Chapter 5
Aggregate Planning
Supply Chain Planning Matrix
longterm
procurement
distribution
sales
Strategic Network Planning
midterm shortterm
production
Master Planning
Material Requirements Planning
Production Planning
Distribution Planning
Scheduling
Transport Planning
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Demand Planning Demand Fulfilment & ATP 44
Supply Chain Planning Matrix
× physical distribution structure
sales
× materials program × supplier selection × cooperations
× personnel planning × material requ. planning × contracts
× master production scheduling × capacity planning
× distribution planning
× mid-term sales planning
× personnel planning × ordering materials
× lot-sizing × machine scheduling × shop floor control
× warehouse replenishement × transport planning
× short-term sales planning
shortterm
× plant location × production system
distribution
longterm
production
midterm
procurement
× product program × strategic sales planning
information flows
flow of goods Production Management
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Aggregate Planning Example: one product (plastic case) two injection molding machines, 550 parts/hour one worker, 55 parts/hour steady sales 80.000 cases/month 4 weeks/month, 5 days/week, 8h/day how many workers? in real life constant demand is rare change demand produce a constant rate anyway vary production Production Management
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Aggregate Planning Influencing demand do not satisfy demand shift demand from peak periods to nonpeak periods produce several products with peak demand in different period Planning Production Production plan: how much and when to make each product rolling planning horizon long range plan intermediate-range plan ⌧units of measurements are aggregates ⌧product family ⌧plant department ⌧changes in workforce, additional machines, subcontracting, overtime,...
Short-term plan
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Aggregate Planning Aspects of Aggregate Planning Capacity: how much a production system can make Aggregate Units: products, workers,... Costs ⌧production costs (economic costs!) ⌧inventory costs(holding and shortage) ⌧capacity change costs
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Aggregate Planning Spreadsheet Methods Zero Inventory Plan Precision Transfer, Inc. Produces more than 300 different precision gears ( the aggregation unit is a gear!). Last year (=260 working days) Precision made 41.383 gears of various kinds with an average of 40 workers. 41.383 gears per year 40 x 260 worker-days/year = 3,98 -> 4 gears/ worker-day Aggregate demand forecast for precision gear: Month Demand
January 2760
February March 3320 3970
April 3540
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May 3180
June 2900
Total 19.670
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Aggregate Planning holding costs: $5 per gear per month backlog costs: $15 per gear per month hiring costs: $450 per worker lay-off costs: $600 per worker wages: $15 per hour ( all workers are paid for 8 hours per day) there are currently 35 workers at Precision currently no inventory Production plan?
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Aggregate Planning Zero Inventory Plan produce exactly amount needed per period adapt workforce
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Aggregate Planning 10 9
Number of Workers (hired / laid off)
8 6 4 2
2 Change in Workforce
0 -1 -2
-2
-4
-4
-6
-6
-8 January February
March
April
May
June
Month
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Aggregate Planning Level Work Force Plan backorders allowed constant numbers of workers demand over the planning horizon gears a worker can produce over the horizon 19670/(4x129)=38,12 -> 39 workers are always needed
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Aggregate Planning Inventory: January: 3276 - 2760 = 516 February: 516 + 3120 – 3320 March: 316 + 3588 – 3670 = -66! -Backorders: 66 x $15 = $990 516
500 400
358 316
300 200
net inventory
100 0
0 -66
-100
-78
-200 -300
-330
ne Ju
M ay
pr il A
M ar ch
y br ua r Fe
nu ar y
-400
Ja
number of units (inventory / back-orders)
600
Month
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Aggregate Planning no backorders are allowed workers= cumulative demand/(cumulative days x units/workers/day) January: 2760/(21 x 4) = 32,86 -> 33 workers February: (2760+3320)/[(21+20) x 4] = 37,07 -> 38 workers. March: 10.050/(64 x 4) =>40 workers April: 13.590/(85 x 4) => 40 workers May: 16.770/(107 x 4) => 40 workers June: 19670/(129 x 4) => 39 workers
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Aggregate Planning Example Mixed Plan The number of workers used is an educated guess based on the zero inventory and level work force plans!
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Spreadsheet Methods Summary Zero-Inv.
Level/BO Level/No BO
Mixed
Hiring cost
4950
1800
2250
3150
Lay-off cost
7800
0
0
4200
59856
603720
619200
593520
Holding cost
0
4160
6350
3890
BO cost
0
7110
0
990
611310
616790
627800
605180
33
39
40
35
Labor cost
Total cost Workers
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Aggregate Planning Linear Programming Approaches to Aggregate Planning
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Aggregate Planning
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Aggregate Planning Decision Variables: Pt K number of units produced in period t Wt K number of workers available in period t H t K number of workers hired in period t Lt K number of workers laid off in period t I t K number of units held in inventory in period t Bt K number of units backordered in period t
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Aggregate Planning
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Aggregate Planning Example: Precision Transfer Planning horizon: 6 months T= 6 Costs do not vary over time CtP = 0 dt : days in month t CtW = $120dt CtH = $450 CtL = $600 CtI = $5 We assume that no backorders are allowed! no production costs and no backorder costs are included! Demand ⌧January 2760
February March 3320 3970
April 3540
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May 3180
June 2900
Total 19.670 62
Linear Program Model for Precision Transfer
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Aggregate Planning LP solution (total cost = $600 191,60)
January February March April May June
Production Inventory Hired Laid off Workers 2940,00 180,00 0,00 0,00 35,00 3232,86 92,86 5,41 0,00 40,41 3877,14 0,00 1,73 0,00 42,14 3540,00 0,00 0,00 0,00 42,14 3180,00 0,00 0,00 6,01 36,14 2900,00 0,00 0,00 3,18 32,95
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Aggregate Planning Rounding LP solution January Days Units/Worker Demand Workers Capacity Capacity - Demand Cumulative Difference Produced Net inventory Hired Laid Off Costs
21 84 2760 35 2940 180 180 2930 170 0 0 89050
February March April May June Total 20 23 21 22 22 129 80 92 84 88 88 516 3320 3970 3540 3180 2900 19670 229 41 42 42 36 33 3280 3864 3528 3168 2904 19684 -40 -106 -12 -12 4 14 140 34 22 10 14 400 3280 3864 3528 3168 2900 19670 130 24 12 0 0 336 6 1 0 0 0 7 0 0 0 6 3 9 101750 116490 105900 98640 88920 600750
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Aggregate Planning Practical Issues 100.000 variables and 40.000 constraints LP/MIP Solvers: CPLEX, XPRESS-MP, ... Extensions Bounds
I t ≤ I tU I Lt ≤ I t ≤ I tU L t ≤ 0.05Wt Training Wt = Wt −1 + H t −1 − Lt Production Management
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Aggregate Planning Transportation Models supply points: periods, initial inventory demand points: periods, excess demand, final inventory ntWt = capacity during period t Dt = forecasted number of units demanded in period t CPt = the cost to produce one unit in period t CIt = the cost to hold one unit in inventory in period t
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Aggregate Planning
initial inventory: 50 final inventory: 75
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Aggregate Planning
Beginning inventory Period 1
Excess capacity
0
2
4
6
0
10
12
14
16
0
350
2
3
50 150
50 -
Period 2
11
75
75
13
15
0
300
12
14
0
350
300 -
Period 3 Demand
Ending inventory
Available capacity 50
1
200
300
350 400
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75
1050
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Aggregate Planning Extension:
t capacity n tW t demand production costs holding costs
1 2 3 350 350 300 400 300 400 10 11 12 2 2 2
⌧overtime: overtime capacity is 90, 90 and 75 in period 1, 2 and 3; ⌧overtime costs are $16, $18 and $ 20 for the three periods respectively; ⌧backorders:units can be backordered at a cost of $5 per unit-month; production in period 2 can be used to satisfy demand in period 1
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Aggregate Planning 1 0
Beginning inventory
Period 1
Regular time Overtime
Period 2
2
Excess capacity
Available capacity
4
6
0
25
10
12
14
16
0
16
18
20
22
0
350 50
40 16
Regular time
11
13
275 23
15
0
22
0
75 18
20
90 22
Regular time
17
12
14
0
20
22
0
300 30
Overtime Demand
Ending inventory
3
25
Overtime
Period 3
2
400
25 300
75 400
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130
50 350 90 350 90 300 75 1305 71
Aggregate Planning Disaggregating Plans aggregate units are not actually produced, so the plan should consider individual products disaggregation master production schedule Questions: In which order should individual products be produced? ⌧e.g.: shortest run-out time Ri = I i / Di
How much of each product should be produced? ⌧e.g.: balance run-out time
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Aggregate Planning Advanced Production Planning Models Multiple Products same notation as before add subscript i for product i Objective function N ⎛ W ⎞ H L P I min ∑ ⎜ Ct Wt + Ct H t + Ct Lt + ∑ Cit Pit + Cit I it ⎟ t =1 ⎝ i =1 ⎠ T
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Aggregate Planning subject to ⎛ 1 ⎜ ∑ i=1 ⎝ nit N
⎞ ⎟ Pit ≤ Wt ⎠
t = 1, 2,..., T
Wt = Wt −1 + H t − Lt
t=1,2,...,T
Iit = I it −1 + Pit − Dit
t=1,2,...,T; i=1,2,...,N
Pit,Wt , H t , Lt , I it ≥ 0
t=1,2,...,T; i=1,2,...,N
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Aggregate Planning Computational Effort: 10 products, 12 periods: 276 variables, 144 constraints 100 products, 12 periods: 2436 variables, 1224 constaints
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Aggregate Planning Example: Carolina Hardwood Product Mix ⌧Carolina Hardwood produces 3 types of dining tables; ⌧There are currently 50 workers employed who can be hired and laid off at any time; ⌧Initial inventory is 100 units for table1, 120 units for table 2 and 80 units for table 3;
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Aggregate Planning ⌧The number of units that can be made by one worker per period:
⌧Forecasted demand, unit cost and holding cost per unit are:
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Aggregate Planning
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Aggregate Planning Multiple Products and Processes
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Aggregate Planning
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Aggregate Planning Example: Cactus Cycles process plan CC produces 2 types of bicycles, street and road; Estimated demand and current inventory:
available capacity(hours) and holding costs per bike: t 1 2 3
Capacity(hours) Holding Machine Worker Street Road 8600 17000 5 6 8500 16600 6 7 8800 17200 5 7 Production Management
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Aggregate Planning process costs ( process1, process2) and resource requirement per unit:
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Aggregate Planning
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Aggregate Planning solution: Objective Function value = $368,756.25
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Aggregate Planning - Extensions Hopp/Spearman, S. 522-540 Notation:
X it ... amount of product i produced in period t ri K net profit from one unit of product i Sit K amount of product i sold in period t a ij K time required on workstation j to produce one unit of product i c jt K capacity of workstation j in period t in units (consistent with a ij )
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Aggregate Planning - Extensions Backorders
t
m
max ∑∑ ri Sit − hi I it+ − π i I t− t =1 i =1
subject to d it ≤ Sit ≤ d it m
∑a
for all i,t
X it ≤ c jt
for all j,t
I it = I it −1 + X it − Sit
for all i,t
I it = I it+ − I it−
for all i,t
X it , Sit , I it+ , I it− ≥ 0
for all i,t
i =1
ij
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Aggregate Planning - Extensions Overtime
l ′j = cost of one hour of overtime at workstation j O jt = overtime at workstation j in period t in hours n ⎧m ⎫ + − max ∑ ⎨∑ (ri Sit − hi I it − π i I it ) − ∑ l ′O jt ⎬ t =1 ⎩ i =1 j =1 ⎭ subject to t
m
∑a i =1
ij
X it ≤ c jt + O jt
for all i,t
X it , Sit , I it+ , I it− , O jt ≥ 0
for all i,t
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Aggregate Planning - Extensions Yield loss
1−α
α
1− β
1− γ
β
γ
α , β , γ K fraction of output that is lost yij K cumulative yield from station j onward (including station j) for product i we must release
d units of i into station j y ij Production Management
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Aggregate Planning - Extensions Basic model + Yield loss extension (no backorders) t
m
max ∑∑ (ri S it − hi I it ) t =1 i =1
subject to d it ≤ Sit ≤ d it m
∑
a ij X it yij
for all i,t
≤ c jt
for all j,t
I it = I it −1 + X it − Sit
for all i,t
X it , Sit , I it ≥ 0
for all i,t
i =1
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Aggregate Planning - WorkforcePlanning Single product, workforce resizing, overtime allocation Notation
b = number of man - hours required to produce one unit of product l = cost of regular time in dollars/man - hour l ′ = cost of overtime in dollars/man - hour e = cost to increase workforce by one man - hour per period e′ = cost to decrease workforce by one man - hour per period Wt = workforce in period t in man - hours of regular time H t = increase in workforce from period t - 1 to t in man - hours Ft = decrease in workforce from period t - 1 to t in man - hours O t = overtime in period t in hours Production Management
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Aggregate Planning - WorkforcePlanning LP formulation: maximize net profit, including labor, overtime, holding, and hiring/firing costs subject to constraints on sales, capacity,...
t
{
max ∑ rS t − h I t − lWt − l ′Ot − eH t − e′Ft
}
t =1
subject to d t ≤ St ≤ d t
for all t
a j X t ≤ c jt
for all j, t
I t = I t −1 + X t − S t
for all t
Wt = Wt −1 + H t − Ft
for all t
bX t ≤ Wt + Ot
for all t
X t , St , I t , Ot ,Wt , Ft , H t , ≥ 0 for all t Production Management
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AP-WP Example Revenue: 1000$ worker capacity: 168h/month initially 15 workers no initial inventory holding costs: 10$/unit/month regular labor costs: 35$/hour overtime: 150% of regular hiring costs: 2500$ (2500/168 ~ 15$ per man-hour) lay-off costs: 1500$ (1500/168 ~ 9$ per man-hour) no backordering demands over 12 months: 200, 220, 230, 300, 400, 450, 320, 180, 170,170, 160, 180 demands must be met! (S=D) Production Management
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AP-WP Example(cont.) Determine over a 12 month horizon: Number of workers: W Output: X Overtime use: O Inventory: I (H, F are additional choice variables in the model)
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Aggregate Planning - WorkforcePlanning
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Aggregate Planning - WorkforcePlanning
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Aggregate Planning - WorkforcePlanning
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Aggregate Planning-Summary The following scenarios have been discussed: single product, single resource, single process find: workforce, output, inventory (w. or w/o backorders) multiple products, single resource, single process find: workforce, all outputs, all inventories (w. or w/o backorders) multiple products, multiple resources, multiple processes (workforce given) find: all outputs, all inventories, use of processes
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Aggregate Planning-Summary The following scenarios have been discussed: multiple products, multiple workstations (workstation capcities given) find: all sales, all outputs, all inventories (w. or w/o backorders) multiple products, multiple workstations find: all sales, all outputs, all inventories (w. or w/o backorders), OT single product, multiple workstations, one resource find: workforce, all sales, all outputs, all inventories (w. or w/o backorders), OT
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Aggregate Planning Work to do: Examples: 5.7, 5.8abcdef, 5.9abcd, 5.10abcd, 5.16abcd, 5.21, 5.22, 5.29, 5.30 Replace capacity columns of table in problem 5.29 with Month Machine Worker 1 1350 19000 2 1270 19000 3 1350 19500
Minicase BF SWING II
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