I need help finding optimal value, solution, slack/surplus.

Question

Maximize 40X + 50Y + 80Z Total Profit
Subject to X + Y + Z > 1000 Total quantity required
X + 3 Y + 2 Z ≤ 3000 Resource 1
2X + Y + 2Z ≤ 2500 Resource 2
And X, Y, Z ≥ 0

Where X, Y, and Z 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.

Linear Programming Results:
Variable      Status      Value
X              NONBasic      0
Y               Basic          250
Z               Basic         1125
surplus 1    Basic          375
slack 2     NONBasic       0
slack 3     NONBasic       0
Optimal Value (Z) 102500

Original Problem with Answers
                     X       Y     Z                               RHS          Dual
Maximize       40      50   80
Constraint      1        1     1 1   >=                  1000              0
Constraint 2   1        3     2      <=                  3000              5
Constraint 3   2        1     2      <=                  2500             35
Solution->      0      250  1125  Optimal Z->     102500


Ranging Results:
Variable      Value     Reduced Cost      Original Val        Lower Bound         Upper Bound
X                  0            35                      40                    -Infinity                      75
Y                250           0                        50                      40                          120
Z               1125          0                        80                      52                          100

Constraint         Dual Value      Slack/Surplus       Original Val        Lower Bound        Upper Bound
Constraint 1        0                       375                    1000                 -Infinity                 1375
Constraint 2        5                        0                       3000                 2500                     7500
Constraint 3       35                       0                       2500                 1000                     3000

 

 

(a) Determine the optimal solution and optimal value and interpret their meanings.
(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.

Answer 1

Hi,

      the optimal solution is x=0, y=250 and z=1125, these are the amounts of each product that will yield the maximum total profit  of 102,500 subject to the constraints given.

 

The slack values are 375 which is the amount in excess of 1000 that the optimal solution requires i.e. 0+250+1125=1375 the other 2 slacks are 0 which is the amount of resource 1 and 2 left. That is to produce 250 of product y and 1125 of product z one would use all of resource1 3000 and all of resource 2 2500.

Regards

Jim