MB0048-Unit-02-Linear ProgrammingUnit 2 Linear Programming Structure: 2.1 Introduction Learning objectives 2.2 Requirements Basic assumptions of linear programming problems 2.3 Linear Programming Canonical forms Case studies of linear programming problems 2.4 Graphical Analysis Some basic definitions 2.5 Graphical Methods to Solve Linear Programming Problems Working rule Solved problems on mixed constraints LP problem Solved problem for unbounded solution Solved problem for inconsistent solution Solved problem for redundant constraint 2.6 Summary 2.7 Terminal Questions 2.8 Answers to SAQs and TQs Answers to Self Assessment Questions Answers to Terminal Questions

2.9 References 2.1 Introduction Welcome to the unit of Operations Research on Linear Programming. Linear programming focuses on obtaining the best possible output (or a set of outputs) from a given set of limited resources. Minimal time and effort and maximum benefit coupled with the best possible output or a set of outputs is the mantra of any decision-maker. Today, decision-makers or managements have to tackle the issue of allocating limited and scarce resources at various levels in an organisation in the best possible manner. Man, money, machine, time and technology are some of these common resources. The managements task is to obtain the best possible output (or a set of outputs) from these given resources. You can measure the output from factors, such as the profits, the costs, the social welfare, and the overall effectiveness. In several situations, you can express the output (or a set of outputs) as a linear relationship among several variables. You can also express the amount of available resources as a linear relationship among various system variables. The managements dilemma is to optimise (maximise or minimise) the output or the objective function subject to the set of constraints. Optimisation of resources in which both the objective function and the constraints are represented by a linear form is known as a linear programming problem (LPP). Learning objectives By the end of this unit, you should be able to: Construct linear programming problem and analyse a feasible region Evaluate and solve linear programming problems graphically 2.2 Requirements of LPP The common requirements of a LPP are as follows. i. Decision variables and their relationship ii. Well-defined objective function iii. Existence of alternative courses of action iv. Non-negative conditions on decision variables 2.2.1 Basic Assumptions of LPP

1. Linearity: You need to express both the objective function and constraints as linear inequalities. 2. Deterministic: All co-efficient of decision variables in the objective and constraints expressions are known and finite. 3. Additivity: The value of the objective function and the total sum of resources used must be equal to the sum of the contributions earned from each decision variable and the sum of resources used by decision variables respectively. 4. Divisibility: The solution of decision variables and resources can be non-negative values including fractions. Self Assessment Questions Fill in the blanks 1. Both the objective function and constraints are expressed in _____ forms. 2. LPP requires existence of _______, _______, ____ and _______. 3. Solution of decision variables can also be ____________. 2.3 Linear Programming The LPP is a class of mathematical programming where the functions representing the objectives and the constraints are linear. Optimisation refers to the maximisation or minimisation of the objective functions. You can define the general linear programming model as follows: Maximise or Minimise: 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 and x1 0, x2 0, xn 0

Where cj, bi and aij (i = 1, 2, 3, .. m, j = 1, 2, 3 - n) are constants determined from the technology of the problem and xj (j = 1, 2, 3 - n) are the decision variables. Here ~ is either (less than), (greater than) or = (equal). Note that, in terms of the above formulation the coefficients cj, bi aij are interpreted physically as follows. If bi is the available amount of resources i, where aij is the amount of resource i that must be allocated to each unit of activity j, the worth per unit of activity is equal to cj. 2.3.1 Canonical forms You can represent the general Linear Programming Problem (LPP) mentioned above in the canonical form as follows: Maximise Z = c1 x1+c2 x2 + + cn Subject to, a11 x1 + a12 x2 + + a1n xn b1 a21 x1 + a22 x2 + + a2n xn b2 am1 x1+am2 x2 + + amn xn bm x1, x2, x3, xn 0. The following are the characteristics of this form. 1. All decision variables are non-negative. 2. All constraints are of type. 3. The objective function is of the maximisation type. You can represent any LPP in the canonical form by using five elementary transformations, which are as follows: 1. The minimisation of a function is mathematically equivalent to the maximisation of the negative expression of this function. That is, Minimise Z = c1 x1 + c2x2 + . + cn xn is equivalent to Maximise Z = c1x1 c2x2 cn xn

2. Any inequality in one direction ( or ) may be changed to an inequality in the opposite direction ( or ) by multiplying both sides of the inequality by 1. For example 2x1+3x2 5 is equivalent to 2x13x2 5 3. An equation can be replaced by two inequalities in opposite direction. For example: 2x1+3x2 = 5 can be written as 2x1+3x2 5 and 2x1+3x2 5 or 2x1+3x2 5 and 2x1 3x2 5 4. An inequality constraint with its left hand side in the absolute form can be changed into two regular inequalities. For example: 2x1+3x2 5 is equivalent to 2x1+3x2 5 and 2x1+3x2 5 or 2x1 3x2 5 5. The variable which is unconstrained in sign ( 0, 0 or zero) is equivalent to the difference between 2 non-negative variables. For example: if x is unconstrained in sign then x = (x+ x) where x+ 0, x 0 Caselet An automobile company has two units X and Y which manufacture three different models of cars - A, B and C. The company has to supply 1500, 2500, and 3000 cars of A, B and C respectively per week (6 days). It costs the company Rs. 1,00,000 and Rs. 1,20,000 per day to run the units X and Y respectively. On a day unit X manufactures 200, 250 and 400 cars and unit Y manufactures 180, 200 and 300 cars of A, B and C respectively per day. The operations manager has to decide on how many days per week should each unit be operated to meet the current demand at minimum cost. The operations manager along with his team uses a LPP model to arrive at the minimum cost solution. 2.3.2 Case Studies of linear programming problems

Self Assessment Questions State True/False 4. One of the characteristics of canonical form in the objective function must be of maximisation. 5. 2x 3y 10 can be written as -2x + 3y -10 2.4 Graphical Analysis

You can analyse linear programming with 2 decision variables graphically. Example Lets look at the following illustration. Maximise Z = 700 x1+500 x2 Subject to 4x1+3x2 210 2x1+x2 90 and x1 0, x2 0 Let the horizontal axis represent x1 and the vertical axis x2. First, draw the line 4x1 + 3x2 = 210, (by replacing the inequality symbols by the equality) which meets the x1-axis at the point A (52.50, 0) (put x2 = 0 and solve for x1 in 4x1 + 3x2 = 210) and the x2 axis at the point B (0, 70) (put x1 = 0 in 4x1 + 3x2 = 210 and solve for x2).

Figure 2.1: Linear programming with 2 decision variables Any point on the line 4x1+3x2 = 210 or inside the shaded portion will satisfy the restriction of the inequality, 4x1+3x2 210. Similarly the line 2x1+x2 = 90 meets the x1-axis at the point C(45, 0) and the x2 axis at the point D(0, 90).

Figure 2.2: Linear programming with 2 decision variables Combining the two graphs, you can sketch the area as follows:

Figure 2.3: Feasible region The 3 constraints including non-negativity are satisfied simultaneously in the shaded region OCEB. This region is called feasible region. 2.4.1 Some basic definitions

Note: The objective function is maximised or minimised at one of the extreme points referred to as optimum solution. Extreme points are referred to as vertices or corner points of the convex regions.

Self Assessment Questions Fill in the blanks 6. The collection of all feasible solutions is known as the ________ region. 7. A linear inequality in two variables is known as a _________. 2.5 Graphical Methods to Solve LPP Solving a LPP with 2 decision variables x1 and x2 through graphical representation is easy. Consider x1 x2 the plane, where you plot the solution space enclosed by the constraints. The solution space is a convex set bounded by a polygon; since a linear function attains extreme (maximum or minimum) values only on the boundary of the region. You can consider the vertices of the polygon and find the value of the objective function in these vertices. Compare the vertices of the objective function at these vertices to obtain the optimal solution of the problem.

2.5.1 Working rule The method of solving a LPP on the basis of the above analysis is known as the graphical method. The working rule for the method is as follows. Step 1: Write down the equations by replacing the inequality symbols by the equality symbols in the given constraints. Step 2: Plot the straight lines represented by the equations obtained in step I. Step 3: Identify the convex polygon region relevant to the problem. Decide on which side of the line, the half-plane is located. Step 4: Determine the vertices of the polygon and find the values of the given objective function Z at each of these vertices. Identify the greatest and least of these values. These are respectively the maximum and minimum value of Z. Step 5: Identify the values of (x1, x2) which correspond to the desired extreme value of Z. This is an optimal solution of the problem

2.5.2 Solved problems on mixed constraints LP problem

In linear programming problems, you may have: i) a unique optimal solution or ii) many number of optimal solutions or iii) an unbounded solution or iv) no solutions.

2.5.3 Solved problem for unbounded solution

2.5.4 Solved problem for inconsistent solution

2.5.5 Solved problem for redundant constraint

Self Assessment Questions State True/False 8. The feasible region is a convex set 9. The optimum value occurs anywhere in feasible region 2.6 Summary In a LPP, you first identify the decision variables with economic or physical quantities whose values are of interest to the management. The problems must have a well-defined objective function expressed in terms of the decision variable.

The objective function is to maximise the resources when it expresses profit or contribution. Here, the objective function indicates that cost has to be minimised. The decision variables interact with each other through some constraints. These constraints arise due to limited resources, stipulation on quality, technical, legal or variety of other reasons. The objective function and the constraints are linear functions of the decision variables. A LPP with two decision variables can be solved graphically. Any non-negative solution satisfying all the constraints is known as a feasible solution of the problem. The collection of all feasible solutions is known as a feasible region. The feasible region of a LPP is a convex set. The value of the decision variables, which maximise or minimise the objectives function is located on the extreme point of the convex set formed by the feasible solutions. Sometimes the problem may be unfeasible indicating that no solution exists for the problem. 2.7 Terminal Questions 1. Use the graphical method to solve the LPP. Maximise Z= 5x1 + 3x2 Subject to: 3x1 + 5x2 15 5x1 + 2x2 10 x1, x2 0 2. Mathematically formulate the problem. A firm manufactures two products; the net profit on product 1 is Rs. 3 per unit and the net profit on product 2 is Rs. 5 per unit. The manufacturing process is such that each product has to be processed in two departments D1 and D2. Each unit of product 1 requires processing for 1 minute at D1 and 3 minutes at D2; each unit of product 2 requires processing for 2 minute at D1 and 2 minutes at D2. Machine time available per day is 860 minutes at D1 and 1200 minutes at D2. How much of products 1 and 2 should be produced every day so that total profit is maximum. Formulate this as a problem in L.P.P. 2.8 Answers to SAQs and TQs Answers to Self Assessment Questions 1. Linear 2. Alternate course of action

3. Fractious 4. True 5. True 6. Feasible 7. Half-plan 8. True 9. False Answers to Terminal Questions

1. 2. Maximise 3x1 + 5x2, subject to x1 + 2x2 800 (minutes) 3x1 + 2x2 1200 (minutes) x1, x2 0