solution that gives the maximum profit while satisfying all the constraints.

More generally, an optimization problem looks as follows:

Determine the decision variablesx1, x2, …, xn so as to optimize an objectivefunctionf (x1, x2, …, xn) satisfying the constraints

gi (x1, x2, …, xn) ≤ bi (i=1, 2, …, m).

An optimization problem is called a Linear Programming Problem (LPP) when the objective function and all the constraints are linear functions of the decision variables, x1, x2, …, xn. We also include the “non-negativity restrictions”, namely xj ≥ 0 for all j=1, 2, …, n. Thus a typical LPP is of the form:

Optimize (i.e. Maximize or Minimize)

z = c1 x1 + c2 x2+ …+ cn xn

subject to the constraints:

a11 x1 + a12 x2 + … + a1n xn ≤ b1

a21 x1 + a22 x2 + … + a2n xn ≤ b2

. . .

am1 x1 + am2 x2 + … + amn xn ≤ bm

x1, x2, …, xn 0

Application Areas of Linear Programming

- LP helps in attaining the optimum use of productive resources. It also indicates how a decision maker can employ his productive factors effectively by selecting and distributing these resources.
- LP technique improves the quality of decisions.
- LP technique provides possible and pratical solutions since there might be other constraints operating operating outside the problem which must be taken into account.

- LP also helps in re-valuation of a basic plan for changing conditions. If conditions change when the plan is partly carried out, they can be determined so as to adjust the remainder of the plan for best results.

- LP treats all relationship s among variables as linear.
- While solving the an LPP, there is no guarntee that we will get integer valued solutions.
- LP model does not take into consideration the effect of time and uncertainnity.

- Parameters appearing in the model are assumed to be constant but in real-life situations, they are frequently neither klnown nor constant.
- It deals with single objective, whereas in real-life situations we may come across conflicting multi-objective problems.

- Agriculture Applications
- Military Operations
- Production Management
- Financial Management
- Marketing Managemant
- Personnel Management

The general LPP with n decision variables and m constraints can be stated as:

Find the values of decision variables…..

- Identify the Decision Variabels
- Identify the Problem data
- Formulate the constraints
- Formulate the Objective Function