Question

Constrained Optimization: One Internal Binding Constraint

Patz Company produces two types of machine parts: Part A and Part B, with unit contribution margins of $500 and $1,000, respectively. Assume initially that Patz can sell all that is produced of either component. Part A requires two hours of assembly, and B requires five hours of assembly. The firm has 500 assembly hours per week.

Required:

1. Express the objective of maximizing the total contribution margin subject to the assembly-hour constraint.

Objective function: Max Z = $500 A + $1,000 B

Subject to:  A +  B ≤

2. Identify the optimal amount that should be produced of each machine part. If none of the components should be produced, enter "0" for your answer.

Component A		units
Component B		units

Identify the total contribution margin associated with this mix.
$

3. What if market conditions are such that Patz can sell at most 125 units of Part A and 100 units of Part B? Express the objective function with its associated constraints for this case.

Objective function: Max Z = $500 A + $1,000 B

Assembly-hour constraint	A +  B ≤
Demand constraint for Part A	A ≤
Demand constraint for Part B	B ≤

Identify the optimal mix and its associated total contribution margin.
$

Answer 1

1.

Objective function: Max Z = $500 A + $1,000 B

Since, part A requires 2 hours of assembly & part B requires 5 hours of assembly, and there are only 500 hours available, hence: 2A + 5 B ≤500

2. Number of unit of part A that can be produced = 500/2= 250

Total contribution in this case = 250*500 = $1,25,000

Number of unit of part B that can be produced = 500/5=100

Total contribution in this case = 100*1000 = $1,00,000

Since, all that is prduced of either component can be sold, 250 units of part A should be produced where contribution would be $1,25,000

3.

Objective function: Max Z = $500 A + $1,000 B

Assembly-hour constraint	2A + 5 B ≤ 500
Demand constraint for Part A	A ≤ 125
Demand constraint for Part B	B ≤ 100

Number of hours reqd to produce 125 units of part A = 125*2 = 250

Hence, hours left for part B = 250 wherein the number of parts that can be produced = 50

Hence, contibution at thiss level = 125*500+50*1000 = $1,12,500