Linear Programming Dr. T. T. Kachwala

Introduction to Linear Programming Slide 2 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.

Example of Product Mix Problem Slide 3 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.

Example of Product Mix problem Three types of data are defined in Product Mix Problem: Requirement of resources per unit of product Availability of resources, i.e., constraints of linear programming problem Objective function 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.”

Introduction to Linear Programming Slide 5 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.

Example of Marketing Mix problem Suppose 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: Defining the decision variables Imposing the restrictions on the decision variables as defined by the constraints 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 Slide 10 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.

Graphical Solution to Linear Programming Working Rules and Guidelines Slide 11 Graphical Solution to Linear Programming Working Rules and Guidelines Formulation of Linear programming problem: This involves: defining the decision variables imposing the restrictions on the decision variables as defined by the constraints defining the objective function in terms of the decision variables. Draw the graph corresponding to each equation of LPP. Identify feasible solution area and the corresponding corner points 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.

Unique, Multiple, Infeasibility and Unbounded Solutions Slide 12 Unique, Multiple, Infeasibility and Unbounded Solutions A problem may have a unique optimal solution, multiple optimal solutions, an unbounded solution, or infeasible solution: Unique optimal solutions means one optimum solution is available for LPP. Multiple optimal solutions means more than one optimum solution is available for LPP. 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. 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 13 Simplex (Mathematical) Solution to Linear Programming Working Rules and Guidelines Formulate the LP problem: (same as Graphical Solution) identification of decision variables. imposing the restrictions on the decision variables as defined by the constraints of LPP. 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.

Slide 14 Simplex (Mathematical) Solution to Linear Programming Working Rules and Guidelines 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 ---.

Slide 15 Simplex (Mathematical) Solution to Linear Programming Working Rules and Guidelines   “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

Slide 16 Simplex (Mathematical) Solution to Linear Programming Working Rules and Guidelines Simplex table (Iterations): Layout the initial Simplex table as per the standard format 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 Initial Simplex table is only a reference table Modify the initial Simplex table through a process of iteration (step by step improvement)

Slide 17 Simplex (Mathematical) Solution to Linear Programming Working Rules and Guidelines Simplex table (Iterations): Modification of the initial Simplex table through a process of iteration involves some mathematical calculations (Replacement ratio, Gauss-Jordan rules for new coefficients etc.). 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.