Constrained Optimization: Multiple Internal Constraints
Fisher Company produces two types of components for airplanes: A and B, with unit contribution margins of $400 and $600, respectively. The components pass through three sequential processes: cutting, welding, and assembly. Data pertaining to these processes and market demand are given below (weekly data).
Resource | Resource Available | Resource Usage (A) | Resource Usage (B) |
Cutting | 300 machine hours | Six hours | Ten hours |
Welding | 308 welding hours | Ten hours | Six hours |
Assembly | 400 labor hours | Four hours | Ten hours |
Market demand (A) | 50 | One unit | Zero units |
Market demand (B) | 40 | Zero units | One unit |
Required:
1. Express Fisher Company's constrained optimization problem as a linear programming model. If an an answer box does not require an entry, enter "0" for your answer.
Objective function: Max Z = $400 A + $600 B
Internal constraints: | A + B ≤ | (cutting) |
A + B ≤ | (welding) | |
A + B ≤ | (assembly) | |
External constraints: | A ≤ | |
B ≤ | ||
Nonnegativity constraints: | A ≥ | |
B ≥ |
2. Select the graph that solves the linear programming model expressed in Requirement 1.
a. | b. |
c. | d. |
Correct answer is .
Which constraints are binding?
3. What if Fisher Company had 10 additional machine hours (cutting) with all other resources held constant? What is the new optimal mix? Enter the appropriate corner point for your answer.
What is the associated total contribution margin? Round the units of A and B to two decimal places, and round intermediate calculations and your final answer to the nearest dollar.
$
What is the incremental benefit per machine hour caused by the additional ten hours, if any?
$ per machine hour
Internal constraints: | 6 A + 10 B ≤ 300 | (cutting) |
10 A + 6 B ≤ 308 | (welding) | |
4 A + 10 B ≤ 400 | (assembly) | |
External constraints: | A ≤ | 50 |
B ≤ | 40 | |
Nonnegativity constraints: | A ≥ | 0 |
B ≥ | 0 |
Cutting and Wielding constraints are
binding
3. F (point F)
Associated total contribution margin
Units of A = 19.06 units
Units of B = 19.56 units
Associated total contribution margin = 19.06*400 + 19.56*600 = $19,360
Incremental benefit = 19360-earlier contribution margin = 19360-18800 = $560.
Incremental benefit per machine hour = 560/10 = $56 per hour
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