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MIT2_854F10_multipart - Single-stage, multiple-part-type...

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Unformatted text preview: Single-stage, multiple-part-type systems Lecturer: Stanley B. Gershwin Setups • Setup: A setup change occurs when it costs more to make a Type j par t after making a Type i par t than after making a Type j par t. • Examples: ⋆ Tool change (when making holes) ⋆ Die change (when making sheet metal par ts) ⋆ Paint color change ⋆ Replacement of reels of components, when populating printed circuit cards Costs Setups • Setup costs can include ⋆ Money costs, especially in labor. Also materials. ⋆ Time, in loss of capacity and delay. • Some machines create scrap while being adjusted during a setup change. • Setups motivate lots or batches: a set of par ts that are processed without interruption by setups. Costs Setups • Problem: ⋆ Large lots lead to large inventories and long lead times. ⋆ Small lots lead to frequent setup changes. • Reduction of setup time has been a ver y impor tant trend in modern manufacturing. Flexibility Setups • Flexibility: a widely-used term whose meaning diminishes as you look at it more closely. (This may be the definition of a buzzword.) • Flexibility: the ability to make many different things — ie, to operate on many different par ts. • Agility is also sometimes used. Flexibility Setups Which is more flexible? • A machine that can hold 6 different cutting tools, and can change from one to another with zero setup time. • A machine that can hold 25 different cutting tools, and requires a 30-second setup time. Which is more flexible? • A final assembly line that can produce all variations of 6 models of cars, and can produce 100 cars per day. • A final assembly line that can produce all variations of 1 model at 800 cars per day? Setup Machines vs Batch Machines Setups • A machine has setups when there are costs or delays associated with changing par t types. Machines that require setups may make one or many par ts at a time. • A machine operates on batches of size n if it operates on up to n par ts simultaneously each time it does an operation. Batch machines may or may not operate on different par t types. If they do, they may or may not require setup changes. (Also, the batches may or may not be homogeneous.) ⋆ Examples: Ovens and chemical chamber operations in semiconductor manufacturing; chemical processing of liquids. Loss of Capacity Setups Assume • there is one setup for ever y Q par ts (Q=lot size), • the setup time is S , • the time to process a par t is τ . Then the time to process Q par ts is S + Qτ . The average time to process one par t is τ + S/Q. Loss of Capacity Setups If the demand rate is d par ts per time unit, then the demand is feasible only if Q 1 1 τ + S/Q < 1/d or d < = < S + Qτ τ + S/Q τ This is not satisfied if S is too large or Q is too small. Deterministic Example Setups • Focus on a single par t type (simplification!) • Shor t time scale (hours or days). • Constant demand. • Deterministic setup and operation times. • Setup/production/(idleness) cycles. • Policy: Produce at maximum rate until the inventor y is enough to last through the next setup time. Deterministic Example Setups Cycle: Cumulative Production and Demand • S = setup period for the par t type • P = period the machine is operating on the par t type • I = period the machine is making or setting up for other par ts, or idle S P I t Deterministic Example Setups Objective — in general Cumulative Production and Demand production P(t) earliness surplus/backlog x(t) demand D(t) t Objective is to keep the cumulative production line close to the cumulative demand line. Deterministic Example Setups Cycle: • Setup period. Duration: S . Production: 0. Demand: S d. Net change of sur plus , ie of P − D is ΔS = −S d. • Production period. Duration: t = Qτ . Production: Q. Demand: td. Net change of P − D is ΔP = Q − td = Q − Qτ d = Q(1 − τ d). Deterministic Example Setups • Idleness period (for the par t we focus on). Duration: I . Production: 0. Demand: I d. Net change of P − D is ΔI = −I d. • Total (desired) net change over a cycle: 0. • Therefore, net change of P − D over whole cycle is ΔS + ΔP + ΔI = −S d + Q(1 − τ d) − I d = 0. Deterministic Example Setups • Since I ≥ 0, Q(1 − τ d) − S d ≥ 0. • If I = 0, Q(1 − τ d) = S d. • If τ d > 1, net change in P − D will be negative. Deterministic Example Setups Production & inventory history S = 3, Q = 10, τ = 1, d = .5 80 • Production period duration = Qτ = 10. Production Demand Inventory/backlog 70 60 • Idle period duration = 7. 50 • Total cycle duration = 20. 40 • Maximum inventor y is Q(1 − τ d) = 5. 30 20 10 0 0 20 40 60 80 t 100 Deterministic Example Setups Not frequent enough S = 3, Q = 30, τ = 1, d = .5 80 Production Demand Inventory/backlog • Production period duration = Qτ = 30. 70 60 • Idle period duration = 27. 50 40 • Total cycle duration = 60. 30 • Maximum inventor y is Q(1 − τ d) = 15. 20 10 0 0 20 40 60 80 100 Deterministic Example Setups Too frequent S = 3, Q = 2, τ = 1, d = .5 • Batches too small – demand not met. 50 Production Demand Inventory/backlog 40 30 • Q(1 − τ d) − S d = −0.5 20 10 • Backlog grows. 0 −10 • Too much capacity spent on setups. −20 −30 0 20 40 60 80 100 Deterministic Example Setups Just right! S = 3, Q = 3, τ = 1, d = .5 80 Production Demand Inventory/backlog 70 • Small batches – small inventories. 60 50 40 • Maximum inventor y is Q(1 − τ d) = 1.5. 30 20 10 0 0 20 40 60 80 100 Deterministic Example Setups Other parameters S = 3, Q = 10, τ = 1, d = .1 70 Production Demand Inventory/backlog 60 50 40 30 20 10 0 0 100 200 300 400 500 600 Deterministic Example Setups Time in the system 400 Time in system −− deterministic 350 • Each batch spends Qτ + S time units in the system if Q(1 − τ d) − S d ≥ 0. 300 250 200 150 100 • Optimal batch size: Q = S d/(1 − τ d) 50 0 0 50 100 150 200 Q Stochastic Example Setups • Batch sizes equal (Q); processing times random. ⋆ Average time to process a batch is Qτ + S = 1/µ. • Random arrival times (exponential inter-arrival times) ⋆ Average time between arrivals of batches is Q/d = 1/λ. • Infinite buffer for waiting batches Stochastic Example Setups 400 Time in system −− deterministic Time in system −− exponential • Treat system as an M /M /1 queue in batches. 350 300 250 200 • Average delay for a batch is 1/(µ − λ). 150 100 50 0 0 50 100 150 200 Q • Variability increases delay . Batch size data from a factory Setups 120 April 97 Batch Sizes 100 80 60 40 20 0 0 20 40 60 80 100 120 140 160 Batch size data from a factory Setups 200 May 97 Batch Sizes 180 160 140 120 100 80 60 40 20 0 0 20 40 60 80 100 120 140 160 Batch size data from a factory Setups 70 January 98 Batch Sizes 60 50 Avg Lot Size=25 Std Dev=27 40 30 20 10 0 0 20 40 60 80 100 120 140 160 Two Part Types Setups • Assumptions: ⋆ Cycle is produce Type 1, setup for Type 2, produce Type 2, setup for Type 1 . ⋆ Unit production times: τ1, τ2. ⋆ Setup times: S1, S2. ⋆ Batch sizes: Q1, Q2. ⋆ Demand rates: d1, d2. ⋆ No idleness. Two Part Types Setups T Cycle: Cumulative Production and Demand Type 1 Type 2 S P S P t Two Part Types Setups Let T be the length of a cycle. Then S1 + τ1Q1 + S2 + τ2Q2 = T To satisfy demand, Q1 = d1T ; Q2 = d2T This implies T= S1 + S2 1 − (τ1d1 + τ2d2) Two Part Types Setups • τidi is the fraction of time that is devoted to producing par t i. • 1 − (τ1d1 + τ2d2) is the fraction of time that is not devoted to production. • We must therefore have τ1d1 + τ2d2 < 1. This is a feasibility condition . Multiple Part Types Setups • New issue: Setup sequence . ⋆ In what order should we produce batches of different par t types? • Sij is the setup time (or setup cost) for changing from Type i production to Type j production. • Problem: ⋆ Select the setup sequence {i1, i2, ..., in} to minimize Si1i2 + Si2i3 + ... + Sin−1in + Sini1 . Multiple Part Types Setups Cases • Sequence-independent setups: Sij = Sj . Sequence does not matter. • Sequence-dependent setups: traveling salesman problem. Multiple Part Types Setups Cases • Paint shop: i indicates paint color number. • Sij is the time or cost of changing from Color i to Color j . • If i > j , i is darker than j and Sij > Sj i. Multiple Part Types Setups Cases • Hierarchical setups. • Operations have several attributes. • Setup changes between some attributes can be done quickly and easily. • Setup changes between others are lengthy and expensive. Dynamic Lot Sizing Setups • Wagner-Whitin (1958) problem • Assumptions: ⋆ Discrete time periods (weeks, months, etc.); t = 1, 2, ..., T . ⋆ Known, but non-constant demand D1, D2, ..., DT . ⋆ Production, setup, and holding cost. ⋆ Infinite capacity. Dynamic Lot Sizing Setups Other notation • ct = production cost (dollars per unit) in period t • At = setup or order cost (dollars) in period t • ht = holding cost; cost to hold one item in inventory from period t to period t + 1 • It = inventor y at the end of period t — the state variable • Qt = lot size in period t — the decision variable Dynamic Lot Sizing Setups Problem minimize �T t=1(Atδ (Qt ) + ctQt + htIt) (where δ (Q) = 1 if Q > 0; δ (Q) = 0 if Q = 0) subject to • It+1 = It + Qt − Dt • It ≥ 0 Dynamic Lot Sizing Setups Wagner-Whitin Property Characteristic of Solution: I Q j k Dynamic Lot Sizing Setups Wagner-Whitin Property Characteristic of Solution: • Either It = 0 or Qt+1 = 0. That is, produce only when inventory is zero. Or, ⋆ If we assume Ij = 0 and Ik = 0 (k > j ) and It > 0, t = j + 1, ..., k, ⋆ then Qj > 0, Qk > 0, and Qt = 0, t = j + 1, ..., k. Dynamic Lot Sizing Setups Wagner-Whitin Property Then • Ij +1 = Qj − Dj , • Ij +2 = Qj − Dj − Dj +1, ... • Ik = 0 = Qj − Dj − Dj +1 − ... − Dk Or, Qj = Dj + Dj +1 + ... + Dk which means produce enough to exactly satisfy demands for some number of periods, star ting now. Dynamic Lot Sizing Setups Wagner-Whitin Property • This is not enough to determine the solution, but it means that the search for the optimal is limited. • It also gives a qualitative insight. Real-Time Scheduling Setups • Problem: How to decide on batch sizes (ie, setup change times) in response to events. • Issue: Same as before. ⋆ Changing too often causes capacity loss; changing too infrequently leads to excess inventor y and lead time. Real-Time Scheduling Setups One Machine, Two Part Types Model: • di = demand rate of Type i • µi = 1/τi = maximum production rate of Type i • S = setup time • ui(t) = production rate of Type i at time t • xi(t) = sur plus (inventor y or backlog) of Type i dxi • = ui(t) − di, i = 1, 2 dt Real-Time Scheduling Setups 0 x1 Heuristic: Corridor Policy • Draw two lines, labeled Setup 1 and Setup 2. Type 1 production −5 −5 −10 −10 • Keep the system in setup i until x(t) hits the Setup j line. −15 −15 Setup 1 −20 −20 −25 −25 −30 −30 Setup 2 −35 −35 −40 −40 −10 −10 • Change to setup j . x2 trajectory −8 −8 −6 −6 −4 −4 −2 −2 0 2 • Etc. Real-Time Scheduling Setups 0 Heuristic: Corridor Policy x1 −5 −5 −10 −10 −15 −15 Setup 1 Setup change −20 −20 −25 −25 −30 −30 Setup 2 −35 −35 −40 −40 −10 −10 x2 trajectory −8 −8 −6 −6 −4 −4 −2 −2 0 2 Real-Time Scheduling Setups 0 Heuristic: Corridor Policy x1 −5 −5 Type 2 production −10 −10 −15 −15 Setup 1 −20 −20 −25 −25 −30 −30 Setup 2 −35 −35 −40 −40 −10 −10 x2 trajectory −8 −8 −6 −6 −4 −4 −2 −2 0 2 Real-Time Scheduling Setups 0 Heuristic: Corridor Policy x1 −5 −5 −10 −10 Setup change −15 −15 Setup 1 −20 −20 −25 −25 −30 −30 Setup 2 −35 −35 −40 −40 −10 −10 x2 trajectory −8 −8 −6 −6 −4 −4 −2 −2 0 2 Real-Time Scheduling Setups 0 Heuristic: Corridor Policy x1 −5 −5 −10 −10 −15 −15 Type 1 production Setup 1 −20 −20 −25 −25 −30 −30 Setup 2 −35 −35 −40 −40 −10 −10 x2 trajectory −8 −8 −6 −6 −4 −4 −2 −2 0 2 Real-Time Scheduling Setups 0 Heuristic: Corridor Policy x1 −5 −5 −10 −10 −15 −15 Setup 1 −20 −20 −25 −25 −30 −30 Setup 2 −35 −35 −40 −40 −10 −10 x2 trajectory −8 −8 −6 −6 −4 −4 −2 −2 0 2 Real-Time Scheduling Setups Heuristic: Corridor Policy • In this version, batch size is a function of time. • Also possible to pick parallel boundaries, with an upper limit. Then batch size is constant until upper limit reached. Real-Time Scheduling Setups Heuristic: Corridor Policy Two possibilities (for two par t types): • Converges to limit cycle — only if demand is within � capacity, ie if i τidi < 1. • Diverges — if ⋆ demand is not within capacity, or ⋆ corridor boundaries are poorly chosen. Real-Time Scheduling Setups More Than Two Part Types Three possibilities (for more than two par t types): • Limit cycle — only if demand is within capacity, • Divergence — if ⋆ demand is not within capacity, or ⋆ corridor boundaries are poorly chosen. • Chaos if demand is within capacity, and corridor boundaries chosen ... not well? MIT OpenCourseWare http://ocw.mit.edu 2.854 / 2.853 Introduction to Manufacturing Systems Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . ...
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.854 taught by Professor Stanleygershwin during the Fall '10 term at MIT.

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