Question

Linear programming.

David, Tracy and Lydia are the sole partners and workers in a company that produces fine clocks. David and Tracy are each available to work 40 hours per week but Lydia is only available for 20 hours per week. The company makes two different kinds of clocks: grandfather clocks and wall clocks and they earn $300 in profit per grandfather clock sold and $200 in profit per wall clock sold. They also have a pre-order for one wall clock that they must satisfy each week. To make either type of clock David assembles the internal mechanical parts of the clock mechanism, Tracy produces the hand-carved wood casing, and Lydia packages and ships the clocks. Note that partial products can be produced, and completed in a future week. The amount of time required per task and type of clock are given below.

	Time required per clock (hrs)
Task	Grandfather clock	Wall clock
Assembling clock mechanism	6	4
Carving wood casing	8	4
Packaging and shipping	3	3

(a) Formally state the problem, if they seek to earn the highest overall weekly profit

(b) Graph the constraints and identify the feasible region. Clearly identify all inter- cept values for constraints and place Grandfather Clocks on the horizontal axis.

(c) The optimal mix is grandfather clocks and wall clocks, and the resulting profit from this choice is $ . Show all work that supports this decision - this means you must evaluate all viable options!

(d) Assuming that they follow your recommended mix, answer for each of the following constraint:

assembly - Binding or non-binding? - Slack =

carving - Binding or non-binding? - Slack =

packaging & shipping - Binding or non-binding - Slack =

(e) Perform the same linear programming problem in Excel and attach the answer report.

Answer 1

A)

 What is the objective for this problem?

The objective of the problem is to maximize the profit earned by the optimal production of the two clocks.

Using your decision variables, formulate the objective function.

Maximize Z = 300x+200y

B)

 What are the constraints in this problem? Using your decision variables, formulate these constraints.

The constraints are the number of hours available with David, Diana and Lydia.

Subject to:

1. David - 6x+4y <= 40 hours

2. Diana - 8x+4y <= 40 hours

3. Lydia - 3x+3y <= 20 hours

C) Optimal solution:

4 grandfather clock and 2 wall clock

Profit = $1600

D)

Grandfather Clock	Wall clock
Decision variable	4	2
Profit	300	200
David	6	4	32	<=	40
Diana	8	4	40	<=	40
Lydia	3	3	18	<=	20
Maximize	1600

ssembly - non-binding? - Slack = 0

carving - Binding - Slack =

packaging & shipping - non-binding - Slack = 0