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 Production Management 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 45 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 46 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 Production Management 47 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 Production Management 48 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 Production Management May 3180 June 2900 Total 19.670 49 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? Production Management 50 Aggregate Planning Zero Inventory Plan produce exactly amount needed per period adapt workforce Production Management 51 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 Production Management 52 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 Production Management 53 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 Production Management 54 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 Production Management 55 Aggregate Planning Example Mixed Plan The number of workers used is an educated guess based on the zero inventory and level work force plans! Production Management 56 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 Production Management 57 Aggregate Planning Linear Programming Approaches to Aggregate Planning Production Management 58 Aggregate Planning Production Management 59 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 Production Management 60 Aggregate Planning Production Management 61 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 Production Management May 3180 June 2900 Total 19.670 62 Linear Program Model for Precision Transfer Production Management 63 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 Production Management 64 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 Production Management 65 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 66 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 Production Management 67 Aggregate Planning initial inventory: 50 final inventory: 75 Production Management 68 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 Production Management 75 75 1050 69 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 Production Management 70 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 Production Management 75 130 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 Production Management 72 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 Production Management 73 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 Production Management 74 Aggregate Planning Computational Effort: 10 products, 12 periods: 276 variables, 144 constraints 100 products, 12 periods: 2436 variables, 1224 constaints Production Management 75 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; Production Management 76 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: Production Management 77 Aggregate Planning Production Management 78 Aggregate Planning Multiple Products and Processes Production Management 79 Aggregate Planning Production Management 80 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 81 Aggregate Planning process costs ( process1, process2) and resource requirement per unit: Production Management 82 Aggregate Planning Production Management 83 Aggregate Planning solution: Objective Function value = $368,756.25 Production Management 84 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 ) Production Management 85 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 Production Management 86 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 Production Management 87 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 88 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 Production Management 89 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 90 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 91 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 92 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) Production Management 93 Aggregate Planning - WorkforcePlanning Production Management 94 Aggregate Planning - WorkforcePlanning Production Management 95 Aggregate Planning - WorkforcePlanning Production Management 96 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 Production Management 97 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 Production Management 98 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 Production Management 99