Chapter 11 – Aggregate Planning and Master Scheduling
We know that we must not have a backlog in the final month.
How many units do we need to subcontract to ensure that we do not have a backlog in the
final month?
to minimize total cost as shown below:
Final Solution – Using Regular Time, Overtime, & Subcontracting
Period
1
2
3
4
5
6
7
Total
Forecast
50
44
55
60
50
40
51
350
Output
Regular
40
40
40
40
40
40
40
280
Part Time
0
Overtime
8
8
8
8
8
8
8
56
Subcontract
2
3
9
14
Output – Forecast
0
4
-4
-3
-2
8
-3
0
Inventory
Beginning
0
0
4
0
0
0
3
Ending
0
4
0
0
0
3
0
Average
0.0
2.0
2.0
0.0
0.0
1.5
1.5
7
Backlog
0
0
0
3
5
0
0
8
Costs:
Regular
@
80
$3,200
$3,200
$3,200
$3,200
$3,200
$3,200
$3,200
$22,400
Part Time
@
$0
$0
$0
$0
$0
$0
$0
$0
Overtime
@
120
$960
$960
$960
$960
$960
$960
$960
$6,720
Subcontract
@
140
$280
$0
$420
$1,260
$0
$0
$0
$1,960
Hire/Layoff
$0
Inventory
@
10
$0
$20
$20
$0
$0
$15
$15
$70
Back orders
@
20
$0
$0
$0
$60
$100
$0
$0
$160
Total
$4,440
$4,180
$4,600
$5,480
$4,260
$4,175
$4,175
$31,310
Chapter 11 – Aggregate Planning and Master Scheduling
11–22
Education.
8. Given:
A planner has developed aggregate forecasts for the next six months shown below:
Month
May
Jun
Jul
Aug
Sep
Oct
Forecast
4,000
4,800
5,600
7,200
6,400
5,000
Use the following information:
Regular production cost
$10 per case
Regular production capacity
5,000 cases
Overtime production cost
$16 per case
Subcontracting cost
$20 per case
Holding cost
$1 per case per month
Beginning inventory
0 cases
a. Use level production. Supplement using overtime as needed. No backlogs are allowed.
Step 1: Determine how much regular production to plan each month. Level Plan Regular
Production per Month = (Total Forecast – Beginning Inventory) / Number of Months =
(33,000 – 0) / 6 = 5,500 units per month. However, regular capacity is 5,000 units per month.
Therefore, we will plan on 5,000 units per month of regular production every month.
Step 2: Supplement with overtime production each month to ensure that there are no backlogs
in every month.
Period
May
Jun
Jul
Aug
Sep
Oct
Total
Forecast
4,000
4,800
5,600
7,200
6,400
5,000
33,000
Output
Regular
5,000
5,000
5,000
5,000
5,000
5,000
30,000
Part Time
0
Overtime
1,600
1,400
3,000
Subcontract
0
Output – Forecast
1,000
200
–600
–600
0
0
0
Inventory
Beginning
0
1,000
1,200
600
0
0
Ending
1,000
1,200
600
0
0
0
Average
500
1,100
900
300
0
0
2,800
Backlog
0
0
0
0
0
0
0
Costs:
Regular
@
10
$50,000
$50,000
$50,000
$50,000
$50,000
$50,000
$300,000
Part Time
@
$0
$0
$0
$0
$0
$0
$0
Overtime
@
16
$0
$0
$0
$25,600
$22,400
$0
$48,000
Subcontract
@
20
$0
$0
$0
$0
$0
$0
$0
Hire/Layoff
$0
Inventory
@
1
$500
$1,100
$900
$300
$0
$0
$2,800
Back orders
@
$0
$0
$0
$0
$0
$0
$0
Total
$50,500
$51,100
$50,900
$75,900
$72,400
$50,000
$350,800
Conclusion: Total cost = $350,800.
Chapter 11 – Aggregate Planning and Master Scheduling
11–23
Education.
b. Use a combination of overtime (500 cases per month maximum), inventory, and
subcontracting (500 cases per month maximum) to handle variations in demand. Backlogs are
not allowed.
Step 1: Determine the least cost option out of regular production, overtime production, and
subcontracting:
Regular production = $10 per case.
Overtime production = $16 per case.
Subcontracting = $20 per case.
Holding cost = $1 per case per month.
Step 2: Compare the costs of overtime production to subcontracting:
Overtime production cost per unit = $16.
Subcontracting cost per unit = $20.
Therefore, using overtime production in the current month always is preferred to
Period
May
Jun
Jul
Aug
Sep
Oct
Total
Forecast
4,000
4,800
5,600
7,200
6,400
5,000
33,000
Output
Regular
5,000
5,000
5,000
5,000
5,000
5,000
30,000
Part Time
0
Overtime
500
500
500
500
500
2,500
Subcontract
500
500
Output – Forecast
1,500
700
–100
–1,700
–400
0
0
Inventory
Beginning
0
1,500
2,200
2,100
400
0
Ending
1,500
2,200
2,100
400
0
0
Average
750
1,850
2,150
1,250
200
0
6,200
Backlog
0
0
0
0
0
0
0
Costs:
Regular
@
10
$50,000
$50,000
$50,000
$50,000
$50,000
$50,000
$300,000
Part Time
@
$0
$0
$0
$0
$0
$0
$0
Overtime
@
16
$8,000
$8,000
$8,000
$8,000
$8,000
$0
$40,000
Subcontract
@
20
$0
$0
$0
$0
$10,000
$0
$10,000
Hire/Layoff
$0
Inventory
@
1
$750
$1,850
$2,150
$1,250
$200
$0
$6,200
Back orders
@
$0
$0
$0
$0
$0
$0
$0
Total
$58,750
$59,850
$60,150
$59,250
$68,200
$50,000
$356,200
Chapter 11 – Aggregate Planning and Master Scheduling
11–24
Education.
c. Use overtime up to 750 cases per month and inventory to handle variations in demand. No
backlogs allowed.
Step 1: Determine the least cost option out of regular production, overtime production, and
subcontracting:
Regular production = $10 per case.
Overtime production = $16 per case.
Holding cost = $1 per case per month.
Regular production costs less than overtime production costs. Comparing regular time
units.
Step 2: Supplement with overtime production each month (maximum of 750 units) to ensure
that there are no backlogs in every month. We may have to plan overtime production early to
cover demand in some months. The key to this problem is determining how many units need
to be produced using overtime and then timing the production of those units to minimize total
cost and to ensure that there are no backlogs in every period.
For how many units do we need to use overtime production?
Total Forecast – Beginning Inventory – Total Regular Production = 33,000 – 0 – 30,000 =
3,000 units.
Period
May
Jun
Jul
Aug
Sep
Oct
Total
Forecast
4,000
4,800
5,600
7,200
6,400
5,000
33,000
Output
Regular
5,000
5,000
5,000
5,000
5,000
5,000
30,000
Part Time
0
Overtime
750
750
750
750
3,000
Subcontract
0
Output – Forecast
1,000
950
150
–1,450
–650
0
0
Inventory
Beginning
0
1,000
1,950
2,100
650
0
Ending
1,000
1,950
2,100
650
0
0
Average
500
1,475
2,025
1,375
325
0
5,700
Backlog
0
0
0
0
0
0
0
Costs:
Regular
@
10
$50,000
$50,000
$50,000
$50,000
$50,000
$50,000
$300,000
Part Time
@
$0
$0
$0
$0
$0
$0
$0
Overtime
@
16
$0
$12,000
$12,000
$12,000
$12,000
$0
$48,000
Subcontract
@
20
$0
$0
$0
$0
$0
$0
$0
Hire/Layoff
$0
Inventory
@
1
$500
$1,475
$2,025
$1,375
$325
$0
$5,700
Back orders
@
$0
$0
$0
$0
$0
$0
$0
Total
$50,500
$63,475
$64,025
$63,375
$62,325
$50,000
$353,700
Chapter 11 – Aggregate Planning and Master Scheduling
11–25
Education.
9. Given:
We have the information shown below. Subcontracting can handle a maximum of 10 units per
month. Beginning inventory is 0. No backorders are allowed.
Month
1
2
3
4
5
6
Demand
160
150
160
180
170
140
Regular
capacity
150
150
150
150
160
160
Overtime
capacity
10
10
0
10
10
10
Cost per Unit
Regular time
$50
Overtime
$75
Subcontract
$80
Holding per month
$4
Step 2: In Months 1 – 5, add overtime production first (cost per unit = $75) and then
subcontracting (cost per unit = $80) so that no backlogs occur.
Period
1
2
3
4
5
6
Total
Forecast
160
150
160
180
170
140
960
Output
Regular
150
150
150
150
160
140
900
Part Time
0
Overtime
10
10
10
10
40
Subcontract
10
10
20
Output – Forecast
0
20
0
–20
0
0
0
Inventory
Beginning
0
0
20
20
0
0
Ending
0
20
20
0
0
0
Average
0
10
20
10
0
0
40
Backlog
0
0
0
0
0
0
0
Costs:
Regular
@
50
$7,500
$7,500
$7,500
$7,500
$8,000
$7,000
$45,000
Part Time
@
$0
$0
$0
$0
$0
$0
$0
Overtime
@
75
$750
$750
$0
$750
$750
$0
$3,000
Subcontract
@
80
$0
$800
$800
$0
$0
$0
$1,600
Hire/Layoff
$0
Inventory
@
4
$0
$40
$80
$40
$0
$0
$160
Back orders
@
$0
$0
$0
$0
$0
$0
$0
Total
$8,250
$9,090
$8,380
$8,290
$8,750
$7,000
$49,760
11–26
Education.
10. Given:
Refer back to the solution in Solved Problem 1 (consider the solution for that problem to be Plan
A). The total cost of Plan A was $20,550. The given information in Solved Problem 1 was:
Current workforce = 20 people, each of whom can produce 10 units of output per period. Regular
production cost per unit = $6. Inventory carrying cost per unit per period = $5. Backlog cost per
Hire one worker at a cost of $200. Make up any shortfall, i.e., reduce backorders, using
subcontracting at $8 per unit, with a maximum of 20 units per period. Ending inventory in period
9 should be 0. Backorders cannot exceed 80 units in any period.
Regular production = regular capacity = (20 + 1) * 10 = 210 units per period.
Therefore, regular production could be used to meet demand of 9 * 210 = 1,890 units. How many
Period
1
2
3
4
5
6
7
8
9
Total
Forecast
190
230
260
280
210
170
160
260
180
1,940
Output
Regular
210
210
210
210
210
210
210
210
210
1,890
Part Time
0
Overtime
0
Subcontract
10
20
20
50
Output – Forecast
30
0
–30
–70
0
40
50
–50
30
0
Inventory
Beginning
30
30
0
0
0
0
20
0
Ending
30
30
0
0
0
0
20
0
0
Average
15
30
15
0
0
0
10.0
10.0
0.0
80
Backlog
0
0
0
70
70
30
0
30
0
200
Costs:
Regular
@
6
$1,260
$1,260
$1,260
$1,260
$1,260
$1,260
$1,260
$1,260
$1,260
$11,340
Part Time
@
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Overtime
@
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Subcontract
@
8
$80
$160
$160
$0
$0
$0
$0
$0
$0
$400
Hire/Layoff
$0
Inventory
@
5
$75
$150
$75
$0
$0
$0
$50
$50
$0
$400
Back orders
@
10
$0
$0
$0
$700
$700
$300
$0
$300
$0
$2,000
Total
$1,415
$1,570
$1,495
$1,960
$1,960
$1,560
$1,310
$1,610
$1,260
$14,140
Chapter 11 – Aggregate Planning and Master Scheduling
11–27
Plan C:
No additional workers are to be hired. Make up any shortfall, i.e., reduce backorders, using
subcontracting at $8 per unit, with a maximum of 20 units per period. Ending inventory in period
9 should be 0. Backorders cannot exceed 80 units in any period.
Regular production = 200 units per period.
Therefore, regular production could be used to meet demand of 9 * 200 = 1,800 units. How many
units do we need to subcontract to ensure that we do not have a backlog in the final month?
Total Forecast – Beginning Inventory – Total Regular Production = 1,940 – 0 – 1,800 = 140
units. The key to this problem is when to plan the 140 units of subcontracting to stay within the
maximum of 20 units per period, to keep the backlog ≤ 80 units (Period 4 will be a challenge
because it has the peak demand), and to minimize total cost.
Period
1
2
3
4
5
6
7
8
9
Total
Forecast
190
230
260
280
210
170
160
260
180
1,940
Output
Regular
200
200
200
200
200
200
200
200
200
1,800
Part Time
0
Overtime
0
Subcontract
20
20
20
20
20
20
20
140
Output – Forecast
30
–10
–40
–60
10
50
40
–40
20
0
Inventory
Beginning
30
20
0
0
0
0
20
0
Ending
30
20
0
0
0
0
20
0
0
Average
15
25
10
0
0
0
10.0
10.0
0.0
70
Backlog
0
0
20
80
70
20
0
20
0
210
Costs:
Regular
@
6
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$10,800
Part Time
@
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Overtime
@
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Subcontract
@
8
$160
$160
$160
$160
$160
$160
$0
$160
$0
$1,120
Hire/Layoff
$0
Inventory
@
5
$75
$125
$50
$0
$0
$0
$50
$50
$0
$350
Back orders
@
10
$0
$0
$200
$800
$700
$200
$0
$200
$0
$2,100
Total
$1,435
$1,485
$1,610
$2,160
$2,060
$1,560
$1,250
$1,610
$1,200
$14,370
11–28
Education.
11. Given:
Refer back to the solution in Solved Problem 1. The total cost in that plan was $20,550. The
given information in Solved Problem 1 was: Current workforce = 20 people, each of whom can
produce 10 units of output per period. Regular production cost per unit = $6. Inventory carrying
cost per unit per period = $5. Backlog cost per unit per period = $10. Forecasts are shown below:
Period
1
2
3
4
5
6
7
8
9
Total
Forecast
190
230
260
280
210
170
160
260
180
1,940
Another option is to use part-time workers during seasonal peaks. Cost per unit (hiring + training)
= $11. A maximum of 10 part-time workers can be used, and the same number of part-time
workers must be used in all periods that have part-time workers. The ending inventory in Period 9
should be 10 units. The limit on backlogs is 20 units per period. Try to make up backlogs as soon
as possible.
Regular production = 200 units per period.
Therefore, regular production could be used to meet demand of 9 * 200 = 1,800 units. How many
units do we need part-time workers to produce? Total Forecast + Ending Inventory Goal –
Beginning Inventory – Total Regular Production = 1,940 + 10 – 0 – 1,800 = 150 units. A part-
time worker can produce 10 units per period. Therefore, we will need to hire 5 part-time workers.
These 5 part-time workers will produce 50 units per month * 3 months = 150 units.
Period
1
2
3
4
5
6
7
8
9
Total
Forecast
190
230
260
280
210
170
160
260
180
1,940
Output
Regular
200
200
200
200
200
200
200
200
200
1,800
Part Time
50
50
50
150
Overtime
0
Subcontract
0
Output – Forecast
10
20
–10
–30
–10
30
40
–60
20
10
Inventory
Beginning
10
30
20
0
0
10
50
0
Ending
10
30
20
0
0
10
50
0
10
Average
5
20
25
10
0
5
30.0
25.0
5.0
125
Backlog
0
0
0
10
20
0
0
10
0
40
Costs:
Regular
@
6
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$10,800
Part Time
@
11
$0
$550
$550
$550
$0
$0
$0
$0
$0
$1,650
Overtime
@
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Subcontract
@
8
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Hire/Layoff
$0
Inventory
@
5
$25
$100
$125
$50
$0
$25
$150
$125
$25
$625
Back orders
@
10
$0
$0
$0
$100
$200
$0
$0
$100
$0
$400
Total
$1,225
$1,850
$1,875
$1,900
$1,400
$1,225
$1,350
$1,425
$1,225
$13,475
Chapter 11 – Aggregate Planning and Master Scheduling
11–29
Education.
12. Given:
Refer back to the solution in Solved Problem 1. The total cost in that plan was $20,550. The
given information in Solved Problem 1 was: Current workforce = 20 people, each of whom can
produce 10 units of output per period. Regular production cost per unit = $6. Inventory carrying
cost per unit per period = $5. Backlog cost per unit per period = $10. Forecasts are shown below:
Period
1
2
3
4
5
6
7
8
9
Total
Forecast
190
230
260
280
210
170
160
260
180
1,940
Prepare an aggregate plan that uses overtime ($9 per unit, maximum = 25 units per period) and
inventory variation. Try to minimize backlogs. The ending inventory in period 9 should be 0
units, and the limit on backlogs is 60 units per period.
Regular production = 200 units per period.
Therefore, regular production could be used to meet 9 * 200 = 1,800 units. How many units do
we need to use overtime to produce? Total Forecast – Beginning Inventory – Total Regular
Production = 1,940 – 0 – 1,800 = 140 units.
The key to this problem is to schedule OT production (140 units) early to minimize backlogs.
Remember: Maximum overtime is 25 units per period.
Period
1
2
3
4
5
6
7
8
9
Total
Forecast
190
230
260
280
210
170
160
260
180
1,940
Output
Regular
200
200
200
200
200
200
200
200
200
1,800
Part Time
0
Overtime
25
25
25
25
25
15
140
Subcontract
0
Output – Forecast
35
-5
–35
–55
15
45
40
–60
20
0
Inventory
Beginning
35
30
0
0
0
0
40
0
Ending
35
30
0
0
0
0
40
0
0
Average
18
33
15
0
0
0
20.0
20.0
0.0
105
Backlog
0
0
5
60
45
0
0
20
0
130
Costs:
Regular
@
6
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$1,200
$10,800
Part Time
@
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Overtime
@
9
$225
$225
$225
$225
$225
$135
$0
$0
$0
$1,260
Subcontract
@
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
Hire/Layoff
$0
Inventory
@
5
$88
$163
$75
$0
$0
$0
$100
$100
$0
$525
Back orders
@
10
$0
$0
$50
$600
$450
$0
$0
$200
$0
$1,300
Total
$1,513
$1,588
$1,550
$2,025
$1,875
$1,335
$1,300
$1,500
$1,200
$13,885
Chapter 11 – Aggregate Planning and Master Scheduling
Education.
13. Given:
Refer to Example 2. The total cost for Example 2 was $4,640. The forecasts for Example 2 are
shown below:
Period
1
2
3
4
5
6
Total
Forecast
200
200
300
400
500
200
1,800
Subcontracting can be used at a maximum rate of 50 units per period as needed. Overtime is not
allowed. Your plan should have ending inventory of 0 units. Ending backlog must equal 0.
Regular capacity = 280 units per period. Regular production can be less than regular capacity.
Step 1: Determine the least cost option out of regular production and subcontracting.
Regular production cost = $2 per unit.
Subcontracting cost = $6 per unit.
Inventory holding cost = $1 per unit per period (based on average inventory).
Backorder cost = $5 per unit per period.
Regular production costs less than subcontracting does. Comparing regular time production to
The cost per unit of producing a unit using regular time production one period late = $2 + (1 x $5)
= $7, which exceeds the cost of subcontracting that unit in the current period. Therefore, given
the choice of meeting demand late using regular time and subcontracting, we would prefer
subcontracting.
Step 2: Determine how much to produce using regular time and how much to subcontract.