Question

A community playhouse needs to determine the​ lowest-cost production budget for an upcoming show. They have to determine which set pieces to construction and which to rent. The organization has only two weeks to construct the set. The theater has two carpenters who work up to 12 hours a​ week, each at ​$12 an hour.​ Additionally, the theater has a scenic artist who can work 15 hours per week to paint as needed at ​$14 per hour. The set needs 20 flats​ (walls), two hanging​ drops, and three wooden tables​ (props). The number of hours required for each piece for carpentry and painting is shown below.​ Flats, hanging​ drops, and props can also be rented at a cost of ​$75​, ​$500​, and ​$400 ​each, respectively. How many of each unit should be built by the theater and how many should be rented to minimize total​ cost?

	Carpentry	Painting
Flats	0.5	2.0
Hanging Drops	3.0	12.0
Props	2.0	4.0

The optimal integer solution is to build __ flat(s) and rent __ ​flat(s); build __hanging​ drop(s) and rent __hanging​ drop(s); build __prop(s) and rent__ prop(s). This solution gives the ▼(minimum, maximum) ​cost, which is ​$__.

Answer 1

Firstly, the decision has to be taken whether it has to be taken on rent or to be constructed. Since, there are both time as well as workforce constraints we have to decide whether it would be constructed or to be taken on rent.

Particulars	Flats	Hanging Drops	Props
Taken on rent	75	500	400
If constructed
Carpenter Cost	6	36	24
Painting Cost	28	168	56
Total Cost (if Constructed)	34	204	80
Savings	41	296	320

Since, in all the above coonstruction leads to lowest cost, we have to go for construction.

Computation of Carpentary Hours required (If all are constructed)
Particulars	No of units required	Hrs required per Unit	Total Hours
Flats	20	0.5	10
Hanging Drops	2	3	6
Props	3	2	6
Total	22

Available is 12hrs*2person*2weeks = 48 Hours. So, this is not a constraint

Computation of Painting Hours required (If all are constructed)
Particulars	No of units required	Hrs required per Unit	Total Hours
Flats	20	2	40
Hanging Drops	2	12	24
Props	3	4	12
Total	76

Available is 15hrs*1person*2weeks = 30hrs. Therefore, painting is a constraint resource. So we have to use this constraints in a beneficial manner by computing per unit savings of constraint hour if construced. Computation is as follows:

Particulars	Flats	Hanging Drops	Props
Savings if constructed	41	296	320
Painting hours required/unit	2	12	4
Savings / painting hour	20.5	25	80
Ranking based on constraint	3rd	2nd	1st

Alloction of Constraint Resources

Total Painting Hours available	30
Utilized for Props (4*3)	12
Balance	18
For Hanging Drops (N-1)	12
Balance	6
For Flats (2*3)	6
Balance	0

N-1. 18hrs cannot be sufficient for 2 hanging drops. So one can be constructed and the other will be taken on rent. For the balance of 6 Hrs, 3 flats can be constructed. Therefore Solution is as follows:

Particulars	Constructed	Rent	Cost for constructed units*	Cost for rented units*	Total cost
Flats	3	17	102	1275	1377
Hanging Drops	1	1	204	500	704
Props	3	0	240	0	240
Total Cost	546	1775	2321

* per unit cost taken from 1st table.

Lowest Cost for upcoming show is $. 2321