Linear Programming for Solving the DSS Problems

Outline Introduction Modeling a Linear Programming Problem Mathematical programming The Linear Programming Model Modeling a Linear Programming Problem A Decision Making Example Developing Linear Programming Model The Simplex Method The Simplex Tableau Sensitivity Analysis Complications in Simplex Method

Introduction Mathematical programming The Linear Programming Model

Mathematical programming Mathematical programming is used to find the best or optimal solution to a problem that requires a decision or set of decisions about how best to use a set of limited resources to achieve a state goal of objectives. Steps involved in mathematical programming Conversion of stated problem into a mathematical model that abstracts all the essential elements of the problem. Exploration of different solutions of the problem. Finding out the most suitable or optimum solution. Linear programming requires that all the mathematical functions in the model be linear functions.

The Linear Programming Model (1) Let: X1, X2, X3, ………, Xn = decision variables Z = Objective function or linear function Requirement: Maximization of the linear function Z. Z = c1X1 + c2X2 + c3X3 + ………+ cnXn …..Eq (1) subject to the following constraints: …..Eq (2) where aij, bi, and cj are given constants.

The Linear Programming Model (2) The linear programming model can be written in more efficient notation as: …..Eq (3) The decision variables, xI, x2, ..., xn, represent levels of n competing activities.

Modeling a LP Problem A Decision Making Example Developing Linear Programming Model The Simplex Method The Simplex Tableau Sensitivity Analysis Complications in Simplex Method

A Decision Making Example A manufacturer has fixed amounts of different resources such as raw material, and equipment. These resources can be combined to produce any one of several different products. The quantity of the ith resource required to produce one unit of the jth product is known. The decision maker wishes to produce the combination of products that will maximize total income.

Developing Linear Programming Model (1) Steps Involved: Determine the objective of the problem and describe it by a criterion function in terms of the decision variables. Find out the constraints. Do the analysis which should lead to the selection of values for the decision variables that optimize the criterion function while satisfying all the constraints imposed on the problem.

Developing Linear Programming Model (2) Example: Product Mix Problem A manufacturer has two products, A and B And has two resources, R1 and R2 Each unit of product A requires 1 unit of R1 and 3 units of R2 Each unit of product B requires 1 unit of R1 and 2 units of R2 The manufacturer has 5 units of R1 and 12 units of R2 Profit: 6$ per each unit of product A sold. 5$ per each unit of product B sold.

Developing Linear Programming Model (3) Represent as table form R1 R2 Profit per unit Product A 1 3 6$ Product B 2 5$ Amount of resources: 5 12

Developing Linear Programming Model (4) Problem formulation ( modeling) has 4 steps: Identify the decision variables Writing the objective function Writing the constraints Writing the non negativity restriction All of them are linear.

Developing Linear Programming Model (5) Define decision variables: x1 be number of units of A produced x2 be number of units of B produced Objective function (A linear function) z = 6 x1 + 5 x2 Constraints x1 + x2 ≤ 5 3 x1 + 2 x2 ≤ 12 Non negativity restriction x1 , x2 ≥ 0

Developing Linear Programming Model (6) Graphical Solution to LP Problem We must maximized z = 6 x1 + 5 x2

The Simplex Method (1) Simplex method is the classic method for solving LP problems, one of the most important algorithms ever invented Invented by George Dantzig in 1947 (Stanford University) When decision variables are more than 2, it is always advisable to use Simplex Method to avoid lengthy graphical procedure. The simplex method is not used to examine all the feasible solutions. It deals only with a small and unique set of feasible solutions, the set of vertex points (i.e., extreme points) of the convex feasible space that contains the optimal solution.

The Simplex Method (2) Standard form of LP problem Must be a maximization problem All constraints (except the non-negativity constraints) must be in the form of linear equations All the variables must be required to be nonnegative Thus, the general linear programming problem in standard form with m constraints and n unknowns (n ≥ m) is maximize c1 x1 + ...+ cn xn subject to ai1x1+ ...+ ain xn = bi , , i = 1,...,m, x1 ≥ 0, ... , xn ≥ 0 Every LP problem can be represented in such form

The Simplex Method (3) z - 6 x1 - 5 x2 = 0 x1 + x2 + x3 = 5 Convert all the inequality constraints into equalities by the use of slack variables (Standard form). z - 6 x1 - 5 x2 = 0 x1 + x2 + x3 = 5 3 x1 + 2 x2 + x4 = 12 x1 , x2 , x3 , x4 ≥ 0

The Simplex Method (4) Steps involved: Locate an extreme point of the feasible region. Examine each boundary edge intersecting at this point to see whether movement along any edge increases the value of the objective function. If the value of the objective function increases along any edge, move along this edge to the adjacent extreme point. If several edges indicate improvement, the edge providing the greatest rate of increase is selected. Repeat steps 2 and 3 until movement along any edge no longer increases the value of the objective function.

The Simplex Method (5) Iteration 1 x3 and x4 are basic variables: x3 = 5 - x1 - x2 x4 = 12 - 3 x1 - 2 x2 z = 6 x1 + 5 x2 x1 can increase the objective function (z) with higher rate Basic variables are not seen in the z equation. +z

The Simplex Method (6) Iteration 2 So now, x1 and x3 are basic variables: 3 x1 = 12 – 2 x2 – x4 x1 = 4 – 2/3 x2 – 1/3 x4 x3 = 1 – 1/3 x2 + 1/3 x4 z = 24 + x2 - 2 x4 x2 can increase the objective function (z) with higher rate

The Simplex Method (7) Iteration 3 So now, x1 and x2 are basic variables: x2 = 3 – 3 x3 + x4 x1 = 2 + 2 x3 - x4 z = 27 – 3 x3 - x4 All of the coefficients are lower than zero, so we are on optimal point. Solution: X1 = 2 X2 = 3 Z = 27

Sensitivity Analysis Sensitivity analysis helps to test the sensitivity of the optimum solution with respect to changes of the coefficients in the objective function, coefficients in the constraints inequalities, or the constant terms in the constraints. For Example in the case study discussed: The actual selling prices (or market values) of the two products may vary from time to time. Over what ranges can these prices change without affecting the optimality of the present solution? Will the present solution remain the optimum solution if the amount of resources is suddenly changed because of shortages, machine failures, or other events? The amount of each type of resources needed to produce one unit of each type of product can be either increased or decreased slightly. Will such changes affect the optimal solution ?

Complications in Simplex Method (1) An objective function to be minimized instead of maximized. Greater-than-or-equal-to constraints. Equalities instead of inequalities for constraints. Decision variables unrestricted in signs. Zero constants on the right-hand side of one or more constraints. Some or all decision variables must be integers. Non-positive constants on the right-hand side of the constraints. More than one optimal solution, that is, multiple solutions such that there is no unique optimal solution.

Complications in Simplex Method (2) The constraints are such that no feasible solution exists. The constraints are such that one or more of the variables can increase without limit and never violate a constraint (i.e., the solution is unbounded). Some or all of the coefficients and right-hand-side terms are given by a probability distribution rather than a single value.