3.2 calculating shadow prices The simplest way to calculate shadow prices for a critical constraint is as follows:
Step 1: add one unit to the constraint concerned, while leaving the other critical constraint unchanged.
Step 2: solve the revised simultaneous equations to derive a new optimal solution.
Step 3: calculate the revised optimal contribution. The increase is the shadow price for the constraint under consideration.
Illustration 7 – Shadow prices来自www.Examw.com
In Hebrus the optimal solution was determined to be x = 4 and y = 4 giving an optimal contribution of $360. This solution was at the intersection of the lines:
Cutting 6x + 3y = 36
Assembly 4x + 8y = 48
Required:
Calculate the shadow prices for cutting and assembly time.
Solution
Suppose one extra hour was available for the cutting process each week. By how much would contribution (and profit) be increased?
We would then need to solve
Cutting 6x + 3y = 37
Assembly 4x + 8y = 48中 华 考 试 网
These give y = 3.888…and x = 4.222…
This gives a revised contribution of C = (50 × 4.222…) + (40 × 3.888…) = $ 366.67.
The increase of $6.67 is the shadow price for cutting time per hour.
A similar calculation can be done for assembly time giving a shadow price of $2.50 per hour.
Test your understanding 7
Using the following data, calculate the shadow price for machining time.
Maximise C = 80x + 75y (contribution), subject to
(ⅰ) 20x + 25y ≦ 500 (machining time)
(ⅱ) 40x + 25y ≦ 800 (finishing time)
The optimal solution at the intersection of the above constraints is: x = 15, y = 8. |