FourRolesOfInventory 4xgs

FourRolesOfInventory 4xgs - Goals of the Game of the Game...

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Unformatted text preview: Goals of the Game of the Game • Experience the dynamics of a distribution the dynamics of distribution The Distribution Game The Distribution Game • • • Peter L. Jackson Professor School of O.R. and I.E. 2/4/2012 Industrial Data and Systems Analysis • system (demand, orders, lead time) Discover rules for managing flow of material Optimize cost tradeoffs Identify causes of inventory: pipeline stock, cycle stock, safety stock Compare different supply chain configurations 2/4/2012 1 The Distribution Game Distribution Game Industrial Data and Systems Analysis 2 Game Parameters Parameters Cost: $70 / unit $70 unit 15 days transport + 1 day pick/pack/ship Supplier Order: $200 Hold:$0.21/$/yr Central Warehouse 5 days transport + 1 day pick/pack/ship Retailer Retailer Order: $2.75 Hold:$0.25/$/yr Retailer Avg. demand: 2 /day (Coeff. of Variation: 1) Customers 2/4/2012 Industrial Data and Systems Analysis 3 2/4/2012 Price: $100 / unit $100 unit Industrial Data and Systems Analysis 4 Demonstration: with Warehouse 2/4/2012 Demonstration: without Warehouse 5 Industrial Data and Systems Analysis 2/4/2012 Industrial Data and Systems Analysis 6 Four Roles of Inventory Roles of Inventory Four Roles of Inventory Roles of Inventory Pipeline Stock Stock Cycle Stock Stock Peter L. Jackson Professor School of O.R. and I.E. Decoupling Stock 2/4/2012 Industrial Data and Systems Analysis 7 2/4/2012 Safety Stock Stock Industrial Data and Systems Analysis 8 In your reading packet In your reading packet Pipeline Stock Stock Covers cycle stock and decoupling stock Input rate, r Jackson, P.L. 2011. Fill-Rate Driven Safety Stock. ORIE 3120 Class Notes. Output rate, r Value-added process time or transport time, T School of Operations Research and Information Engineering, Cornell University Ithaca NY University, Ithaca, NY. Little’s Law: Average pipeline stock = rT Covers safety stock To reduce pipeline stock, reduce the duration of th value-added activities 2/4/2012 Industrial Data and Systems Analysis 9 Decoupling Stock Stock 2/4/2012 Industrial Data and Systems Analysis 10 A Useful Queueing Model Useful Queueing Model Jobs arriving Waiting queue 1 Completed jobs departing Workstation • Decoupling stock: queues of work • • • • between workstations caused by variability in processing times 2/4/2012 Industrial Data and Systems Analysis 11 Focus on a single workstation on single workstation Jobs arrive at rate λ Jobs can be completed at rate μ can be completed at rate λ < μ so the workstation is sometimes idle 2/4/2012 Industrial Data and Systems Analysis 12 No Decoupling Stock in Deterministic Case Variability is the Enemy is the Enemy • If the time between arrivals is constant the time between arrivals is constant • Queues develop when the arrival rate temporarily exceeds the processing rate and the processing time is constant then there will never be a queue arrival busy arrival arrival arrival busy idle • Assume independent and identically id • Utilization rate: fraction of time the distributed (i.i.d.) inter-arrival times, and i.i.d. process times i.i.d. process times workstation is busy; = busy/(busy+idle) 2/4/2012 Industrial Data and Systems Analysis idle 13 2/4/2012 Industrial Data and Systems Analysis Interpreting the Utilization Rat te How Shall We Measure Variability? • Let ρ = λ/μ • ρ is the utilization rate, the average fraction of 14 • Let ma = mean interarrival time (ma = 1/ λ) mean interarrival time (m 1/ 2 = variance of interarrival time • Let σ a • Let ca = σa / ma, the coefficient of the coefficient of time the workstation is in use • 1- ρ is the probability you will find the queue empty and the workstation idle if you observe it at random time at a random time • μ- λ is the average rate at which queues decrease, once a queue has formed • ρ/(1- ρ)= λ/(μ- λ) a unit-less measure of the (lack of) ability of workstation to work off queues 2/4/2012 Industrial Data and Systems Analysis variation of interarrival time • ca is a unit-less measure of variability unit measure of variability • ca ≥ 1 corresponds to high variability • ca = 1 is characteristic of the exponential is characteristic of the exponential distribution • ca ≤ 1 for our examples 15 2/4/2012 Industrial Data and Systems Analysis 16 Variability Variability Variability of Effective Process Ti Time • c2a is called the squared coefficient of • Process times are usually not highly variation (SCV) of the interarrival process variable on their own • Or, “the arrival SCV” • Time to make a part if everything goes well doesn’t change much from piece to piece doesn’t change much from piece to piece • Breakdowns, rework, setups make effective process time more variable • Similarly, let me = mean effective processing time (me = 1/ μ) • Let σ2e = variance of effective processing variance of effective processing time • Why “effective”? Recall our adjustments effective Recall our adjustments to the throughput rate (speed loss, breakdowns, etc.) 2/4/2012 Industrial Data and Systems Analysis • Let ce = σe / me, the coefficient of variation of effective process time • c2e is called the process SCV th SCV 2 is not easy to estimate • Unfortunately, c e 17 2/4/2012 • Let Wq = expected waiting time in queue expected waiting time in queue • Then, a reasonable approximation is • Let Lq = expected number of units in expected number of units in queue • Little’s Law: Lq = λWq • So, a reasonable approximation is c 2 + c 2 ρ Wq = a e 2 1 − ρ me c 2 + c 2 ρ 2 Lq = a e 2 1 − ρ • Suppose interarrival times and effective interarrival times and effective process times are exponentially distributed (c distributed (ca = ce = 1). Then, 1). Then, ρ λme mean arrivals during service time Wq = 1 − ρ me = μ − λ = rate at which queue is reduced rate which Industrial Data and Systems Analysis 18 Decoupling Stock: Expected Queue Length Kingman’s Approximation Approximation 2/4/2012 Industrial Data and Systems Analysis 19 • What happens as ρ approaches 1? 2/4/2012 Industrial Data and Systems Analysis 20 The Impact of Utilization and Variability on Decoupling Stock St • Decrease utilization utilization Decoupling Stock: Expected Queue Length Kingman's Approximation to the G/G/1 Queue th G/G/1 • I.e. increase μ relative to λ (throughput improvements) • Reduce λ (the DGR) • Buy extra capacity (extra machines to increase increase μ) 100 80 60 40 20 • Reduce variability 0 0 0.2 0.4 0.6 0.8 • Increase MTTF, reduce MTTR MTTF reduce MTTR • Reduce or eliminate rework • Reduce or eliminate setup times or eliminate setup times 1 Utilization Rate High Variability 2/4/2012 Medium Variability Low Variability 21 Industrial Data and Systems Analysis Series of Queues of Queues 1 2 Industrial Data and Systems Analysis 22 • Throughput rate = min(λ,μ) • If λ>μ then this workstation is a bottleneck and it 3 will have to work overtime to catch up (or λ will have to be reduced) have to be reduced) • In general, we will assume λ<μ (i.e. ρ<1). • In a series of workstations with ρi <1 for each station i, the throughput rate at each station will th th be λ • The bottleneck will be the workstation with the highest utilization, ρi.. Since the throughput is the same at each station (λ), the bottleneck will be the workstation, i, with the lowest processing rate, μi another • Variability in departure times from one station becomes variability in arrival times at the next station Industrial Data and Systems Analysis 2/4/2012 Interpreting Utilization Utilization • Output of one workstation is input to 2/4/2012 How to Reduce Decoupling St Stock? 23 2/4/2012 Industrial Data and Systems Analysis 24 Data We Assume We Know Data We Assume We Know ma=1/ λ c2a =σ 2a λ 2 2 me1=1/ μ1 c2 e1 • Input Data Data λ 1 λ Example =σ 2 e1 μ1 2 3 me2=1/ μ2 c2 e2 =σ 2 e2 μ2 2 Station Time per piece (seconds) Coefficient of variation 0 (Inter-arrivals) 40 0.25 1 2 25 1.25 3 38 0.75 37 0.75 me3=1/ μ3 c2 e3 =σ 2e3 μ32 • Given the input rate, λ, the arrival SCV, c2a, the processing rates, μi , and the process SCV’s process SCV’s, c2ei, for each station i, we for each station we would like to estimate the expected queue length in front of each station 2/4/2012 Industrial Data and Systems Analysis 25 2/4/2012 Industrial Data and Systems Analysis Estimating Departure Variability bilit Two Formulas to Remember Formulas to Remember • Let c2d denote the departure SCV from a 26 • Kingman’s Approximation for Queue queue (the squared coefficient of variation of inter-departure times) • A good simple approximation is good simple approximation is Length 2 2 cd = ρ 2 ce2 + (1 − ρ 2 )ca • Approximation for Departure Time Variability bilit • If utilization is high, the process SCV 2 2 cd = ρ 2 ce2 + (1 − ρ 2 )ca dominates (and is transmitted to the next dominates (and is transmitted to the next station) • If utilization is low, the arrival SCV dominates 2/4/2012 Industrial Data and Systems Analysis 2 ca + ce2 ρ 2 Lq = 2 1 − ρ • These two formulas are sufficient to estimate queue lengths at each station estimate queue lengths at each station 27 2/4/2012 Industrial Data and Systems Analysis 28 Solution Solution Station Time per piece (seconds) Coefficient of variation of variation Station Process Rates (pieces per second) Utilization SCV Departure SCV Performance View View 0 (Inter-arrivals) 40 0.25 1 25 1.25 38 0.75 37 0.75 0 (Inter-arrivals) 0.025 1 0.0400 0.6250 1.5625 0.6484 2 0.0263 0.9500 0.5625 0.5709 3 0.0270 0.9250 0.5625 0.5637 Variability Term Utilization Term Capacity Term Expected Waiting Time in Queue Decoupling Stock c 2 + c 2 ρ Wq = a e 2 1 − ρ me 2/4/2012 0.0625 0.0625 2 3 Total Expected Process Time Expected Process Time Total Expected Time in Queue Queue Factor • Most of the time a unit spends in factory is 0.8125 0.6055 0.5667 1.6667 19.0000 12.3333 25.0000 38.0000 37.0000 33.8542 437.1484 258.5993 0.8464 10.9287 6.4650 Lq = λWq 2 2 cd = ρ 2 ce2 + (1 − ρ 2 )ca Industrial Data and Systems Analysis 100.0000 729.6019 8.2960 29 Cycle Stock Stock queue time • One example: Total processing time for a ti small metal part: 55 seconds; Total order flow time: 55 days flow time: 55 days • Lots of opportunities for improvement! 2/4/2012 30 Industrial Data and Systems Analysis Cycle Stock Stock Cumulative Units Order Size Throughput rate Average Cycle Stock • Cycle stock: inventory in the system Average Queue Time because of batching operations because of batching operations Time 2/4/2012 Industrial Data and Systems Analysis 31 2/4/2012 Industrial Data and Systems Analysis 32 Cycle Stock Calculation Cycle Stock Calculation Reasons for Batching for Batching • If Q is the order (batch) size, then the average inventory on hand will be Q/2 • By Little’s Law, the average queue time due to batching = Q/(2*Throughput rate) • To reduce queue time (and cycle stock), reduce Q • But there is always a reason for batching 2/4/2012 Industrial Data and Systems Analysis 33 A Simple Model for Economic Order Quantit tity 34 • The average number of orders placed per year is dollars per order placed. • Let h denote the cost of holding one unit in denote the cost of holding one unit in inventory for one year. • Let λ denote the demand rate, measured in units per year. If you want to measure demand on a shorter term basis, such as units per month, then you must change the units of h , the holding cost, to be on the same time basis. th ti • Let Q denote the order size, in units. This is the decision variable we are trying to choose. Industrial Data and Systems Analysis Industrial Data and Systems Analysis The Basic Tradeoff Basic Tradeoff • Let K denote the fixed order cost, measured in 2/4/2012 2/4/2012 • • • • 35 λ / Q. The average annual cost of placing orders is K* K* λ / Q. Assuming that demand occurs constantly and continuously throughout the year at rate continuously throughout the year at rate λ , the the average inventory is given by Q/2. The average annual inventory cost is h*Q/2. Let f(Q) denote the total average annual cost of placing orders and holding inventory. Then, f(Q) = K* λ / Q + h*Q/2. 2/4/2012 Industrial Data and Systems Analysis 36 The EOQ The EOQ Safety Stock Stock • Solution: the optimal economic order the optimal economic order quantity (EOQ) is given by Q* = 2λ K h • Insight: to reduce cycle stock, reduce K, to reduce cycle stock, reduce K, • Safety stock: the expected net inventory th the fixed cost of the operation • What are the benefits to reducing cycle What are the benefits to reducing cycle stock? 2/4/2012 (on hand less backorders) at the time of delivery of replenishment order delivery of a replenishment order 37 Industrial Data and Systems Analysis Order Points and Lead Time Points and Lead Time 2/4/2012 38 Industrial Data and Systems Analysis Order Point, Order Quantity Point, Order Quantity Inventory Level Order More! Q R L Q L Q L B Time • ‘Inventory Position’ = on hand Stockout! 2/4/2012 Industrial Data and Systems Analysis backorders + on order on order • When inventory position falls below R, order order Q additional units additional units Delivery 39 2/4/2012 Industrial Data and Systems Analysis 40 Safety Stock Safety Stock Analysis • Mark cycle from order placement times cycle from order placement times • Total demand that occurs in one cycle = Q • Let B denote the number of customer denote the number of customer Inventory Level Q R L Q L Q backorders that occur during one cycle • Assume there is at most one order there is at most one order outstanding (ok if Q is large relative to Lμ) • The number of customer backorders is The number of customer backorders is unaffected by Q L B Time • Safety stock: expected net inventory (on hand – backorders) just before a delivery • Assume daily demand is normally distributed (mean variance distributed (mean μ, variance σ2) 2/4/2012 Industrial Data and Systems Analysis 41 Order Point Problem Point Problem 42 • Order lead time is L days lead time is days • Let DL be demand over L days • Backorders exist if lead time demand exceeds that can be satisfied without a stockout • Let f denote the fill rate: f = (Q-E[B]) / Q • Suppose we want to achieve some minimum fill rate fill rate, fmin : choose R to ensure that choose to ensure that (Q-E[B]) / Q ≥ fmin • That is, choose R to satisfy is choose to satisfy E[B] = Q(1- fmin) Industrial Data and Systems Analysis Industrial Data and Systems Analysis Order Point Analysis Point Analysis • ‘Fill rate’ is the expected fraction of demand rate is the expected fraction of demand 2/4/2012 2/4/2012 the order point: B = max(DL-R,0) • Assume daily demands are independent and daily demands are independent and identically distributed • Recall: variance of sum of independent random variables is the sum of the variances variables is the sum of the variances • Then, DL is normally distributed: N(Lμ,Lσ2) 43 2/4/2012 Industrial Data and Systems Analysis 44 ‘Normalize’ the Demand the Demand ‘Normalize’ the Reorder Point the Reorder Point • Let Z = (DL - Lμ)/L1/2σ (D • Then Z has a standard normal • Let k = (R - Lμ)/L1/2σ ) • Then, E[B] = L1/2σE[max(Z-k,0)] • Choose k to satisfy E[max(Z-k,0)] = Q(1- fmin)/(L1/2σ) • After some calculus: E[max(Z-k,0)] = φ(k) - k(1-Φ(k)) distribution: Z~N(0,1) where • φ(k) is the standard normal density function and • Φ(k) is the standard normal cumulative distribution function function 4 3 3.5 2 2.5 1 1.5 0.5 0 -1 -0.5 -2 -1.5 -3 -2.5 -4 2/4/2012 -3.5 Standard normal probability density function Industrial Data and Systems Analysis 45 Order Point Solution Technique 46 • μ = 15 units per day; σ2 = 16 • Q = 45 units; L = 2 days • What value of R will ensure that the fill φ(k) - k(1-Φ(k)) = Q(1- fmin)/(L1/2σ) • In Microsoft Excel: Microsoft Excel: rate is at least 98%? (ie. fmin=0.98) • Calculate r.h.s. of equation: Q(1- fmin)/(L1/2σ)= 45 * (0.02) / (2 1/2 *4) = 0.159 • Search for solution to: φ(k) - k(1-Φ(k)) = 0.159 φ(k)=EXP(-(k^2)/2)/ SQRT(2*PI()) Φ(k)=NORMSDIST(k) • Call the solution, k* • Value of R that ensures fill rate ≥ fmin: of that ensures fill rate R = Lμ + k*L1/2σ Industrial Data and Systems Analysis Industrial Data and Systems Analysis Example • Search for k that solves: for that solves: 2/4/2012 2/4/2012 47 2/4/2012 Industrial Data and Systems Analysis 48 Solution Solution Formulas to Remember to Remember 0.4 0.35 • Find k to satisfy 0.3 0.25 l.h.s. 0.2 φ(k) - k(1-Φ(k)) = Q(1- fmin)/(L1/2σ) k(1 Q(1 • Reorder point is given by R = Lμ + k*L1/2σ • Safety stock is given by SS = k*L1/2σ r.h.s. 0.15 0.1 0.05 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0 0.2 0 k • k* = 0.63. Now, solve for R: R = Lμ + k*L1/2σ =2*15 + 0.63 * (2 1/2 *4) k 0.63 (2 4) = 30 + 3.56 = 33.56 • Order point should be at least 33.56 units (34 if whole units must be ordered) whole units must be ordered) 2/4/2012 Industrial Data and Systems Analysis Std. dev. of lead time demand 49 Observations Concerning FillRate Driven Safety Stock St 2/4/2012 Industrial Data and Systems Analysis 50 Four Roles of Inventory Roles of Inventory • Safety stock is proportional to the stock is proportional to the standard deviation of lead time demand • Safety stock is proportional to the square stock is proportional to the square root of the lead time • Only the tail of the probability distribution the tail of the probability distribution is used • Safety stock depends inversely on the stock depends inversely on the order quantity, Q! 2/4/2012 Industrial Data and Systems Analysis Pipeline Stock Stock Cycle Stock Stock Decoupling Stock 51 2/4/2012 Safety Stock Stock Industrial Data and Systems Analysis 52 ...
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This note was uploaded on 02/09/2012 for the course ORIE 3120 taught by Professor Jackson during the Spring '09 term at Cornell.

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