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 $200 and $400, 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 200 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 = $200 A + $400 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 50 units of Part A and 40 units of Part B? Express the objective function with its associated constraints for this case.

Objective function: Max Z = $200 A + $400 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.
$

Constrained Optimization: One Internal Binding Constraint

Patz Company produces two types of machine parts: Part A and Part B, with unit contribution margins of $200 and $400, 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 200 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 = $200 A + $400 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 50 units of Part A and 40 units of Part B? Express the objective function with its associated constraints for this case.

Objective function: Max Z = $200 A + $400 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.Max Z=200A+400B

2A +   5B ?  200

2.Contribution par hour for A=(200/2)=$100

Contribution per hour for B=(400/5)=$80

Contribution per hour for A is higher than B

Hence, maximum assembly hours should be allocated to A

Hence A should be produced (200/2)=100 numbers, there is no constraint in ability to sell.

Component A	100	units
Component B	0	units

Total Contribution=Z=200*100=$20,000

3.

Assembly-hour constraint	2A + 5  B ?  200
Demand constraint for Part A	A ?  50
Demand constraint for Part B	B ?  40

If maximum 50 units of A can be sold, 50 units of A should be manufactured

Balance Capacity (200-(2*50))=100 assembly hours should be used for B

Part B required 5 hours of assembly

Hence, Number of units of B manufactured=100/5=20 units

Patz can sell 40 units of B. hence it will not be a constraint to sell 20 units of B

Optimal mix:

Component A	50	units
Component B	20	units

Total Contribution=Z=50*200+400*20= $   18,000