LINEAR PROGRAMMINGExample 1 MaximiseI = x + 0.8y subject tox + y  x + y  x + 2y  2400 Initial solution: I = 0 at (0, 0)

LINEAR PROGRAMMINGExample 1 MaximiseI = x + 0.8y subject to x + y  x + y  x + 2y  2400 MaximiseI whereI - x - 0.8y = 0 subject to x + y + s 1 = x + y + s 2 = x + 2y + s 3 = 2400

Ixys1s1 s2s2 s3s3 RHS SIMPLEX TABLEAU I = 0, x = 0, y = 0, s 1 = 1000, s 2 = 1500, s 3 = 2400 Initial solution

Ixys1s1 s2s2 s3s3 RHS PIVOT 1Choosing the pivot column Most negative number in objective row

Ixys1s1 s2s2 s3s3 RHS / / /3 PIVOT 1Choosing the pivot element Ratio test: Min. of 3 ratios gives 2 as pivot element

Ixys1s1 s2s2 s3s3 RHS PIVOT 1Making the pivot Divide through the pivot row by the pivot element

Ixys1s1 s2s2 s3s3 RHS PIVOT 1Making the pivot Objective row + pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 1Making the pivot First constraint row - pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 1Making the pivot Third constraint row – 3 x pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 1New solution I = 750, x = 750, y = 0, s 1 = 250, s 2 = 0, s 3 = 150

LINEAR PROGRAMMINGExample MaximiseI = x + 0.8y subject tox + y  x + y  x + 2y  2400 Solution after pivot 1: I = 750 at (750, 0)

Ixys1s1 s2s2 s3s3 RHS PIVOT 2 Most negative number in objective row Choosing the pivot column

Ixys1s1 s2s2 s3s3 RHS / / /0.5 PIVOT 2Choosing the pivot element Ratio test: Min. of 3 ratios gives 0.5 as pivot element

Ixys1s1 s2s2 s3s3 RHS PIVOT 2Making the pivot Divide through the pivot row by the pivot element

Ixys1s1 s2s2 s3s3 RHS PIVOT 2Making the pivot Objective row x pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 2Making the pivot First constraint row – 0.5 x pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 2Making the pivot Second constraint row – 0.5 x pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 2New solution I = 840, x = 600, y = 300, s 1 = 100, s 2 = 0, s 3 = 0

LINEAR PROGRAMMINGExample MaximiseI = x + 0.8y subject tox + y  x + y  x + 2y  2400 Solution after pivot 2: I = 840 at (600, 300)

Ixys1s1 s2s2 s3s3 RHS PIVOT 3Choosing the pivot column Most negative number in objective row

Ixys1s1 s2s2 s3s3 RHS / / PIVOT 3Choosing the pivot element Ratio test: Min. of 2 ratios gives 1 as pivot element

Ixys1s1 s2s2 s3s3 RHS PIVOT 3Making the pivot Divide through the pivot row by the pivot element

Ixys1s1 s2s2 s3s3 RHS PIVOT 3Making the pivot Objective row x pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 3Making the pivot Second constraint row – 2 x pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 3Making the pivot Third constraint row + 3 x pivot row

Ixys1s1 s2s2 s3s3 RHS PIVOT 3Optimal solution I = 880, x = 400, y = 600, s 1 = 0, s 2 = 100, s 3 = 0

LINEAR PROGRAMMINGExample MaximiseI = x + 0.8y subject tox + y  x + y  x + 2y  2400 Optimal solution after pivot 3: I = 880 at (400, 600)