MD 021 - Management and Operations
Aggregate Planning
Outline
· Aggregate planning - definitions and strategies
· Linear programming (LP)
· Aggregate planning LP problem
· Chase and level strategy problems
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Definitions of Aggregate Planning
Aggregate planning is the “big picture” approach to planning for the intermediate term ( 1 year). The goal of aggregate planning is to achieve a production plan that will effectively utilize the organization’s resources to satisfy expected demand.
Planners must make decisions on:
· Output rates
· Employment levels and changes
· Inventory levels and changes
· Backorders
· Subcontracting/outsourcing
In a manufacturing firm, the aggregate plan links the strategic goals with plans for the individual products (i.e., the master production schedule).
In a service firm, the aggregate plan links the strategic goals with detailed work-force schedules.
Aggregation
Reason for aggregation - A planner can devise a course of action, consistent with strategic goals and objectives, without having to deal with a lot of detail.
Three dimensions of aggregation:
Product families - Based on similar demand requirements and common processing, labor, and materials requirements.
Labor - Considerations include work-force flexibility, physical and geographic locations.
Timing - The planning horizon is the length of time covered by the aggregate plan.
Planning Strategies
Chase strategy – Matching capacity to demand; the planned output for a period is set at the expected demand of the period.
Level strategy – Maintaining a steady rate of regular time output while meeting the variations in demand by a combination of inventories, overtime, part-time workers, subcontracting, and back-orders.
Pros / ConsChase strategy / · Low inventory investment and backlogs / · Expense of adjusting output rates and/or work-force
· Alienation of work-force
· Loss of productivity
· Lower quality
Level strategy / · Level output rates
· Stable work-force / · Increased inventory investment
· Increased undertime and overtime expense
· Increased backlogs
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Issues to Consider in Aggregate Planning
Production / Workforce / InventoryCapacity / Minimum level / Minimum level
Demand / Maximum level / Maximum level
Material cost / Overtime / Holding cost
Labor cost / Subcontracting
Overhead cost / Hiring cost
Service level / Firing/layoff costs
The Specific Motors Company (SMC), which manufactures only one model of car, wants to plan its production and inventory levels for the next 4 months. The following table provides the relevant data for each month, where the inventory levels in the last two columns refer to the levels at the end of the month:
Month / Demand / Production cost/unit / Maximum production level / Minimum production level / Maximum inventory level / Minimum inventory level1 / 10,000 / $10,800 / 25,000 / 3,000 / 15,000 / 2,000
2 / 15,000 / 11,000 / 35,000 / 3,000 / 15,000 / 2,000
3 / 25,000 / 11,000 / 30,000 / 3,000 / 15,000 / 2,000
4 / 20,000 / 11,300 / 10,000 / 3,000 / 15,000 / 2,000
SMC estimates that the cost to hold one car in inventory for one month is $150. To estimate a month's inventory costs, SMC multiplies the average of the month's starting and ending inventory levels by $150. SMC currently has an inventory level of 3000 cars. SMC wants to meet its demand with no backlogging; that is, all demand must be met in the month it occurs. Formulate an LP model for SMC.
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Let X1 = Number of cars to be produced in month 1
X2 = Number of cars to be produced in month 2
X3 = Number of cars to be produced in month 3
X4 = Number of cars to be produced in month 4
I1 = Inventory at the end of month 1
I2 = Inventory at the end of month 2
I3 = Inventory at the end of month 3
I4 = Inventory at the end of month 4
Min Z = 10800X1+11000X2+11000X3+11300X4 + 75I0 +
150I1+ 150I2 + 150I3 + 75I4
Subject to:
I0 = 3000
I0+X1- I1 = 10000
I1 + X2 - I2 = 15000
I2 + X3 - I3 = 25000
I3 + X4 - I4 = 20000
X1 25000 X1 3000
X2 35000 X2 3000
X3 30000 X3 3000
X4 10000 X4 3000
I1 15000 I1 2000
I2 15000 I2 2000
I3 15000 I3 2000
I4 15000 I4 2000
X1, X2, X3, X4, I1, I2, I3, I4 0
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Microsoft Excel 7.0 Answer ReportTarget Cell (Min)
Cell / Name / Original Value / Final Value
$K$4 / Total cost / 0 / 763075000
Adjustable Cells
Cell / Name / Original Value / Final Value
$B$2 / Cars produced in month 1 / 0 / 22000
$C$2 / Cars produced in month 2 / 0 / 10000
$D$2 / Cars produced in month 3 / 0 / 30000
$E$2 / Cars produced in month 4 / 0 / 7000
$F$2 / Initial ending inventory / 0 / 3000
$G$2 / Inventory end of month 1 / 0 / 15000
$H$2 / Inventory end of month 2 / 0 / 10000
$I$2 / Inventory end of month 3 / 0 / 15000
$J$2 / Inventory end of month 4 / 0 / 2000
Constraints
Cell / Name / Cell Value / Formula / Status / Slack
$K$6 / Demand1 LHS / 10000 / $K$6=$L$6 / Binding / 0
$K$7 / Demand2 LHS / 15000 / $K$7=$L$7 / Binding / 0
$K$8 / Demand3 LHS / 25000 / $K$8=$L$8 / Binding / 0
$K$9 / Demand4 LHS / 20000 / $K$9=$L$9 / Binding / 0
$K$10 / MaxProd1 LHS / 22000 / $K$10<=$L$10 / Not Binding / 3000
$K$11 / MaxProd2 LHS / 10000 / $K$11<=$L$11 / Not Binding / 25000
$K$12 / MaxProd3 LHS / 30000 / $K$12<=$L$12 / Binding / 0
$K$13 / MaxProd4 LHS / 7000 / $K$13<=$L$13 / Not Binding / 3000
$K$14 / MinProd1 LHS / 22000 / $K$14>=$L$14 / Not Binding / 19000
$K$15 / MinProd2 LHS / 10000 / $K$15>=$L$15 / Not Binding / 7000
$K$16 / MinProd3 LHS / 30000 / $K$16>=$L$16 / Not Binding / 27000
$K$17 / MinProd4 LHS / 7000 / $K$17>=$L$17 / Not Binding / 4000
$K$18 / MaxInv1 LHS / 15000 / $K$18<=$L$18 / Binding / 0
$K$19 / MaxInv2 LHS / 10000 / $K$19<=$L$19 / Not Binding / 5000
$K$20 / MaxInv3 LHS / 15000 / $K$20<=$L$20 / Binding / 0
$K$21 / MaxInv4 LHS / 2000 / $K$21<=$L$21 / Not Binding / 13000
$K$22 / MinInv1 LHS / 15000 / $K$22>=$L$22 / Not Binding / 13000
$K$23 / MinInv2 LHS / 10000 / $K$23>=$L$23 / Not Binding / 8000
$K$24 / MinInv3 LHS / 15000 / $K$24>=$L$24 / Not Binding / 13000
$K$25 / MinInv4 LHS / 2000 / $K$25>=$L$25 / Binding / 0
$K$26 / Inv0 LHS / 3000 / $K$26=$L$26 / Binding / 0
$B$2 / Cars produced in month 1 / 22000 / $B$2>=0 / Not Binding / 22000
$C$2 / Cars produced in month 2 / 10000 / $C$2>=0 / Not Binding / 10000
$D$2 / Cars produced in month 3 / 30000 / $D$2>=0 / Not Binding / 30000
$E$2 / Cars produced in month 4 / 7000 / $E$2>=0 / Not Binding / 7000
$F$2 / Initial ending inventory / 3000 / $F$2>=0 / Not Binding / 3000
$G$2 / Inventory end of month 1 / 15000 / $G$2>=0 / Not Binding / 15000
$H$2 / Inventory end of month 2 / 10000 / $H$2>=0 / Not Binding / 10000
$I$2 / Inventory end of month 3 / 15000 / $I$2>=0 / Not Binding / 15000
$J$2 / Inventory end of month 4 / 2000 / $J$2>=0 / Not Binding / 2000
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Sensitivity Analysis
1. Objective function coefficients
· Holding all else the same, what is the range that each objective function coefficient can change without changing the optimal solution?
2. Right-hand-side parameters
· Holding all else the same, what is the change in the value of the objective function if we add one more unit of each constrained resource? In other words, what is the shadow price?
· What is the range over which the right-hand-side can vary while its shadow price remains valid?
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Microsoft Excel 7.0 Sensitivity ReportChanging Cells
Final / Reduced / Objective / Allowable / Allowable
Cell / Name / Value / Cost / Coefficient / Increase / Decrease
$B$2 / Cars produced in month 1 / 22000 / 0 / 10800 / 50 / 1E+30
$C$2 / Cars produced in month 2 / 10000 / 0 / 11000 / 0 / 50
$D$2 / Cars produced in month 3 / 30000 / 0 / 11000 / 150 / 1E+30
$E$2 / Cars produced in month 4 / 7000 / 0 / 11300 / 1E+30 / 0
$F$2 / Initial ending inventory / 3000 / 0 / 75 / 1E+30 / 1E+30
$G$2 / Inventory end of month 1 / 15000 / 0 / 150 / 50 / 1E+30
$H$2 / Inventory end of month 2 / 10000 / 0 / 150 / 0 / 150
$I$2 / Inventory end of month 3 / 15000 / 0 / 150 / 0 / 1E+30
$J$2 / Inventory end of month 4 / 2000 / 0 / 75 / 1E+30 / 11375
Constraints
Final / Shadow / Constraint / Allowable / Allowable
Cell / Name / Value / Price / R.H. Side / Increase / Decrease
$K$6 / Demand1 LHS / 10000 / 10800 / 10000 / 3000 / 19000
$K$7 / Demand2 LHS / 15000 / 11000 / 15000 / 25000 / 7000
$K$8 / Demand3 LHS / 25000 / 11150 / 25000 / 5000 / 7000
$K$9 / Demand4 LHS / 20000 / 11300 / 20000 / 3000 / 4000
$K$10 / MaxProd1 LHS / 22000 / 0 / 25000 / 1E+30 / 3000
$K$11 / MaxProd2 LHS / 10000 / 0 / 35000 / 1E+30 / 25000
$K$12 / MaxProd3 LHS / 30000 / -150 / 30000 / 7000 / 5000
$K$13 / MaxProd4 LHS / 7000 / 0 / 10000 / 1E+30 / 3000
$K$14 / MinProd1 LHS / 22000 / 0 / 3000 / 19000 / 1E+30
$K$15 / MinProd2 LHS / 10000 / 0 / 3000 / 7000 / 1E+30
$K$16 / MinProd3 LHS / 30000 / 0 / 3000 / 27000 / 1E+30
$K$17 / MinProd4 LHS / 7000 / 0 / 3000 / 4000 / 1E+30
$K$18 / MaxInv1 LHS / 15000 / -50 / 15000 / 3000 / 13000
$K$19 / MaxInv2 LHS / 10000 / 0 / 15000 / 1E+30 / 5000
$K$20 / MaxInv3 LHS / 15000 / 0 / 15000 / 4000 / 3000
$K$21 / MaxInv4 LHS / 2000 / 0 / 15000 / 1E+30 / 13000
$K$22 / MinInv1 LHS / 15000 / 0 / 2000 / 13000 / 1E+30
$K$23 / MinInv2 LHS / 10000 / 0 / 2000 / 8000 / 1E+30
$K$24 / MinInv3 LHS / 15000 / 0 / 2000 / 13000 / 1E+30
$K$25 / MinInv4 LHS / 2000 / 11375 / 2000 / 3000 / 2000
$K$26 / Inv0 LHS / 3000 / -10725 / 3000 / 19000 / 3000
Sensitivity Analysis
Aggregate Planning Problem
1. What changes in the aggregate plan would cause costs to go down?
2. Why does an increase in the minimum inventory level at the end of month 4 cause costs to go up? Why is this deceptive?
3. According to the sensitivity report, increasing the initial inventory by 1 results in costs going down by $10,725. Why is this deceptive?
Level and Chase Strategies Example
Bob Carlton’s golf camp estimates the following workforce requirements for its services over the next two years.
Quarter / 1 / 2 / 3 / 4Demand / 4200 / 6400 / 3000 / 4800
Quarter / 5 / 6 / 7 / 8
Demand / 4400 / 6240 / 3600 / 4800
Each certified instructor puts in 480 hours per quarter regular time and can work an additional 120 hours overtime. Regular-time wages and benefits cost Carlton $7200 per employee per quarter for regular time worked up to 480 hours, with an overtime cost of $20 per hour. Unused regular time for certified instructors is paid at $15 per hour. There is no cost for unused overtime capacity. The cost of hiring, training, and certifying a new employee is $10,000. Layoff costs are $4000 per employee. Currently, eight employees work in this capacity.
Level Strategy
Find a level work-force plan that allows for no delay in service and minimizes undertime. What is the total cost of this plan?
Chase Strategy
Use a chase strategy that varies the work-force level without using overtime or undertime. What is the total cost of this plan?
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