The Bull Grin Company produces a feed supplement for animal foods produced by a number of companies. Sales are seasonal, and Bull Grin’s customers refuse to stockpile the supplement during slack sales periods. In other words, the customers want to minimize inventory, insist on shipments according to their schedules and will not accept backorders.
Bull Grin employs manual, unskilled laborers who require little or no training. Producing 1,000 pounds of supplement costs $810 on regular time and $900 on overtime. These amounts include materials, which account for more than 80 percent of the cost. Overtime is limited to production of 30,000 pounds per quarter. In addition, subcontractors can be hired at $1,100 per thousand pounds, but only 10,000 pounds per quarter can be produced this way. Under time is unpaid.
The current level of inventory is 40,000 pounds and management wants to end the year at that level. Holding 1,000 pounds of feed supplement in inventory costs $110 per quarter. The latest annual forecast follows:
Quarter Demand
1 .......... 100,000
2 .......... 410,000
3 .......... 770,000
4 .......... 440,000
Total 1,720,000

The firm currently has 180 workers, a number of whom management wants to keep in quarter 4. Each worker can produce 2,000 pounds per quarter, so regular-time production costs $1,620 per worker. Idle workers must be paid at that same rate. Hiring one worker costs $1000, and laying off a worker costs $600.
Write the objective function and constraints describing this production planning problem after fully defining the decision variables.