Question

1) (25pts) An engineer is making aggregate planning for the next 4 months. Demand forecasts in aggregate items are 600, 350, 400, 700 for months 1 to 4. The company has an initial workforce of 80 and capacity can be adjusted by hiring (at a cost of 150 liras per worker) or firing (at a cost of 900 liras per worker). Each worker is capable of producing 0.2 aggregate items in one workday. The available workdays are given as 25, 23, 24, 26 for months 1 to 4 respectively. Unit production cost is 14 liras per aggregate item. It is possible to manufacture during overtime with an additional production cost of 6 liras per item. Inventory holding cost is 3 liras per item per month. Backordering is not allowed. Beginning inventory is 150 aggregate items. Ending inventory at the end of fourth month is planned to be 200 items. Underutilization of existing capacity has an undertime cost of 4 liras per aggregate item. The maximum amount of overtime production is limited to 50 aggregate items per month. Formulate the aggregate planning problem as an LP. State all variables and costs explicitly. While answering this question (and the question below) make use of the following decision variables: Wi, W2, W3, W4 Н, Н, Н, Н, F1, F2, F3, F4 P1, P2, P3, P4 О, Ог, Оз, О, U1, U2, U3, U4 I, I2, 13, 14 Number of employed workers in months 1, 2, 3, 4 Number of hired workers in months 1, 2, 3, 4 Number of fired workers in months 1, 2, 3, 4 Production quantities in months 1, 2, 3, 4 Overtime production quantities in months 1, 2, 3, 4 Undertime production units in months 1, 2, 3, 4 Inventory-on-hand at the end of months 1, 2, 3, 4 2) (15pts) Making use of an LP software determine an optimal solution of the formulation developed in Q.1. Include the printed text output of the problem formulation and output of the solution. Explicitly state what should be the values of the decision variables for the optimal solution. If the optimal solution has noninteger quantities for number of workers or aggregate quantities round them to nearest integer quantities.

Answer 1

Q. 1

LP model is following:

Minimize Z =

150H1+150H2+150H3+150H4 (hiring cost)

+900F1+900F2+900F3+900F4   (firing cost)

+14P1+14P2+14P3+14P4 (production cost)

+20O1+20O2+20O3+20O4 (overtime cost)

+3I1+3I2+3I3+3I4   (inventory holding cost)

+4U1+4U2+4U3+4U4 (undertime cost)

s.t.

W1-H1+F1=80

W2-H2+F2-W1=0

W3-H3+F3-W2=0

W4-H4+F4-W3=0

P1-I1=600-150

P2-I2+I1=350

P3-I3+I2=400

P4-I4+I3=700

I4=200

P1-25*.2*W1-O1+U1=0

P2-23*.2*W2-O2+U2=0

P3-24*.2*W3-O3+U3=0

P4-26*.2*W4-O4+U4=0

Oi <= 50

All variables >= 0