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3. A company which produces a single product has definite orders for this product over the next six months as follows: Month 12 3 4 5 6 Demand 350 680 475 590 460 390 The company ended the previous month (month 0) with an inventory of 200 units, and the previous month production level was 400 units. The company wishes to end this six-month planning period with an inventory of at least 50 units. It costs $3.50 per unit to increase the production level from one month to the next, and $6.00 per unit to decrease it. The cost of holding inventory from one month to the next is $4.80 per unit per month. No shortages are permitted. In each month, the company wants to produce an integer number of units. The company wishes to minimize the sum of production level change costs and inventory costs over the six-month planning horizon. Let, for j=1,2,3,4,5,6 Xi = Production volume in month-j Ej = Ending inventory level of month-j Hj = Increase of production level from month-(i-1) to month-i Lj = Decrease of production level from month-(i-1) to month-j Minimize Z = total cost incurred in the six months = 3.5*(H1 + H2 + H3 + H4 + H5 + H6) + 6*(L1 + L2 + L3 + L4 + 15 + L6) + 4.8*(E1 + E2 + 3 + 4 + 55+ E6) Subject to [Production + Beginning Inventory - Ending Inventory = Demand) for each month. So, X1 + 200 - E1 = 350 or, X1 - E1 = 150 X2 + E1 - E2 = 350 X3 + E2 - E3 = 350 X4 + E3-E4 = 350 X5 + E4-E5 = 350 X6 + E5 - E6 = 350 [Production volume = Volume of previous month + Hj - Lj] for each month. So, P1 = 400 + H1 - L1 or, P1 - H1 + L1 - 400 = 0 P2 - H2 + L2 - P1 = 0 P3-H3 + L3 - P2 = 0 P4 - H4 + L4 - P3 = 0 P5 - H5 + L5 - P4 = 0 P6 - H6 + L6 - P5 = 0 E6 >= 50 (the ending inventory at the end of the 6 months period should be at least 50) K, Hj, Lj, Ej >= 0 for j=1,2,3,4,5,6