GQ6.¶
Statement¶
For the following questions, please read this problem statement:
A computer manufacturer must determine what product mix to produce. A server requires 4 CPU’s and 8 Memory modules. A desktop computer requires 1 CPU and 4 Memory modules. Each server is sold for $1850 and each desktop is sold for $925. The manufacturer must produce a quantity of both units to keep both lines in production so the quantity of servers and desktops produced must both be greater than 0.
The manufacturer can only get a supply of 1250 CPU’s and 3800 memory modules due to shortages in the supply chain. Using the Simplex algorithm, determine the number of servers and desktops that should be built to maximize the profits of the manufacturer and determine how much revenue will be generated.
- How many servers should be built?
- How many Desktop computers should be built?
- How much Revenue (money in dollars received by selling the desktops and servers) will be generated?
Answer¶
Problem Formulation¶
Let’s denote s as the number of servers and d as the number of desktops. The objective function is to maximize the profit, which can be represented as:
P = 1850s + 925d
And the constraints are:
4s + d <= 1250 (CPU constraint) 8s + 4d <= 3800 (Memory constraint) s >= 0 d >= 0
Hence, the equations can be represented as:
-1850s - 925d + P = 0 4s + d + s1 = 1250 8s + 4d + s2 = 3800
Initial Simplex Table¶
And the simplex table can be represented as:
| s | d | s1 | s2 | P | R | |
|---|---|---|---|---|---|---|
| s1 | 1 | 0.25 | 1 | 0 | 0 | 1250 |
| s2 | 2 | 0.5 | 0 | 1 | 0 | 3800 |
| P | -925 | -1850 | 0 | 0 | 1 | 0 |
Iteration 1¶
Now, the pivot column is d (the column with the most negative value for P) and hence we divide R by the corresponding d to get the ratios:
| s | d | s1 | s2 | P | R | |
|---|---|---|---|---|---|---|
| s1 | 1 | 0.25 | 1 | 0 | 0 | 1250 |
| s2 | 2 | 0.5 | 0 | 1 | 0 | 3800 |
| d | 3700 | 7400 | 0 | 0 | 1 | 0 |