Chapter 3. Aggregate Planning (Steven Nahmias)

Hierarchy of Production Decisions Long-range Capacity Planning

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

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.

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.

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

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?

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.)

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.

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)

Example of fictitious aggregate units. (Example 3.1) 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?

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) + . . . + .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.

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.

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

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.

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.

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

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

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.

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.

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(420-200) 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(125+100) 1940

Step 2. Graph Cumulative Net Demand to Find Plans Graphically

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.

Level vs. Chase Strategy

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.

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.

Monthly Production = 1940/8 = 242. 2 or rounded to 243/month Monthly Production = 1940/8 = 242.2 or rounded to 243/month. But: there are stockouts.

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:

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) 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

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.

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

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) = .3421 average number of units produced by one worker in one day.

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 = 22+16+23 = 61. We find that: 960/61 = 15.7377 units/day 15.7377/.3421 = 46 workers required Actually 46.003 – 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

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?

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 1150.00 27.71 41.50 May 21.00 7.18 310.00 1460.00 34.89 41.84 Jun 145.00 1605.00 42.42 37.84 Jul 110.00 1715.00 49.60 34.57 Aug 225.00 1940.00 57.13 33.96

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!

Constant Work Force Production Plan 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

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

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: (126+98+0+. . .+687) = 2093. 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 + $18,640.50 + $4800 = $599,590.50

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.

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.

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)

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

Changing the Level of Work Force Period # hired #fired 1 10 2 21 3 8 4 31 5 15 6 23 7 5 8 15

An Alternative is called the “Chase Plan” Inventory cost is essentially 165*8.5 = $1402.50 Employment costs: $428325 Chase Plan Total: $558677.50 It is better than the “Constant Workforce Plan” by: 599590.50 – 558677.50 = 40913 But will this be good for your image? Can we find a better plan?

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.

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

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

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.

Aggregate Plan to Master Schedule Aggregate Planning Disaggregation Master Schedule

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.

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.

Master Production Schedule

Develop capacity requirements for the following combinations:

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)

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.

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.

Work Force Size Planning Forecasted Demand Quarter (standard units of work) 1 6,000 2 4,500 3 4,200 4 5,500 1 year 20,200

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?

Answers a. Quarter 1:6000 x 2 = 12,000 hours 12,000 / (180 x 3 months) = 22.22 ---> 23 employees Quarter 2: 4,500 x 2 = 9,000 9,000/(180 x 3 months) = 16.66 ---> 17 employees

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) ($24.59 -$20)/$20 = 0.23 or 23% higher

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.

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.

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

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

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

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

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

Example

LP formulation for the problem above: Min 20X111 + 23X112 + 25X211 + 28X212 + 30X311 + 33X312 + 20X122 + 25X222 + 30X322 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 = 300 xijk>00 i:

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.