A small manufacturing process has two phases: printing department and assembly. The cycle time/unit in printing is P1 minutes and in assembly P2 minutes. The printed units queue in front of assembly line to be assembled into finished goods. This is called Work in Progress Inventory, WIP. Demand for next week is D(t) units for each day, t=1, 2...5. Capacity per shift is CAP minutes. The company can either work up to 3 shifts per day or do overtime with only CAP/2 minutes overtime capacity per day. Overtime cost/unit is OC and a shift of work cost SC/ shift. All demand must be met at the end of 5 days. Holding cost per day per unit is H1/unit/day for WIP and H2/unit/day for finished product. Write a mathematical model that optimizes labor and holding costs while adhering to capacity constraints. Note that the space for WIP each day is restricted by space limit WCAP.
Variable Notation:
X….: Production quantities during shifts
O….: Production quantities during overtime
I…...: Inventory levels
Y..…: number of shifts
Z….: indicates if overtime is done
Add subscripts or more “letters” as necessary to represent the manufacturing stages and time.