Question

Maximize 12X1 + 10X2 + 8X3             Total Profit

Subject to      X1 + X2 + X3 > 160        At least a total of 160 units of all three products needed

                 X1 + 3X2 + 2X3 ≤ 450         Resource 1

                 2X1 + X2 + 2X3 ≤ 300         Resource 2

               2X1 + 2X2 + 3X3 ≤ 400         Resource 3

                  And X1, X2, X3 ≥ 0

Where X1, X2, and X3 represent the number of units of Product 1, Product 2, and Product 3 to be manufactured.

The QM for Windows output for this problem is given below.

Solution List:

Variable           Status   Value

X1       Basic    100

X2       Basic    100

X3       NONBasic       0

surplus 1          Basic    40

slack 2 Basic    50

slack 3 NONBasic       0

slack 4 NONBasic       0

Optimal Value (Z)        2200

Linear Programming Results:

X1       X2       X3                  RHS     Dual

Maximize         12        10        8                                 

Constraint 1     1          1          1          >=        160      0

Constraint 2     1          3          2          <=        450      0

Constraint 3     2          1          2          <=        300      2

Constraint 4     2          2          3          <=        400      4

Solution           100      100      0                      2200   

Ranging Results:

Variable           Value   Reduced Cost   Original Val     Lower Bound   Upper Bound

X1 100      0 12 10        20

X2 100 0 10 6          12

X3 0          8 8 -Infinity           16

            Dual Value       Slack/Surplus   Original Val     Lower Bound   Upper Bound

Constraint 1     0 40 160      -Infinity 200

Constraint 2     0 50 450 400 Infinity

Constraint 3     2 0 300 275 400

Constraint 4     4 0 400 320 420

(a) Determine the optimal solution and the optimal value and interpret their meanings.

(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.

Answer 1

(a) Optimal solution can be read from the Ranging results.

X1 = 100

X2 = 100

X3 = 0

Optimal value = 12*100+10*100+8*0 = 2200

Interpretation: 100 units of product 1 and 2 each should be produced. this will give a total profit of $ 2200

b)

Surplus value for constraint 1 = 40

Slack value for constraint 2 = 50

Slack for constraint 3 and 4 = 0 each

Interpretation : Surplus value of constraint 1 means that 40 units more than required are produced

Slack value of constraint 2 indicates that 50 less units of resource 1 are used, than the maximum units available.