Tools › Operations Research › Optimization › Linear Programming › Example
Hillier and Lieberman: Wyndor Glass Co. product mix with shadow prices and ranging
| Tool | Linear Programming |
| Source | Hillier, F. S., and Lieberman, G. J. (2015). Introduction to Operations Research, 10th ed. McGraw-Hill, New York. Sections 3.1, 4.4 and 7.2, Table 4.8. |
| Level | Undergraduate |
Problem
Wyndor Glass makes doors (x1, profit 3 per batch) and windows (x2, profit 5) in three plants with 4, 12 and 18 hours available: x1 ≤ 4, 2 x2 ≤ 12, 3 x1 + 2 x2 ≤ 18. Find the product mix that maximizes profit, the shadow price of each plant's capacity and the ranges over which the solution stays optimal.
Inputs to Enter
- max 3 x1 + 5 x2
- plant1: x1 <= 4 ; plant2: 2 x2 <= 12 ; plant3: 3 x1 + 2 x2 <= 18 ; x1, x2 >= 0 (loaded by default)
- Show Simplex Tableaus and Sensitivity Analysis checked
Answers
- x1 = 2, x2 = 6, Z = 36 after two simplex iterations (final tableau row 0: 0, 0, 0, 1.5, 1 | 36)
- Shadow prices y1 = 0, y2 = 1.5, y3 = 1; slacks 2, 0, 0 (plant 1 non-binding)
- RHS ranges: plant 1 [2, ∞), plant 2 [6, 18], plant 3 [12, 24]
- Objective ranges: c1 [0, 7.5], c2 [2, ∞); at c1 = 7.5 the optimum is shared with (4, 3)
- Dual: min 4 y1 + 12 y2 + 18 y3 with y1 + 3 y3 ≥ 3, 2 y2 + 2 y3 ≥ 5; b·y = 36
Notes
The default model box holds this problem; press Submit. The test script also re-solves at the ends of each range to confirm the basis stays the same.