Homework 8-1

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Supply Chain Management: Strategy, Planning, and Operation

Seventh Edition

Chapter 8

Aggregate Planning in a Supply Chain

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Learning Objectives

8.1 Describe aggregate planning and its importance as a supply chain activity.

8.2 Explain the basic trade-offs to consider when creating an aggregate plan.

8.3 Model and solve the aggregate planning problem as a linear program.

8.4 Formulate and solve basic aggregate planning problems using Microsoft Excel.

Aggregate Planning and Its Role in a Supply Chain

Capacity has a cost and lead times are often long

Aggregate planning:

Given the demand forecast for each period in the planning horizon, determine the production level, inventory level, capacity level (internal and outsourced), and any backlogs (unmet demand) for each period that maximize the firm’s profit over the planning horizon.

How can a firm best use the facilities it has?

Role of Aggregate Planning in a Supply Chain

Identify operational parameters over the specified time horizon

Production rate

Workforce

Overtime

Machine capacity level

Subcontracting

Backlog

Inventory on hand

All supply chain stages should work together on an aggregate plan that will optimize supply chain performance

The Aggregate Planning Problem

Given the demand forecast for each period in the planning horizon, determine the production level, inventory level, and the capacity level for each period that maximizes the firm’s (supply chain’s) profit over the planning horizon

Specify the planning horizon (typically 3-18 months)

Specify the duration of each period

Specify key information required to develop an aggregate plan

Information Needed for An Aggregate Plan

Aggregate demand forecast Ft for each Period t over T periods

Production costs

Labor costs, regular time ($/hr) and overtime ($/hr)

Subcontracting costs ($/hr or $/unit)

Cost of changing capacity – hiring or layoff ($/worker), adding or reducing machine capacity ($/machine)

Labor/machine hours required per unit

Inventory holding cost ($/unit/period)

Stockout or backlog cost ($/unit/period)

Constraints – overtime, layoffs, capital available, stockouts, backlogs, from suppliers

Outputs of Aggregate Plan

Production quantity from regular time, overtime, and subcontracted time

Inventory held

Backlog/stockout quantity

Workforce hired/laid off

Machine capacity increase/decrease

A poor aggregate plan can result in lost sales, lost profits, excess inventory, or excess capacity

Summary of Learning Objective 1 (1 of 2)

To create an aggregate plan, a planner needs a demand forecast, cost and production information, and any supply constraints. The demand forecast consists of an estimate of demand for each period of time in the planning horizon. The production and cost data consist of capacity levels and costs to raise and lower them, production costs, costs to store the product, costs of stocking out the product, and any restrictions that limit these factors. Supply constraints determine limits on outsourcing, overtime, or materials. The aggregate plan then determines capacity, production, and inventory decisions over the next 3 to 18 months

Summary of Learning Objective 1 (2 of 2)

Good aggregate planning is done in collaboration with both customers and suppliers because accurate input is required from both stages. The quality of these inputs, in terms of both the demand forecast to be met and the constraints to be dealt with, determines the quality of the aggregate plan. The results of the aggregate plan must also be shared across the supply chain because they influence activities for both customers and suppliers. For suppliers, the aggregate plan determines anticipated orders; for customers, the aggregate plan determines planned supply.

Basic Tradeoffs in Aggregate Planning

Trade-off between capacity, inventory, backlog/lost sales

Chase strategy – using capacity as the lever

Flexibility strategy – using utilization as the lever

Level strategy – using inventory as the lever

Tailored or hybrid strategy – a combination of strategies

Chase Strategy

Vary machine capacity or hire and lay off workers as demand varies

Often difficult to vary capacity and workforce on short notice

Expensive if cost of varying capacity is high

Negative effect on workforce morale

Results in low levels of inventory

Used when inventory holding costs are high and costs of changing capacity are low

Utilization Flexibility Strategy

Use excess machine capacity

Workforce stable, number of hours worked varies

Use overtime or a flexible work schedule

Flexible workforce, avoids morale problems

Low levels of inventory, lower utilization

Used when inventory holding costs are high and capacity is relatively inexpensive

Level Strategy

Stable machine capacity and workforce levels, constant output rate

Inventory levels fluctuate over time

Inventories carried over from high to low demand periods

Better for worker morale

Large inventories and backlogs may accumulate

Used when inventory holding and backlog costs are relatively low

Summary of Learning Objective 2

The basic trade-offs in aggregate planning involve balancing the cost of capacity, the cost of inventory, and the cost of stockouts to maximize profitability. Increasing any one of the three allows the planner to lower the other two.

Aggregate Planning Using Linear Programming

Maximize profits while respecting supply chain constraints

Red Tomato Tools

Capacity determined mainly by workforce size

Identifying Aggregate Units of Production

Aggregate unit should be identified in a way that the resulting production schedule can be accomplished in practice

Focus on the bottlenecks when selecting the aggregate unit and identifying capacity and production times

Account for activities such as setups and maintenance

Red Tomato Tools (1 of 8)

Table 8-1 Costs, Revenues, and Times at Red Tomato Tools

Family	Material Cost/ Unit ($)	Revenue/ Unit ($)	Setup Time/Batch (hour)	Average Batch Size	Production Time/ Unit (hour)	Net Production Time/Unit (hour)	Percentage Share of Units Sold
A	15	54	8	50	5.60	5.76	10
B	7	30	6	150	3.00	3.04	25
C	9	39	8	100	3.80	3.88	20
D	12	49	10	50	4.80	5.00	10
E	9	36	6	100	3.60	3.66	20
F	13	48	5	75	4.30	4.37	15

Red Tomato Tools (2 of 8)

Weighted average approach

Material cost per aggregate unit

= (15 × 0.10) + (7 × 0.25) + (9 × 0.20)

+ (12 × 0.10) + (9 × 0.20) + (13 × 0.15)

= $10

Similarly

Revenue per aggregate unit = $40

Net production time per aggregate unit = 4.00 hours

Demand and Costs (1 of 3)

Highly seasonal demand

Options

Adding workers during peak times

Subcontract

Build up inventory

Develop a forecast

Demand and Costs (2 of 3)

Table 8-2 Demand Forecast

Month	Demand Forecast
January	1,600
February	3,000
March	3,200
April	3,800
May	2,200
June	2,200

Demand and Costs (3 of 3)

Starting inventory in January = 1,000

20 working days each month

Employees earn $4/hour regular time

Regular time = 8 hours/day, then overtime

Maximum 10 hours overtime/employee/month

End June with 500 units in inventory

Red Tomato Tools (3 of 8)

Table 8-3 Costs for Red Tomato

Item	Cost
Material cost	$10/unit
Inventory holding cost	$2/unit/month
Marginal cost of stockout/backlog	$5/unit/month
Hiring and training costs	$300/worker
Layoff cost	$500/worker
Labor hours required	4/unit
Regular time cost	$4/hour
Overtime cost	$6/hour
Cost of subcontracting	$30/unit

Red Tomato Tools Decision Variables

For t = 1, ..., 6

Wt = Workforce size for month t

Ht = Number of employees hired at the beginning of month t

Lt = Number of employees laid off at the beginning of month t

Pt = Production in month t

It = Inventory at the end of month t

St = Number of units stocked out at the end of month t

Ct = Number of units subcontracted for month t

Ot = Number of overtime hours worked in month t

Red Tomato Tools Objective Function

Minimize

(Regular-time labor cost) + (Overtime labor cost) + (Cost of hiring and layoffs) + (Cost of holding inventory) + (Cost of stocking out) + (Cost of subcontracting) + (Material cost)

Red Tomato Tools Constraints

All for t = 1,..., 6

Workforce, hiring, and layoff constraints

Capacity constraints

Inventory balance constraints

Overtime limit constraints

Red Tomato Tools (4 of 8)

Total cost over planning horizon = $422,660

Table 8-4 Aggregate Plan for Red Tomato

Period, t	No. Hired, Ht	No. Laid Off, Lt	Workforce Size, Wt	Overtime, Ot	Inventory, It	Stockout, St	Subcontract, Ct	Total Production, Pt	Demand, Dt
0	0	0	80	0	1,000	0	0	Blank	Blank
1	0	16	64	0	1,960	0	0	2,583	1,600
2	0	0	64	0	1,520	0	0	2,583	3,000
3	0	0	64	0	880	0	0	2,583	3,200
4	0	0	64	0	0	220	140	2,583	3,800
5	0	0	64	0	140	0	0	2,583	2,200
6	0	0	64	0	500	0	0	2,583	2,200

Red Tomato Tools (5 of 8)

Higher demand variability

Table 8-5 Demand Forecast with Higher Seasonal Fluctuation

Month	Demand Forecast
January	1,000
February	3,000
March	3,800
April	4,800
May	2,000
June	1,400

Red Tomato Tools (6 of 8)

Total cost over planning horizon = $433,080

Table 8-6 Optimal Aggregate Plan for Demand in Table 8-5

Period, t	No. Hired, Ht	No. Laid Off, Lt	Workforce Size, Wt	Overtime, Ot	Inventory, It	Stockout, St	Subcontract, Ct	Total Production, Pt	Demand, Dt
0	0	0	80	0	1,000	0	0	Blank	Blank
1	0	16	64	0	2,560	0	0	2,560	1,000
2	0	0	64	0	2,120	0	0	2,560	3,000
3	0	0	64	0	880	0	0	2,560	3,800
4	0	0	64	0	0	1,220	140	2,560	4,800
5	0	0	64	0	0	660	0	2,560	2,000
6	0	0	64	0	500	0	0	2,560	1,400

Red Tomato Tools (7 of 8)

Lower hiring and layoff costs – $50

Total cost over planning horizon = $412,770

Table 8-7 Optimal Aggregate Plan for Hiring and Layoff Cost of $50/Worker

Period, t	No. Hired, Ht	No. Laid Off, Lt	Workforce Size, Wt	Overtime, Ot	Inventory, It	Stockout, St	Subcontract, Ct	Total Production, Pt	Demand, Dt
0	0	0	80	0	1,000	0	0	Blank	Blank
1	0	35	45	0	1,200	0	0	1,800	1,600
2	0	0	45	0	0	0	0	1,800	3,000
3	42	0	87	0	280	0	0	3,480	3,200
4	1	0	88	0	0	0	0	3,520	3,800
5	0	27	61	0	240	0	0	2,440	2,200
6	0	0	61	0	500	0	20	2,440	2,200

Red Tomato Tools (8 of 8)

Building a Rough Master Production Schedule

Table 8-8 Disaggregating the Aggregate Plan at Red Tomato Tools for Period 1

Product	Setup Time/ Batch (hour)	Average Batch Size	Production Time/ Unit (hour)	Production Quantity	Number of Setups	Setup Time (hours)	Production Time (hours)
A	8	50	5.60	256	5	40	1,433.6
B	6	150	3.00	640	4	24	1,920.0
C	8	100	3.80	512	5	40	1,945.6
D	10	50	4.80	256	5	50	1,228.8
E	6	100	3.60	512	5	30	1,843.2
F	5	75	4.30	384	5	25	1,651.2

Planned production and setup = 10,231.4 hrs Available production time = 10,240 hrs

Summary of Learning Objective 3

Given the goal of maximizing profits (or minimizing costs) subject to supply chain constraints, the aggregate planning problem can be modeled as a linear program. The first step is to identify a suitable aggregate unit of production and forecast demand in terms of the aggregate unit. The next step is to identify the various costs (such as material, inventory, production) and constraints in the supply chain. We then identify the set of decision variables and construct the objective function and constraints in terms of the decision variables. Linear programming then allows us to optimize the objective function subject to the specified constraints. The aggregate plan should then be converted to a feasible master production schedule.

Aggregate Planning in Excel

Table 8-9 Building the Basic Aggregate Planning Spreadsheet

Output	Cells	Relationship to inputs	Formula in Row 5	Copied to Calls
Workforce	D5:D10	W sub t = W sub t minus 1 + H sub t minus L sub t.	= D4 + B5 – C5	D6:D10
Production	I5:I10	P sub t = 40 times W sub t plus start fraction O sub t over 4 end fraction.	= 40 times D 5 + left parenthesis E 5 over 4 right parenthesis	I6:I10
Inventory	F5:F10	I sub t = max of left parenthesis I sub t minus 1 plus P sub t + C sub t minus D sub t minus S sub t minus 1, 0 right parenthesis	= max left parenthesis F 4 + I 5 + H 5 minus G 4 minus J 5, 0 right parenthesis	F6:F10
Stockout	G5:G10	S sub t minus max left parenthesis 0, S, sub t minus 1 + D sub t + I sub t minus 1 minus P sub t minus c sub t	Equals max of left parenthesis 0, J 5 + G 4 minus I 5, minus H 5 minus F 4	G6:G10

Aggregate Planning Using Solver (1 of 8)

Figure 8-1 Basic Aggregate Planning Spreadsheet

Aggregate Planning Using Solver (2 of 8)

For t = 1, ..., 6

Wt = Workforce size for Month t

Ht = Number of employees hired at the beginning of Month t

Lt = Number of employees laid off at the beginning of Month t

Pt = Production in Month t

It = Inventory at the end of Month t

St = Number of units stocked out at the end of Month t

Ct = Number of units subcontracted for Month t

Ot = Number of overtime hours worked in Month t

Aggregate Planning Using Solver (3 of 8)

Figure 8-2 Spreadsheet Area for Decision Variables

Aggregate Planning Using Solver (4 of 8)

Figure 8-3 Spreadsheet Area for Constraints

Aggregate Planning Using Solver (5 of 8)

Figure 8-4 Spreadsheet Area for Cost Calculations

Aggregate Planning Using Solver (6 of 8)

Set Target Cell: C22

Equal to: Select Min

By Changing Cells: B5:I10

Subject to the constraints:

B5:C10 = integer {Number of workers hired or laid off is integer}

Aggregate Planning Using Solver (7 of 8)

{Inventory at end of Period 6 is at least 500}

G10 = 0 {Stockout at end of Period 6 equals 0}

Aggregate Planning Using Solver (8 of 8)

Figure 8-5 Solver Parameters Dialog Box

Forecast Error in Aggregate Plans (1 of 2)

Forecast errors must be considered, flexibility must be built in

Safety inventory

Build and carry extra inventories to satisfy higher than forecasted demand

Forecast Error in Aggregate Plans (2 of 2)

Safety capacity

Capacity used to satisfy higher than forecast demand

Use overtime as a form of safety capacity

Carry extra workforce permanently as a form of safety capacity

Use subcontractors as a form of safety capacity

Build and carry extra inventories as a form of safety inventory

Purchase capacity or product from an open or spot market as a form of safety capacity

The Role of Software in Aggregate Planning

The ability to handle large amounts of data

Develop optimal solutions using linear programming

The ability to handle complex problems (often using linear approximations of nonlinear functions)

Stability and data accuracy are important

Summary of Learning Objective 4

Aggregate planning problems can be solved in Excel by setting up cells for the objective function and the constraints and using the Solver tool to produce the solution. It is best if these plans can account for forecast error, resulting in a plan that has some degree of stability.

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made, one can determine the workforce (Wt), production (Pt), inventory (It) and stockout (St) for each month (see Table 8-8), thus completing the aggregate plan. Figure 8-1 shows the final spreadsheet (available as Chapter8-trial-aggplan try different inputs for each decision variable in the appropriate cells in worksheet Planning. It is

* Wt and is shown in cells

each period. For each set of inputs, the outputs are calculated as shown in Table 8-8.

= *sum(D :D1 ) + *sum(E :E1 ) + 3 *sum(B :B1 ) + *sum(C :C1 ) + *sum(F :F1 ) + *sum(G :G1 ) + 1 *sum(I :I1 ) + 3 *sum(H :H1 )

The goal is to build an aggregate plan by changing inputs in a way that minimizes the total cost

Building an Aggregate Planning Spreadsheet Using Solver

To access Excel’s linear programming capabilities, use Solver (Data ! Analysis ! Solver). To

Chapter8,9-examples), containing the following decision variables:

Wt = workforce size for Month t, t = Ht = number of employees hired at the beginning of Month t, t = Lt = number of employees laid off at the beginning of Month t, t = Pt = number of units produced in Month t, t = It = inventory at the end of Month t, t = St = number of units stocked out at the end of Month t, t = Ct = number of units subcontracted for Month t, t = Ot = number of overtime hours worked in Month t, t =

FIGURE 8-1 Basic Aggregate Planning Spreadsheet

TABLE 8-8 Building the Basic Aggregate Planning Spreadsheet Output Cells Relationship to Inputs Formula in Row 5 Copied to Cells

Workforce D5:D10 Wt = Wt-1 + Ht - Lt = D4 + B5 - C5 D6:D10 Production I5:I10 Pt = 40 * Wt + Ot/4 = 40*D5 + (E5/4) I6:I10 Inventory F5:F10 It = max(It-1 + Pt + Ct - Dt - St-1, 0) = max(F4 + I5 + H5 - G4 - J5,0) F6:F10 Stockout G5:G10 St = max(0, St-1 + Dt - It!1 - Pt - Ct) = max(0,J5 + G4 - I5 - H5 - F4) G6:G10

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222

Planning) illustrates what this table should look like. The deci-

Also note that column J contains the actual demand. The demand information is included because it is required to calculate the aggregate plan.

- straint table may be constructed as shown in Figure 8-3.

the six periods. Each constraint will eventually be written in Solver as

Cell value5 … , = , or Ú 6 In our case, we have constraints

M :M1 = , N :N1 Ú , O :O1 = , : 1 Ú

FIGURE 8-2 Spreadsheet Area for Decision Variables

Cell

Equation

8.2

8.3

8.4

8.5

Copied to

M6:M10

N6:N10

O6:O10

P6:P10

Cell Formula

=D5-D4-B5+C5

=40*D5+E5/4-I5

=F4-G4+I5+H5-J5-F5+G5

=-E5+10*D5

FIGURE 8-3 Spreadsheet Area for Constraints

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Chapter 8 • Aggregate Planning in a Supply Chain 225

Cell

Equation

8.2

8.3

8.4

8.5

Copied to

M6:M10

N6:N10

O6:O10

P6:P10

Cell Formula

=D5 - D4 - B5 + C5

=40*D5 + E5/4 -I5

=F4-G4+I5+H5-J5-F5+G5

=-E5 + 10*D5

FIGURE 8-2 Spreadsheet Area for Constraints

The third step is to create a cell containing the objective function, which is how each solution is judged. This cell need not contain the entire formula but can be written as a formula using cells with intermediate cost calculations. For the Red Tomato example, the spreadsheet area for cost calculations is shown in Figure 8-3. Cell B15, for instance, contains the hiring costs incurred in Period 1. The formula in cell B15 is the product of cell B5 and the cell containing the hiring cost per worker, which is obtained from Table 8-3. Other cells are filled similarly. Cell C22 contains the sum of cells B15 to I20, representing the total cost.

The fourth step is to use Data Analysis Solver to invoke Solver. Within the Solver Parameters dialog box, enter the following information to represent the linear programming model:

Set Target Cell: C22

Equal to: Select Min

By Changing Cells: B5:I10

Subject to the constraints:

{All decision variables are nonnegative}

{Inventory at end of Period 6 is at least 500}

{Stockout at end of Period 6 equals 0}G10 = 0 F10 Ú 500 B5:I10 Ú 0

FIGURE 8-3 Spreadsheet Area for Cost Calculations

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Chapter 8 • Aggregate Planning in a Supply Chain 225

Cell

Equation

8.2

8.3

8.4

8.5

Copied to

M6:M10

N6:N10

O6:O10

P6:P10

Cell Formula

=D5 - D4 - B5 + C5

=40*D5 + E5/4 -I5

=F4-G4+I5+H5-J5-F5+G5

=-E5 + 10*D5

FIGURE 8-2 Spreadsheet Area for Constraints

The fourth step is to use Data Analysis Solver to invoke Solver. Within the Solver Parameters dialog box, enter the following information to represent the linear programming model:

Set Target Cell: C22

Equal to: Select Min

By Changing Cells: B5:I10

Subject to the constraints:

{All decision variables are nonnegative}

{Inventory at end of Period 6 is at least 500}

{Stockout at end of Period 6 equals 0}G10 = 0 F10 Ú 500 B5:I10 Ú 0

FIGURE 8-3 Spreadsheet Area for Cost Calculations

M08_CHOP3952_05_SE_C08.QXD 10/27/11 4:41 PM Page 225

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Chapter 8 ◆ Aggregate Planning in a Supply Chain 221

• Use subcontractors as a form of safety capacity. • Build and carry extra inventories as a form of safety inventory. • Purchase capacity or product from an open or spot market as a form of safety capacity.

In addition to the suggestions listed above, a manager should perform sensitivity analysis on the inputs into an aggregate plan. For example, if the plan recommends expanding expensive capacity while facing uncertain demand, examine the outcome of a new aggregate plan when demand is higher and lower than expected. If this examination reveals a small savings from expanding capacity when demand is high but a large increase in cost when demand is lower than expected, deciding to postpone the capacity investment decision is a potentially attractive option. Using sensitivity analysis on the inputs into the aggregate plan enables a planner to choose the best solution for the range of possibilities that may occur.

Even though an aggregate plan may provide a map for the next 3 to 18 months, it does not mean that a firm should run aggregate plans only once every 3 to 18 months. As inputs such as demand forecasts change, managers should use the latest values of these inputs and rerun the aggregate plan. Be careful, however, to modify plans in a way that limits volatility. Frequent changes may diminish the extent to which supply chain partners trust the aggregate plan.

The Role of Software in Aggregate Planning

Aggregate planning is arguably the supply chain area in which software has been used the most. The earliest supply chain software products were aggregate planning modules, often called fac- tory, production, or manufacturing planning. Some of the early modules focused only on obtain- ing a feasible production plan subject to constraints arising from demand and available capacity. Later modules provided tools that chose an optimal solution from among the feasible production plans, based on objectives such as increased output or lowered cost.

Figure 8-5 Solver Parameters Dialog Box

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