6-437-LP(simplex solution)

# 6-437-LP(simplex solution) - QMT437 Pn Paezah Hamzah...

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QMT437 Pn Pa’ezah Hamzah Linear Programming Part 2 The Simplex Method (for LP problems in 2 variables) Learning Outcomes Students should be able to: 1. Solve LP problems by the simplex method (focusing on maximization problems) 2. Interpret the optimal simplex tableau. I. Introduction The simplex method can be used to solve LP problems in two or more decision variables . For illustration, we will solve an LP problem in 2 variables using both the graphical and simplex procedures. Example: XYZ Company produces two types of toys: tricycles and scooters. Tricycles need to be assembled at Workstation 1, and scooters at Workstation 2. However, both products require processing time at Workstation 3. The following table summarizes the data of the problem. Hours per batch Workstation tricycles scooters 1 1 0 2 0 2 3 3 2 The available time for Workstation 1, Workstation 2, and Workstation 3 are 4, 12, and 18 hours, respectively. The profits from one batch of tricycles and one batch of scooters are RM3,000 and RM5,000, respectively. The LP model for XYZ problem is Let X 1 = number of batches of tricycles, X 2 = number of batches of scooters Maximize profit (Z) = 3X 1 + 5X 2 (RM’000) subject to: X 1 4 ;Workstation 1 (hours) 2X 2 12 ;Workstation 2 (hours) 3X 1 + 2X 2 18 ;Workstation 3 (hours) X 1 ,X 2 0 Corner Point solution Corner points Z=3X 1 +5X 2 (RM000) (0,6) (0,0) 0 (0,6) 30 (2,6) 36 (4,3) 27 (4,0) 12 (4,0) 6 (2, 6) Feasible region 3X 1 +2X 2 =18 (4,3) X 1 =4 2X 2 =12 X 2 9 X 1 (0,0) Optimal solution : produce 2 batches of tricycles and 6 batches of scooters. Maximum profit = RM36,000 1

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II. The Simplex Method The following diagram illustrates the procedure of the simplex method. The Simplex Solution Procedure Start Optimal Convert LP model to Standard Form Obtain an Initial Feasible Solution Test for Optimality Interpre t Stop Not Optimal Determine Entering Variable Determine Leaving Variable Obtain an Improved Solution Perform elementary row operations Choose variable from the pivot row (row with the smallest positive RHS-to- pivot column ratio) Choose variable from the pivot column (column with the MOST NEGATIVE Z -row value) Testing for Optimality (Maximization): NO NEGATIVE z-row value Optimal Testing for Optimality (Minimization): NO POSITIVE z-row value Optimal 2
Elementary row operations (ERO) Multiply any row by a nonzero number.

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