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Ch 2: The Simplex MethodMATH2230/MATH3205/Mark Lau1. (Initialization) We choose (x1, x2) = (0,0) to be the initial CFP.2. (Optimality test) This CFP is not optimal sincezincreases when eitherx1orx2increases, in order to satisfy the first constraint.3. (Initial BF solution) Set up the initial tabular formz(x1)(x2)x3x4x5x6RHS1-3-20000001210006021010080-110010100100012and see that (0,0,6,8,1,2) is our first BF solution.4. (Iteration 1, step 1) We choosex1as the entering variable, sincezincreases faster inthe direction ofx1.5. (Iteration 1, step 2) We choosex4as the leaving variable, so thatx1yields the greatestincrease without leaving the feasible set. Note thatx5cannot be the leaving variablesince-x1+x5= 1, and hencex1would be negative ifx5= 0.6. (Iteration 1, step 3) We perform row operations on the simplex tableau in 3, in orderto obtain a BF solution:zx1(x2)x3(x4)x5x6RHS10-1/203/20012011/201/2004003/21-1/2002003/201/210500100012The BF solution found in the current iteration is (4,0,2,0,5,2). More iterations areneed since the objective function value can be improved further by moving to one ofthe adjacent BF solutions.7. (Iteration 2, step 1) Assignx2as the entering variable.Last updated: October 7, 2014Page 10 of 18
Ch 2: The Simplex MethodMATH2230/MATH3205/Mark Lau8. (Iteration 2, step 2) Assignx3the leaving variable.9. (Iteration 2, step 3) Perform row operations on the tabular form in Iteration 1 to getzx1x2(x3)(x4)x5x6RHS1001/33/20038/3010-1/32/30010/30012/3-1/3004/3000-11103000-2/31/3012/3The BF solution found in the current iteration is (10/3,4/3,0,0,3,2/3). The optimalsolution is found with objective function value 38/3. Keep in mind that the algorithmstops when an optimal solution has been obtained. In problems having multiple optimalsolutions, one can perform more iterations to obtain other optimal solutions.2.1Solving Minimization ProblemsThe following minimizationminc1x1+c2x2+· · ·+cnxnsubject to a set of constraints can be converted to a maximization,max-c1x1-c2x2- · · · -cnxn,over the same set of constraints. For example, the following minimization problemminx1,x2∈R0.4x1+ 0.5x2s.t.0.3x1+ 0.1x2≤2.7,0.5x1+ 0.5x2= 6,0.6x1+ 0.4x2≥6,x1, x2≥0Last updated: October 7, 2014Page 11 of 18