1. Chapter 3. Aggregate Planning (Steven Nahmias)

2. Hierarchy of Production Decisions
Long-range Capacity Planning

3. Planning Horizon
Aggregate planning: Intermediate-range capacity planning, usually covering 2 to 12 months.
Short
range
Intermediate range
Long range
Now
2 months
1 Year

4. Aggregate planning
Aggregate planning is intermediate-range capacity planning used to establish employment levels, output rates, inventory levels, subcontracting, and backorders for products that are aggregated, i.e., grouped or brought together. It does not specifically focus on individual products but deals with the products in the aggregate.

5. Concept of Aggregate Product
For example, imagine a paint company that produces blue, brown, and pink paints; the aggregate plan in this case would be expressed as the total amount of the paint without specifying how much of it would be blue, brown or pink.
Such an aggregate plan may dictate, for example, the production of 100,000 gallons of paint during an intermediate-range planning horizon, say during the whole year. The plan can later be disaggregated as to how much blue, brown, or pink paint to produce every specific time period, say every month.

6. Why Aggregate Planning Is Necessary
Fully load facilities and minimize overloading and underloading
Make sure enough capacity available to satisfy expected demand
Plan for the orderly and systematic change of production capacity to meet the peaks and valleys of expected customer demand
Get the most output for the amount of resources available

7. Aggregate Planning Strategies
Should inventories be used to absorb changes in demand during planning period?
Should demand changes be accommodated by varying the size of the workforce?
Should part-timers be used, or should overtime and/or machine idle time be used to absorb fluctuations?
Should subcontractors be used on fluctuating orders so a stable workforce can be maintained?
Should prices or other factors be changed to influence demand?

8. Introduction to Aggregate Planning
Goal: To plan gross work force levels and set firm-wide production plans so that predicted demand for aggregated units can be met.
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 handout and in slide below.)

9. Aggregation Method Suggested by Hax and Meal
Hax and Meal suggest grouping products into three categories:
1. items, 2. families, and 3. types.
Items are the finest level in the product structure and correspond to individual Stock-Keeping Units (SKU). For example, a firm selling refrigerators would distinguish white from almond in the same refrigerator as different items.
A family in this context would be refrigerators in general.
Types are natural groupings of families; kitchen appliances might be one type.

10. Aggregate Units
The method is based on notion of aggregate units. They may be
Actual units of production
Weight (tons of steel)
Dollars (Value of sales)
Fictitious aggregate units(See example 3.1)

11. Example of fictitious aggregate units. (Example 3.1)
One plant produced 6 models of washing machines:
Model # hrs Price % sales A K L L M M
Question: How do we define an aggregate unit here?

12. Example continued
Notice: Price is not necessarily proportional to worker hours (i.e., cost): why?
One method for defining an aggregate unit: requires: .32(4.2) + .21(4.9) (5.8) = 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.

13. Aggregate Planning
Aggregate planning might also be called macro production planning.
Whether a company provides a service or product, macro planning begins with the forecast of demand.
Aggregate planning methodology is designed to translate demand forecasts into a blueprint for planning :
- staffing and
- production levels
for the firm over a predetermined planning horizon.

14. Aggregate Planning
The aggregate planning methodology discussed in this chapter assumes that the demand is deterministic and dynamic.
This assumption is made to simplify the analysis and allow us to focus on the systematic and predictable changes in the demand pattern.
Aggregate planning involves competing objectives:
- react quickly to anticipated changes in demand
- retain a stable workforce
- develop a production plan that maximizes profit over the planning horizon subject to constraints on capacity

15. Nature of Demand Demand I. Deterministic Static Dynamic
II. Probabilistic
Stationary
Non-Stationary
In aggregate production planning, we assume that demand is deterministic and dynamic.

16. Costs in Aggregate Planning
Smoothing Costs
changing size of the work force
changing number of units produced
Holding Costs
primary component: opportunity cost of investment in inventory
Shortage Costs
Cost of demand exceeding stock on hand.
Other Costs: payroll, overtime, idle cost, subcontracting.

17. Cost of Changing the Size of the Workforce
Fig. 3-2
Cost of Changing the Size of the Workforce

18. Holding and Back-Order Costs
Fig. 3-3
Back-orders Positive inventory
Slope = CP
Slope = Ci
$ Cost
Inventory

19. Overview of the Aggregate Production Problem
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.

20. Prototype Aggregate Planning Example (this example is not in the handout)
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.

21. Step 1: Determine “net” demand
Step 1: Determine “net” demand. (subtract starting inventory from period 1 forecast and add ending inventory to period 8 forecast.)
Month Net Predicted Cum. Net
Demand Demand
1(Jan) 220( )
2(Feb)
3(Mar)
4(Apr)
5(May)
6(June)
7(July)
8(Aug) 225( )

22. Step 2. Graph Cumulative Net Demand to Find Plans Graphically

23. Basic Strategies Constant Workforce (Level Capacity) strategy:
Maintaining a steady rate of regular-time output while meeting variations in demand by a combination of options.
Zero Inventory (Matching Demand, Chase) strategy:
Matching capacity to demand; the planned output for a period is set at the expected demand for that period.

24. Level vs. Chase Strategy

25. Advantages and Disadvantages
Chase Strategy
Reduced inventory costs.
High levels of worker utilization.
Cost of fluctuating workforce levels.
Potential damage to employee morale.
Level Strategy
Worker levels and production output are stable.
High inventory costs.
Increased labor costs.
(Source: Aggregate Planning)
Since the chase method does not rely on inventory level to meet demand all cost associated with carrying inventory will be less than that of a level strategy. Since the firm wishes to produce the most goods with the fewest employees possible (to keep labor costs down) workers tend to be used as efficiently as possible.

26. Constant Work Force Plan
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.

27. Monthly Production = 1940/8 = 242. 2 or rounded to 243/month
Monthly Production = 1940/8 = or rounded to 243/month. But: there are stockouts.

28. How can we have a constant work force plan with no stockouts?
Answer: using the graph, find the straight line that goes through the origin and lies completely above the cumulative net demand curve:

29. Month Cum. Net. Dem. Cum. Prod. Invent. 1(Jan) 220 320 100
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)
2(Feb)
3(Mar)
4(Apr.)
5(May)
6(June)
7(July)
8(Aug)

30. But - may not be realistic for several reasons:
Since all months do not have the same number of workdays, a constant production level may not translate to the same number of workers each month.

31. To Overcome These Shortcomings:
Assume number of workdays per month is given (reasonable!)
Compute a “K factor” given by:
K = number of aggregate units produced by one worker in one day

32. Finding K
Suppose we are told that over a period of 40 days, 520 units were produced with 38 workers. It follows that:
K= 520/(38*40) = average number of units produced by one worker in one day.

33. Computing Constant Work Force -- Realistically
Assume we are given the following # of working days per month: 22, 16, 23, 20, 21, 22, 21, 22.
March is still the critical month.
Cum. net demand thru March = 960.
Cum # working days = = 61.
We find that:
960/61 = units/day
/.3421 = 46 workers required
Actually – here we truncate because we are set to build inventory so the low number should work (check for March stock out) – however we must use care and typically ‘round up’ any fractional worker calculations thus building more inventory

34. Why again did we pick on March?
Examining the graph we see that March was the “Trigger point” where our constant production line intersected the cumulative demand line assuring NO STOCKOUTS!
Can we “prove” this is the best?

35. Tabulate Days/Production Per Worker Versus Demand To Find Minimum Numbers
Month
# Work Days
#Units/worker
Forecast Demand net
Cum. Net Demand
Cum.Units/
Worker
Min # Workers
Jan
22.00
7.53
220.00
29.23
Feb
16.00
5.47
280.00
500.00
13.00
38.46
Mar
23.00
7.87
460.00
960.00
20.87
46.00
Apr
20.00
6.84
190.00
27.71
41.50
May
21.00
7.18
310.00
34.89
41.84
Jun
145.00
42.42
37.84
Jul
110.00
49.60
34.57
Aug
225.00
57.13
33.96

36. What Should We Look At?
Cumulative Demand says March needs most workers – this can be interpretted as building inventories in Jan + Feb to fulfill the greater March demand
However, if we keep this number of workers we will continue to build inventory through the rest of the plan!

37. Constant Work Force Production Plan
Mo # wk days Prod. Cum Cum Nt End Inv
Level Prod Dem
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug

38. Addition of Costs Holding Cost (per unit per month): $8.50
Hiring Cost per worker: $800
Firing Cost per worker: $1,250
Payroll Cost: $75/worker/day
Shortage Cost: $50 unit short/month

39. Cost Evaluation of Constant Work Force Plan
Assume that the work force at the end of Dec. was 40.
Cost to hire 6 workers: 6*800 = $4800
Inventory Cost: accumulate ending inventory: ( ) = Add in 100 units netted out in Aug = 2193.
Hence Inv. Cost = 2193*8.5=$18,640.50
Payroll cost:
($75/worker/day)(46 workers )(167days) = $576,150
Cost of plan: $576,150 + $18, $4800 = $599,590.50

40. Cost Reduction in Constant Work Force Plan (Mixed Strategy)
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.

41. Zero Inventory Plan (Chase Strategy)
Here the idea is to change the workforce each month in order to reduce ending inventory to nearly zero by matching the workforce with monthly demand as closely as possible. This is accomplished by computing the # of units produced by one worker each month (by multiplying K by #days per month) and then taking net demand each month and dividing by this quantity. The resulting ratio is rounded up to avoid shortages.

42. An Alternative is called the “Chase Plan”
Here, we hire and fire (layoff) workers to keep inventory low!
We would employ only the number of workers needed each month to meet the demand
Examining our chart (earlier) we need:
Jan: 30; Feb: 51; Mar: 59; Apr: 27; May: 43 Jun: 20; Jul: 15; Aug: 30
Found by: (monthly demand)  (monthly production/worker), for Jan= 220/(22*.3425)

43. An Alternative is called the “Chase Plan”
So we hire or Fire (lay-off) monthly
Jan (starts with 40 workers): Fire 10 (cost $8000)
Feb: Hire 21 (cost $16800)
Mar: Hire 8 (cost $6400)
Apr: Fire 31 (cost $38750)
May: Hire 15 (cost $12000)
Jun: Fire 23 (cost $28750)
Jul: Fire 5 (cost $6250)
Aug: Hire 15 (cost $12000)
Total Personnel Costs: $128950

44. Changing the Level of Work Force
Period # hired #fired

45. An Alternative is called the “Chase Plan”
Inventory cost is essentially 165*8.5 = $
Employment costs: $428325
Chase Plan Total: $
It is better than the “Constant Workforce Plan” by:
– = 40913
But will this be good for your image?
Can we find a better plan?

46. Example
Demand for Quantum Corporation’s action toy series follows a seasonal pattern – growing through the fall months and culminating in December, with smaller peaks in January (for after-season markdowns, exchanges, and accessory purchases) and July (for Christmas-in-July specials).
MONTH
DEMAND (CASES)
January
1000
July
500
February
400
August
March
September
April
October
1500
May
November
2500
June
December
3000
Each worker can produce on average 100 cases of action toys each month. Overtime is limited to 300 cases, and subcontracting is unlimited. No action toys are currently in inventory. The wage rate is $10 per case for regular production, $15 for overtime production, and $25 for subcontracting. No stockouts are allowed. Holding cost is $1 per case per month. Increasing the workforce costs approximately $1,000 per worker. Decreasing the workforce costs $500 per worker.

47. Example – Level Production
Input: Beg. Wkrs 10 Regular $10 Hiring $1,000
Units/Wkr 100 Overtime $15 Firing $500 Cost: $146,000
Beg. Inv. 0 Subk $25 Inventory $1
Month
Demand
Reg OT
Subk
Inv
#Wkrs
#Hired
#Fired
Jan
1000
1,000 10
Feb
400
600
Mar
1,200
Apr
1,800
May
2,400
Jun
3,000
Jul
500
3,500
Aug
4,000
Sept
Oct
1500
Nov
2500
2,000
Dec
3000
Total
12,000
26,000

48. Example – Chase Demand Input: Beg. Wkrs 10 Regular $10 Hiring $1,000
Units/Wkr 100 Overtime $15 Firing $500 Cost: $149,000
Beg. Inv. 0 Subk $25 Inventory $1
Month
Demand
Reg OT
Subk
Inv
#Wkrs
#Hired
#Fired
Jan
1000 10
Feb
400 4 6
Mar
Apr
May
Jun
Jul
500 5 1
Aug
Sept
Oct
1500 15
Nov
2500 25
Dec
3000 30
Total
12,000 26

49. Disaggregating The Aggregate Plan
Disaggregation of aggregate plans mean converting an aggregate plan to a detailed master production schedule for each individual item (remember the hierarchical product structure given earlier: items, families, types).
Keep in mind that unless the results of the aggregate plan can be linked to the master production schedule, the aggregate planning methodology could have little value.

50. Aggregate Plan to Master Schedule
Aggregate Planning
Disaggregation
Master Schedule

51. Rough-cut Capacity Planning
Aggregate planning is based on a general production plan that deals with how much capacity will be available and how it will be allocated. A rough-cut capacity plan can be developed to evaluate the work load that a production plan imposes on work centers.

52. Below is a bill of labor which lists the hours
Below is a bill of labor which lists the hours required in each department to make one unit of product.

53. Master Production Schedule

54. Develop capacity requirements for the following combinations:

55. The solution is outlined below:
240(1.1) + 220(0.3) + 480(0.7) + 330(0.6) = 864 hours (Month 2)
420(0.4) + 230(0.2) + 440(0.7) + 410(0.5) = 727 hours (Month 4)
260(0.5) + 200(0.0) + 400(0.4) + 500(0.6) = 590 hours (Month 6)
300(0.1) + 240(0.6) + 450(0.4) + 380(0.9) = 696 hours (Month 3)

56. Excess Demand
For example, we must have at least 864 hours available in Department 33 for Month 2 to meet capacity requirements. Suppose that we only have 640 man hours available in Department 33 in Month 2. Then, we can use aggregate planning strategies such as hiring, overtime, etc. to bring the capacity up to the required amount of 864 man or machine hours in order to comply with master production schedule.

57. Work Force Size Planning
In aggregate planning the major objective is to determine feasible and possibly optimal production quantities and the corresponding capacity (work force size) to accommodate such production requirements. An example of determining the appropriate work force size follows.

58. Work Force Size Planning
Forecasted Demand
Quarter (standard units of work)
,000
,500
,200
,500
1 year ,200

59. Questions
a. Assume employees contribute 180 regular working hours each month, and each unit requires 2 hours to produce. How many employees will be needed during Quarter 1 and Quarter 2?
b. What will be the average labor cost for each unit if the company pays employees $10/hour and maintains for the entire year a sufficient staff to meet the peak demand?
c. What percentage above the standard-hour cost is the company's average labor cost per unit in this year due to excess staffing for all but the peak quarterly period?

60. Answers a. Quarter 1:6000 x 2 = 12,000 hours
12,000 / (180 x 3 months) = > 23 employees
Quarter 2: 4,500 x 2 = 9,000
9,000/(180 x 3 months) = > 17 employees

61. Solution
b. (180 hours/employ.-month x 12 months x 23 employ. x $10/hour)/20200= = 496,800/20200= $24.59/unit c. 2 hours x $10/hour = $20/unit (standard cost) ($ $20)/$20 = 0.23 or 23% higher

62. Optimal Solutions to Aggregate Planning Problems Via Linear Programming
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 workers 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.

63. Optimal Solutions to Aggregate Planning Problems Via Linear Programming
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.

64. Exploring the Optimal (L.P.) Approach
We need an Objective Function for cost of the aggregate plan (target is to minimize costs):
Here the ci’s are cost for hiring, firing, inventory, production, etc
HT and FT are number of workers hired and fired
IT, PT, OT, ST AND UT are numbers units inventoried, produced on regular time, on overtime, by ‘sub-contract’ or the number of units that could be produced on idled worker hours respectively

65. Exploring the Optimal (L.P.) Approach
This objective Function would be subject to a series of constraints (one of each type for each period)
‘Number of Workers’ Constraints:
Inventory Constraints:
Production Constraints:
Where: nt * k is the number of units produced by a worker in a given period of nt days

66. Real Constraint Equation (rewritten for L.P.):
Employee Constraints:
Inventory Constraints:

67. Real Constraint Equations (rewritten for L.P.):
Production Constraints:

68. Real Constraint Equations (rewritten for L.P.):
Finally, we need constraints defining:
Initial Workforce size
Starting Inventory
Final Desired Inventory
And, of course, the general constraint forcing all variables to be  0

69. Example

70. LP formulation for the problem above:
Min 20X X X X X X X X X322
Subject To
X111 + X112 <=200
X211 + X212 <=80
X311 + X312 <=100
X122 <=180
X222 <=60
X322 <=100
X111 + X211 + X311 = 340
X112 + X212 + X312 + X122 + X222 + X322 = xijk>00 i:

71. Solution
For example, for X312 the first subscript (3) stands for the type of capacity (i.e. subcontract). The middle subscript (1) stands for the supplying period or production period (i.e. period 1). The last subscript (2) stands for the receiving period or consumption period (i.e. period 2).
The optimal values of the variables such as x111, x112, x211, etc. would show how much regular capacity, overtime capacity, etc. to use in each time period in order to minimize the cost of the planned aggregate production.