AGGREGATE PLANNING AND MASTER SCHEDULING

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Clifford McCormick
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1 AGGREGATE PLANNING AND MASTER SCHEDULING (Part 2) Al-Naimi Assistant Professor Industrial Engineering Branch Department of Production Engineering and Metallurgy University of Technology Baghdad - Iraq dr.mahmoudalnaimi@uotechnology.edu.iq dr.mahmoudalnaimi@yahoo.com

5.5 MATHEMATICAL PLANNING MODELS Mathematical models attempt to refine or improve upon the trial-and-error approaches. Table 5.13 identifies four mathematical approaches.

2 5.5 MATHEMATICAL PLANNING MODELS Mathematical models attempt to refine or improve upon the trial-and-error approaches. Table 5.13 identifies four mathematical approaches. The value from some of these models is more theoretical than practical. The LDR is not easily understood, nor are the outputs always realistic. The management coefficients model is non-optimal and not easily transferable, whereas the computer search models do not necessarily yield a global minimum cost. Table 5.13 A summary of some mathematical aggregate planning models A useful version of the linear-programming model (the transportation algorithm) views the aggregate planning problem as one of allocating capacity (supply) to meet forecast requirements (demand) where supply consists of the inventory on hand and units that can be produced using regular time (RT), overtime (OT), and subcontracting (SC), etc. Demand consists of individual-period requirements plus any desired ending inventory. Costs associated with producing units in the given period or producing them and carrying them in inventory until a later period are entered in the small boxes inside the cells in the matrix, as is done in the standard transportation linear-programming format. ILLUSTRATION 9: Given the accompanying supply, demand, cost, and inventory data (Tables 5.14, and 5.15) for a firm that has a constant work force and wishes to meet all demand (that is, with no back orders), allocate production capacity to satisfy demand at minimum cost. 1

3 Table 5.14 Supply capacity (units) *50 per cent of cost is labour. Table 5.15 Demand and inventory The initial linear-programming matrix in units of capacity is shown in table 5.16, with entries determined as explained below. Because total capacity exceeds demand, a slack demand of unused capacity is added to achieve the required balance in supply versus demand. Table 5.16 Linear programming format for scheduling 2

4 Initial inventory: There are 20 units available at no additional cost if used in period 1. Carrying cost is Rs. 2 per unit per period if units are retained until period 2, Rs. 4 per unit until period 3, and so on. If the units are unused during any of the four periods, the result is Rs.6 per-unit cost, plus Rs. 2 per unit to carry it forward to the next planning horizon, for Rs. 8 total if unused. Regular time: Cost per unit-month is Rs.100 if units are used in the month produced; otherwise, a carrying cost of Rs. 2 per unit-month is added on for each month the units are retained. Unused regular time costs the firm 50 per cent of Rs. 100 = Rs. 50. Overtime: Cost per unit is Rs. 125 if the units are used in the month produced; otherwise, a carrying cost of Rs. 2 per unit-month is incurred, as in the regular-time situation. Unused overtime has zero cost. Subcontracting: Cost per unit is Rs.130 plus any costs for units carried forward. This latter situation is unlikely, however, because any reasonable demand can be obtained when needed, as indicated by the arbitrarily high number (1,000) assigned to subcontracting capacity. There is no cost for unused capacity here. Note: If the initial allocations are made so as to use regular time as fully as possible, the solution procedure is often simplified. Overtime and subcontracting amounts can also be allocated on a minimum-cost basis. Final inventory: The final-inventory requirement (25 units) must be available at the end of period 4 and has been added to the period 4, demand of 80 units to obtain a total of 105 units. Since no back orders are permitted, production in subsequent months to fill demand in a current month is not allowed. These unavailable cells, along with the cells associated with carrying forward any subcontracted units, may therefore be blanked out, since they are infeasible. The final solution, following normal methods of distribution linear programming, is shown in table This result flows from a leastcost allocation. 3

5 Table 8.17 Master Schedule for Furniture Company The optimal solution values can be taken directly from the cells. Thus in period 2, the planners will schedule the full 50 units to be produced on regular time plus 12 units on overtime to be carried forward to period 4. This leaves 3 units of unused overtime capacity and no subcontracting during that period. Because of the similar carrying cost for units produced on regular time or overtime, it does not matter which physical units are carried forward, once overtime production is required. Thus, different optimal solutions (but with identical costs) may be obtained. 4

6 ILLUSTRATION: From the following forecast determine the monthly inventory balances required to follow a plan of letting the inventory absorb all fluctuations in demand. In this case, we have a constant work force, no idle time or overtime, no back orders, no use of subcontractors, and no capacity adjustment. Assume that the firm does not use safety stock or cushion inventory to meet the demand. Solution:- The firm can satisfy demand by producing at an average requirement (14 units per day) and by accumulating inventory during periods of slack demand and depleting it during periods of strong demand. Disregarding any safety stock, the inventory balance is: Inventory balance = Σ (production demand) See Table below for the solution. The pattern of demand is such that column 4 reveals a maximum negative balance of 566 units at the end of July, so 566 additional units must be carried in stock initially if demand is to be met. Column 5 shows the resulting inventory balances required. 5

AGGREGATE PLANNING PROBLEMS

AGGREGATE PLANNING PROBLEMS Problem 1. ABC Corporation has developed a forecast for a group of items that has the following seasonal demand pattern. Quarter Demand Cumulative Demand 1 22 22 2 17 39 3 4