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)?
1 answer:
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
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