13
C H A P T E R
Aggregate Planning
END-OF-CHAPTER PROBLEMS 13.1
180
Production Month Days
Forecast Demand
Jan Feb Mar Apr May Jun July Aug Sep Oct Nov Dec
1,000 1,100 1,200 1,300 1,350 1,350 1,300 1,200 1,100 1,100 1,050 900 13,950
22 18 22 21 22 21 21 22 21 22 20 20 252
Needed Production Each Day 45.5 61.1 54.5 61.9 61.4 64.3 61.9 54.5 52.4 50.0 52.5 45.0 55.4 (on average)
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(a) Plan 5 Month Jan Feb Mar Apr May Jun
Expected Demand
Production Days
Demand Per Day
900 700 800 1,200 1,500 1,100 6,20 0
22 18 21 21 22 20 124
41 39 38 57 68 55
Average daily production requirement =
Mont h
Expected Demand
Production (@ 35/day)
900 700 800 1,200 1,500 1,100
770 630 735 735 770 700
Jan Feb Mar Apr May Jun
6,200
C R
Constant workforce of 6 persons; subcontract to meet extra demand: Subcontract cost = $20/unit
= 7 persons × $80 × 124 = $69,440
Subcontracting: C SC
=6×
8 1.6
130 70 65 465 730 400 1,86 0
Plan 6 Cost analysis: Regular production:
124 = 50 units/day
Production rate/day = Persons ×
Subcontract
Hours/day
= 1,860 units × $20 = $37, 200
Total cost: C T = 69,440 + 37,200 = $106,640
Hours/unit
= 30 units/day
Plan 2 is still preferable, but Plan 6 has lower cost than Plan 5. Comparing:
Month
Expected Demand
Jan Feb Mar Apr May Jun
900 700 800 1,200 1,500 1,100
Production (@ 30/day) Subcontrac t 660 540 630 630 660 600
240 160 170 570 840 500 2,48 0
Plan 5 Cost analysis: Regular production: C R =
6 persons × $80 × 124 = $59,520
Subcontract cost @ $10/unit: C SC
= 2, 480 units × $20 / u nit = $49, 600
Total cost: C T = $59, 520 + $49, 600 = $109,120 (not preferable to Plan 2 at $105,152, but preferable to Plan 4 at $113,488).
(b)
Plan 6 Constant workforce of 7 persons; subcontract to meet extra demand: Labor → 1.6 hours/unit Hours / day Production rate /day = Persons × Hours /unit 8 = 7× = 35 units / day 1.6
Plan 1 Plan 2 Plan 3 Plan 4 Plan 5 Plan 6 Carrying cost Reg. time Overtime Subcont. Hire Layoff Total cost
9,250 0 0 400 0 0 99,200 75,392 99,200 79,360 59,520 69,440 0 0 0 33,728 49,600 0 0 29,870 0 0 0 37,200 0 0 9,000 0 0 0 0 0 9,600 0 0 0 108,45 105,15 117,800 113,48 109,12 106,64 0 2 8 0 0
Based simply upon total cost, Plan 2 is preferable. From a practical viewpoint, Plans 1, 5, and 6 will likely have equivalent costs. Practical implementation of Plan 2 may, for example, require the employment of eight full-time employees, rather than seven fulltime and one part-time employee. When several plans have roughly equivalent costs, other parameters gain importance—such as the amount of control one would have over production and excess wear on equipment and personnel. Plan 3 should be avoided. 13.3
Period 1 2 3 4 5 6 7 8
Expected Demand 1,400 1,600 1,800 1,800 2,200 2,200 1,800 1,400 14,20 0
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13.3
AGGREGATE PLANNING
182
(cont’d) Plan A
Period
Demand
Productio n (Result of Previous Inventory Stockout Hire LayoffPersonnel Month) (Units) (Units) (Units (Units)Cost )
1 (Jan)
1,400
1,600
400
2 (Feb)
1,600
1,400
200
3 4 5 6 7 8
1,800 1,800 2,200 2,200 1,800 1,400
1,600 1,800 1,800 2,200 2,200 1,800
(Mar) (Apr) (May) (June) (July) (Aug)
200 200 400
400
400 800 1,800 @ $20 =$36,000
200 $15,000 ←(cost to go from 1,600 in Jan to 1,400 in Feb) 10,000 ←(cost to go from 1,400 in Feb to 1,600 in Mar) 10,000 — 20,000 — 400 30,000 400 30,000 ←(cost to go from 1,800 in August to 1,400 in Sept)
$400 Total
$115,000
@ $100 Personnel Cost: =$40,0 00
Note: December demand was 1,600, and because our strategy is chasing prior-period demand, our January production is 1,600. So 200 units remain in inventory, and January production adds 200 units to this inventory, for a total of 400 units. Inventory units: Jan. 400 + Feb. 200 + July 400 + Aug. 800 (400 from July and 400 from August) = 1,800 units at $20 = $36,000. Stockout units: May 400 units at $100 = $40,000. Hiring and layoff costs = $115,000. Total costs = $36,000 + $40,000 + $115,000 = $191,000.
Plan C Period
13.5
(a)
0 1 2 3 4 5 6 7 8
Demand Production*
1,400 1,600 1,800 1,800 2,200 2,200 1,800 1,400
1,775 1,775 1,775 1,775 1,775 1,775 1,775 1,775
Ending Inv. 200 575 750 725 700 275 0 0 375
Stockouts (Units)
150 25
Extra Cost
$11,500 15,000 14,500 14,000 5,500 15,000 2,500 7,500 Total Extra Cost:
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*(14,200/8) = 1,775 average. All other things being equal, it would appear that Plan C, with a cost of $85,500 and stockout costs ignored, should be recommended over Plan A (cost = $224,000) or Plan B (cost = $214,000).
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CHAPTER 13
(b)
AGGREGATE PLANNING
Graph of Plan C
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13.7
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Month
AGGREGATE PLANNING
Expected Demand
Jul Aug Sep Oct Nov
400 500 550 700 800
Production per person per day: 8 hr/person ÷ 4 hours/d is k Therefore, each person can produce 2 disks per day, or 40 disks per month. (a) Aggregate plan, hiring/layoff only:
Unit
Beg. Inventor y Over
Perio Demand (or d Short) Jun Jul Aug Sep Oct Nov
Hours
Productio n Over
Units
Require at 20 Personnel Units d days Require at 4 at 8 hrs on staff Produce (or Short) d each d
150 150 –10 10 20 0
400 500 550 700 800
Personnel Required
250 510 540 680 800
1,000 2,040 2,160 2,720 3,200
6.25 12.75 13.50 17.00 20.00
8 6 13 14 17 20
240 520 560 680 800
–10 10* 20* 0 0
Costs Layoff Hire: 40 Hire $40
$80 Layoff: 80
2 7 1 3 3
$160 $280 $40 $120 $120
* Inventory (August = 10 and Sept. = 20) = 30 × 8 = $240 Inventory Cost = 30 × 8 = $240 Hiring/Layoff Cost = 960 $1,200 Note: In computing cost, we assumed that, if the capacity of a fraction of a worker was needed (was excess), one worker was hired (layed off). Solution by POM for Windows, in which the increase cost is $1 per unit and the decrease cost is $2 per unit, yields a similar result, with a total extra cost of $890. (b) Aggregate plan, overtime only:
Period
Demand
Production
Production
Ending
(Regular)
(Overtime)
Inv.
Jun Jul 400 320 Aug 500 320 Sep 550 320 Oct 700 320 Nov 800 320 Dec 320 (OT) cost + 1,580 $37,920 = Extra total × ($72 – $48) =700
↑
Inventory Holding Cost @ $8/unit/month
150 70
560
110 230 380 480 380 $560 holding cost = $38,480
↑
Uni ts ma de on $72 = 4 hr overtime (OT) each × $18
$48 = 4 hr each × $12
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AGGREGATE PLANNING
13.9
Month Jul Aug Sep Oct Nov Dec
Expected Demand 1,000 1,200 1,400 1,800 1,800 1,600
(a) Plan A: Minimum rate of 1,000/month, subcontract for additional. Plan A Period Jul Aug Sep Oct Nov Dec
Demand
Production
Ending Inv.
1,000 1,200 1,400 1,800 1,800 1,600
1,000 1,000 1,000 1,000 1,000 1,000
0 0 0 0 0 0
Subcont. (Units) Extra Cost — 200 400 800 800 600
0 12,000 24,000 48,000 48,000 36,000
Total Extra Cost: $168,000
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Plan B: Vary workforce. Plan B Period Jul Aug Sep Oct Nov Dec
13.11
Demand
Production (Existing)
Hire (Units)
Layoffs (Units)
1,000 1,200 1,400 1,800 1,800 1,600
1,300 1,000 1,200 1,400 1,800 1,800
— 200 200 400 — —
300
Extra Cost
$18,000 6,000 6,000 12,000 — 200 12,000 Total Extra Cost: $54,000
Initial data: Costs (per unit)
Reg Time Overtime Subcontract Holding Stockout Hiring Layoffs
Initial inventory
=
0 = $ 30 Units last period = 1,500 = $ 15 extra per unit = not available = 10 = 50 = 40 = 80
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AGGREGATE PLANNING
188
(a) The Chase plan:
Period Quarter Quarter Quarter Quarter Total
Demand 1 2 3 4
1,400 1,200 1,500 1,300 5,400
Reg. Time Producti on
Change
1,400 1,200 1,500 1,300 5,40 0 @$30/unit $162,000
Hiring
Layoffs
0 0 300 0 300
100 200 0 200 500
@$40/unit $12,000
@$80/unit $40,000
–100 –200 300 –200
Cost Overtime production = $0 Subcontract = $0 and Inventory holding and shortage cost = $0
(b) The Level plan:
Period Quarter Quarter Quarter Quarter Total
Demand 1 2 3 4
1,400 1,200 1,500 1,300 5,40* 0
Cost
Reg. Time Production 1,350 1,350 1,350 1,350 5,40 0 $162,00 0
Inventory –50 100 –50 0
Holding 0 100 0 0 10 0 $1,00 0
Shortage 50 0 50 0 10 0 $5,00 0
Change
Hiring
–150 0 0 0
0 0 0 0 0
Total Cost:
(c) A Level plan will cost $180,000, while a Chase plan will cost $214,000. 13.13
Assuming that back orders are not permitted, the solution is:
Total cost = $11,790
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$0
Layoffs 150 0 0 15 0 $12,00 0
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Assuming that back orders are not permitted, the solution is:
Total cost = $627,100
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CHAPTER 13
AGGREGATE PLANNING
An alternative solution is:
Total cost = $627,100 Total cost = $100,750
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(a) The cost matrix and the optimal plan are shown below: Cost Matrix:
Quarter 1
Quarter 2
Beg. inv.
0.2
0.4
0.6
0.8
1
250
Reg. time 1 Overtime 1 Subcontract 1
1 1.5 2
1.2 1.7 2.2
1.4 1.9 2.4
1.6 2.1 2.6
1.8 2.3 2.8
400 80 100
Reg. time 2 Overtime 2 Subcontract 2
1.5 2 2.5
1 1.5 2
1.2 1.7 2.2
1.4 1.9 2.4
1.6 2.1 2.6
400 80 100
Reg. time 3 Overtime 3 Subcontract 3
2 2.5 3
1.5 2 2.5
1 1.5 2
1.2 1.7 2.2
1.4 1.9 2.4
800 160 100
Reg. time 4 Overtime 4 Subcontract 4 Demand
2.5 3 3.5 500
2 2.5 3 750
1.5 2 2.5 900
1 1.5 2 450
1.2 1.7 2.2
400 80 100 2600/305
Quarter 3
Quarter 4
Optimal Plan:
Quarter 1
Quarter 2
Beg. inv.
100
150
Reg. time 1 Overtime 1 Subcontract 1
400
Quarter 3
Quarter 4
Ending Inv.
Ending Inv.
Supply
Dummy
80 100
Reg. time 2 Overtime 2 Subcontract 2
400 80 100
Reg. time 3 Overtime 3 Subcontract 3
40
800 100
Reg. time 4 Overtime 4 Subcontract 4
20 100 400 50
500
750
900
450
Optimal cost = $2,641
(b) The cost of the optimal plan is $2,641. Alternate optimal solutions are possible. (c) All regular time is used. (d) 40 units are backordered in Quarter 2 and produced on overtime in Quarter 3 at a cost of $.50 each for a total cost of $20.
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