Chapter 3. Aggregate Planning (Steven Nahmias). Hierarchy of Production Decisions. Long-range Capacity Planning. Long range. Intermediate range. Short range. Now. 2 months. 1 Year. Planning Horizon.
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Long-range Capacity Planning
Concept is predicated on the idea of an “aggregate unit” of production. May be actual units, or may be measured in weight (tons of steel), volume (gallons of gasoline), time (worker-hours), or dollars of sales. Can even be a fictitious quantity. (Refer to example in text and in slide below.)
- staffing and
- production levels
for the firm over a predetermined planning horizon.
- react quickly to anticipated changes in demand
- retaining a stable workforce
- develop a production plan that maximizes profit over the planning horizon subject to constraints on capacity
Suppose that D1, D2, . . . , DT are the forecasts of demand for aggregate units over the planning horizon (T periods.) The problem is to determine both work force levels (Wt) and production levels (Pt ) to minimize total costs over the T period planning horizon.
Planning Horizon. The planning horizon is the number of periods of demand forecast used to generate the aggregate plan. If the horizon is too short, there may be insufficient time to build inventories to meet future demand surges and if it is too long the reliability of the demand forecasts is likely to be low. (ın practice, rolling schedules are used)
Treatment of Demand. Assume demand is known. Ignores uncertainty to focus on the predictable/systematic variations in demand, such as seasonality.
The method is based on notion of aggregate units. They may be
One plant produced 6 models of washing machines:
Model # hrs. Price % sales
A 5532 4.2 285 32
K 4242 4.9 345 21
L 9898 5.1 395 17
L 3800 5.2 425 14
M 2624 5.4 525 10
M 3880 5.8 725 06
Question: How do we define an aggregate unit here?
One method for defining an aggregate unit: requires: .32(4.2) + .21(4.9) + . . . + .06(5.8) = 4.8644 worker hours. This approach for this example is reasonable since products produced are similar. When products produced are heterogeneous, a natural aggregate unit is sales dollars.
The washing machine plant is interested in determining work force and production levels for the next 8 months. Forecasted demands for Jan-Aug. are: 420, 280, 460, 190, 310, 145, 110, 125. Starting inventory at the end of December is 200 and the company would like to have 100 units on hand at the end of August. Find monthly production levels.
Month Net Predicted Cum. Net
1(Jan) 220 220
2(Feb) 280 500
3(Mar) 460 960
4(Apr) 190 1150
5(May) 310 1460
6(June) 145 1605
7(July) 110 1715
8(Aug) 225 1940
Suppose that we are interested in determining a production plan that doesn’t change the size of the workforce over the planning horizon. How would we do that?
One method: In previous picture, draw a straight line from origin to 1940 units in month 8: The slope of the line is the number of units to produce each month.
Answer: using the graph, find the straight line that goes through the origin and lies completely above the cumulative net demand curve:
From the previous graph, we see that cum. net demand curve is crossed at period 3, so that monthly production is 960/3 = 320. Ending inventory each month is found from:
Month Cum. Net. Dem. Cum. Prod. Invent.
1(Jan) 220 320 100
2(Feb) 500 640 140
3(Mar) 960 960 0
4(Apr.) 1150 1280 130
5(May) 1460 1600 140
6(June) 1605 1920 315
7(July) 1715 2240 525
8(Aug) 1940 2560 620
K = number of aggregate units produced by one worker in one day
= average number of units produced by one worker in one day.
Mo # wk days Prod. Cum Cum Nt End Inv
Level Prod Dem
Jan 22 346 346 220 126
Feb 16 252 598 500 98
Mar 23 362 960 960 0
Apr 20 315 1275 1150 125
May 21 330 1605 1460 145
Jun 22 346 1951 1605 346
Jul 21 330 2281 1715 566
Aug 22 346 2627 1940 687
($75/worker/day)(46 workers )(167days) = $576,150
In the original cum net demand curve, consider making reductions in the work force one or more times over the planning horizon to decrease inventory investment.
Linear Programming provides a means of solving aggregate planning problems optimally. The LP formulation is fairly complex requiring 8T decision variables(1.workforce level,2. production level, 3. inventory level, 4. # of workers hired, 5. # of workres fired, 6. overtime production, 7. idletime, 8. subcontracting) and 3T constraints (1. workforce, 2. production, 3. inventory), where T is the length of the planning horizon. (See section 3.5, pg.125)
Clearly, this can be a formidable linear program. The LP formulation shows that the modified plan we considered with two months of layoffs is in fact optimal for the prototype problem.
Refer to the latter part of Chapter 3 and the Appendix following the chapter for details.
Where: nt * k is the number of units produced by a worker in a given period of nt days