1 Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow SMU EMIS 5300/7300 NTU SY-521-N NTU SY-521-N SMU EMIS 5300/7300 Statistical Analysis Other Linear Programming Methods updated

2 The Simplex Method

3 Introduction The simplex method is an iterative algorithm for efficiently solving large linear programming problems. Currently the simplex method and its variations have been programmed and coded for practically all makes and types of computers.

4 The Simplex Process Step 1 Standardize the program into a linear programming tableau. Step 2 Generate an initial solution Step 3 Test the solution for optimality. If the solution is not optimal, improve (go to step 4); otherwise go to step 6 Step 4 Identify one variable that will leave the solution and one variable will enter the solution Step 5 Generate an improved solution. The improved solution is checked for optimality. If it is not optimal, then steps 4 and 5 are repeated. If it is optimal, step 6 is undertaken. Step 6 Find if more than one optimal solution exists.

5 Schematic presentation of the simplex method Is Solution Optimal? Identify Incoming Variables Identify Outgoing Variables Generate Improved Solution Find All Multivariable Optimal Solutions, If they Exist Stop Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Start Standardize Problem Format: Slack, Surplus, and Artificial Variables Generate Initial solution (initial tableau) No Yes

6 Using Lindo (from

7 Maximization Problem - Using Lindo Type in a new project. (where x = x 1 and y = x 2 ) max 300x + 250y st 2x+y<=40 x+3y<=45 x<=12 x>=0 y>=0 end

8 Maximization Problem - Using Lindo The results that it returned are: LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1) VARIABLE VALUE REDUCED COST X Y so, x 1 = 12 and x 2 = 11 resulting in z = 6350.

9 Using Excel

10 Maximization Problem - Using Excel A company produces two models of color TV sets, designated A and B. The company’s objective is profit maximization. The profit realized is $300 from set A and $250 from set B. The limitations are: 1. Only 40 hours of labor each day are available. Labor constraint 2. Only 45 hours of machine time are available. Machining constraint 3. Only 12 sets of model A may be sold each day. Marketing constraint Determine how many sets of each model to produce each day so that total profit is maximized. Use Excel!

11 Formulation The Decision Variables: x 1 = number of model A produced X 2 = number of model B produced The Objective Function (total profit): z = 300x x 2 This is the function to be maximized, as was demonstrated using a graphical technique last week in class.

12 Constraints Labor Constraint: 2x 1 + x 2  40 Machine Constraint: x 1 + 3x 2  45 Marketing Constraint: x 1  12 Non-negativity Constraint (no negative sales): x 1  0x 2  0

13 Solver in Excel First, make the spreadsheet look nice so we can understand what the heck you are doing.

14 Solver in Excel From the Tools menu in Excel select the Solver. It will spawn the menu on the next slide, into which you put your constraints.

15 Solver in Excel The Target Cell is where we want the answer.

16 Solver in Excel Once the constraints are input, push Solve and the cell will change from the previous value, to the value the Solver solved for.