Carco uses robots to manufacture cars. The following demands for cars must be me
ID: 384657 • Letter: C
Question
Carco uses robots to manufacture cars. The following demands for cars must be met (not necessarily on time, but all demands must be met by end of quarter 4): Quarter 1 Quarter 2 Quarter 3 Quarter 4 Demand 600 800 500 400 At the beginning of the quarter, Carco has 2 robots. Robots can be purchased at the beginning of any quarter for $5,000 per robot, but a maximum of 2 per quarter can be purchased. Each robot can build as many as 200 cars per quarter. Each quarter, a robot incurs $500 in maintenance costs (even if it is not used to build any cars). Robots can also be sold at the beginning of any quarter for $3,000. At the end of each quarter, a holding cost of $200 per car is incurred. If any demand is backlogged, then a cost of $300 per car is incurred for each quarter the demand is backlogged. At the end of quarter 4, Carco must have at least 2 robots. We would like to formulate an integer program to minimize the total cost incurred in meeting the next four quarters’ demands for cars.
Explanation / Answer
Decision Variables :
Rt : robots available during quarter t (after robots are bought or sold for the quarter)
Bt : robots bought during quarter t
St : robots sold during quarter t
It : cars in inventory at end of quarter t
Ct : cars produced during quarter t
Dt : backlogged demand for cars at end of quarter t
LP formulation :
MIN 500( R1 + R2 + R3 + R4 ) + 200( I1 + I2 + I3 + I4 ) + 5000( B1 + B2 + B3 + B4 ) – 3000( S1 + S2 + S3 + S4 ) + 300( D1 + D2 + D3 + D4 )
s.t.
R1 = 2 + B1 - S1
R2 = R1 + B2 - S2
R3 = R2 + B3 - S3
R4 = R3 + B4 - S4
I1 - D1 = C1 – 600
I2 – D2 = I1 - D1 + C2 – 800
I3 – D3 = I2 - D2 + C3 – 500
I4 – D4 = I3 - D3 + C4 – 400
R4 >= 2
C1 <= 200 R1
C2 <= 200 R2
C3 <= 200 R3
C4 <= 200 R4
D4 = 0 B1 <= 2
B2 <= 2
B3 <= 2
B4 <= 2
All variables >= 0