Question

Cully Furniture buys two products for resale: king beds (K) and queen beds (Q). Each king bed costs $500 and requires 100 cubic feet of storage space, and each queen bed costs $250 and requires 80 cubic feet of storage space. The company has $80,000 to invest in shelves this week, and the warehouse has 30,000 cubic feet available for storage. Profit for each king bed is $400 and for each queen bed is $200. 1). (10’) Please set up the LP model for the above situation.2). (20’) How many king beds (K) and how many queen bed (Q) should be purchased at optimal? 3). (10’) Whether we have slack or surplus variable applicable in this question? If we do, what is the value of slack (or/and surplus) variable? What is the meaning behind it?4). (10’) If the furniture company purchases no king beds and 300 queen beds, which resources will be completely used (at capacity)?

Answer 1

Answer:

1) 500K + 250Q ≤ 80,000

  100K+80Q≤ 30,000

K≥0

Q≥0

P= 400K + 200Q

2) zero king beds and 320 queen beds

or zero queen beds and 160 king beds

3) surplus variable

surplus variable is storage space

values of surplus variable is 14,000 cubic feet if zero queen beds and 160 king beds

value of surplus variable is 4,400 cubic feet if zero king beds and 320 queen beds.

surplus is the extra amount available after all resources have been utilized to theri maximum.

4) none of the resources will be completely used. There will be surplus of both

Explanation:

1) Implicit variables: K≥0 ;   Q ≥ 0

Explicit variables:

500K + 250Q ≤ 80,000-------------------------- from investment constraint

  100K+80Q≤ 30,000 ---------------------- from storage space constraint

LP:  P= 400K + 200Q

2) See the attachment for profit maximization graph

3) From graph in the attachment, it can be inferred that storage space is the surplus variable since graph of that equation lies completely outside of optimal area.

Also,

if K=160 and Q=0, the inequality of investment gives

500(160) + 250(0)= 80,000

if K=0 and Q=320, the inequality of investment gives

500(0)+ 250(320) =80,000

if K=0 and Q = 320, the inequality of space gives

100(0) + 80(320)= 25,600

surplus= 30,000- 25,600= 4,400

if K=160 and Q=0, the inequality of space gives

100(160) +250(0)= 16000

surplus= 30000-16000= 14000

4) K=0 and Q=300

investment inequality gives

500(0) + 250(300) ≤ 80,000

75,000≤ 80,000

space inequality gives

100(0) + 80(300) ≤ 30,000

24,000≤ 30,000

Both space and investment are in surplus