Question

A company must meet (on time) the following demands: quarter 1 -30 units; quarter 2—20 units; quarter 3—40 units. Each quarter, up to 27 units can be produced with regular-time labor, at a cost of $40 per unit. During each quarter, an unlimited number of units can be produced with overtime labor, at a cost of $60 per unit. Of all units produced, 25% are unsuitable and cannot be used to meet demand. Also, at the end of each quarter, 10% of all units on hand spoil and cannot be used to meet any future demands. After each quarter's demand is satisfied and spoilage is accounted for, a cost of $15 per unit is assessed against the quarter's ending inventory. Formulate an LP that can be used to minimize the total cost of meeting the next three quarters' demands. Assume that 20 usable units are available at the beginning of quarter 1. (Hint: Define inventory variables to keep track of the number of usable units.)

Answer 1

Choice Variables: Let's characterize them as follows

Xi = number of units that are delivered with normal time work during these 3 quarters, 1,2,3

Yi = number of units that are delivered with extra time work during these 3 quarters, 1,2,3

Zi = number of units that are held to starting with one quarter then onto the next quarter

\RightarrowTotal Cost = 40 Xi + 60 Yi + 15 Zi (i=1,2,3) ........ (1)

At most 27 units can be delivered with normal time work, Xi\leq 27 for i=1,2,3 .....(2)

To locate the quantity of usable units that are left by end of quarter 1, we have 20 units during the beginning of quarter 1, assume X1+Y1 are delivered and 80% are usable, and request is 30. Henceforth,

Z1 = 20 + 0.8 (X1+Y1) - 30 ....(3)

Additionally for quarter 2, number of usable units to begin will be Z1, X2 + Y2 are prdouced with 80% ease of use and request is 20,

Z2 = Z1 + 0.8 (X2 + Y2) - 20 ....(4)

for quarter 3, number of usable units to begin will be Z2, X3 + Y3 are prdouced with 80% ease of use and request is 40,

Z3 = Z2 + 0.8 (X3 + Y3) - 40 ....(5)

All factors should be non negative, thus

Xi, Yi, Zi\geq 0 for i=1,2,3 ... (6)

Consequently, the LP plan is

Limit 40 Xi + 60 Yi + 15 Zi (i=1,2,3

subject to Xi\leq 27 for i=1,2,3

Z1 = 20 + 0.8 (X1+Y1) - 30

Z2 = Z1 + 0.8 (X2 + Y2) - 20

Z3 = Z2 + 0.8 (X3 + Y3) - 40

Xi, Yi, Zi\geq 0 for i=1,2,3