INDEPENDENT DEMAND SYSTEMS: PROBABILISTIC MODELS Introduction

Transcription

1 INDEPENDENT DEAND SYSTES: PROAILISTIC ODELS Introduction Demand and Lead-Time are treated as random variable The models in this section assumes that the average demand remains approximately constant with time Probability distribution o demand is available The inventory can be divided into working stock and saety stock see ig. 1 S UANTITY S Working Stock Working Stock Saety Stock TIE Fig. 1 Working stock and saety stock in a -system o inventory control Working stock is the quantity expected to be used during a given time period The average working stock is one-hal the order quantity lot size, which may be determined by the EO ormula or some variant o it. Saety stock is the quantity used to protect against higher than expected demand levels It is the extra inventory kept on hand as a cushion against stockouts due to random perturbations o nature or the environment It has two eects on a irm s cost: decreases the cost o stockouts or increases holding costs Organisation can use countermeasures to prevent, avoid or mitigate stockouts. Typical countermeasures are expediting, emergency shipping, special handling, rescheduling, overtime, and substitution The prevention cost can be considered as stockout costs even though the stockout does not occur Customer s reaction to a stockout condition can result in a back order or a lost sales 1 Department o echanical Engineering

2 Saety stock determination is depends on the type o inventory control system used FIXED ORDER UANTITY SYSTE OR -SYSTE Inventory variation in an ideal and realistic situation is given in igs and 3 respectively S UANTITY S Lead Time Reorder Point Saety Stock Order Lot Order Lot Placed Received Placed Received TIE Fig. Ideal inventory model S UANTITY S Lead Time Stockout Lead Time Lead Time TIE Fig. 3 Realistic inventory model Department o echanical Engineering

3 Saety stock is needed to protect against a stockout ater the reorder point is reached and prior to receipt o an order This period is usually called lead-time The reorder point is composed o the mean lead-time demand plus saety stock Average inventory level on hand just beore the receipt o a replenishment order is the saety stock. Over many cycles, the inventory level will sometimes be more than the saety stock and sometimes less, but it should average to the saety stock Larger the order quantity, ewer the annual orders, which means the ewer opportunities or stockout to occur Saety stocks are dependent on stockout cost or service level, holding cost, demand variation and lead-time variation Working stock quantity is determined beore considering saety stock In the order quantity ormulations, it is assumed that the order quantity can be determined by an economic balance o the relevant cost, and that it is independent o the reorder point Average inventory Expected level o inventory beore receiving the order is saety stock Expected level o inventory immediately ater receiving the order is Saety stock Average inventory is the average o these two values is There are two approaches or saety stock calculation: 1 Known stockout cost explicit costs can be allocated to shortages Saety stock Unknown stockout costs management speciies a service level based on some probability distribution o demand during the lead-time Statistical considerations Notations lead-time demand in units a random variable - mean lead-time demand - standard deviation o lead-time demand probability density unction o lead time demand P probability o a lead-time demand o units max - maximum lead-time demand reorder points in units P> probability o stockout 3 Department o echanical Engineering

4 Department o echanical Engineering 4 E> expected stockout in units during lead-time Continuous Distribution 0 d 0 d > d P > d E Discrete Distribution max 0 P max 0 m P > max 1 P P > max 1 P E Lead-time demand distribution is required or inventory analysis in -system Frequently the demand distribution is expressed on a time basis that is dierent rom the lead-time Convolution technique is used to get the demand distribution or varying length o time Standard Distributions Normal Distribution Parameters are mean and standard deviation π e or < < - > d E

5 Z 1 F Z [1] S Z [] Equations [1] and [] uses standard normal variate values Another way o estimating E > is given below Standard Normal Distribution ean 0, standard deviation 1 Z Z e Z π Transormations or normal distribution to standard normal distribution Let a value o normal random variable be and the corresponding value or standard normal random variable Z be t. t t E Z > t E t E > For standard normal random variate, the values or the ollowing are tabulated Z, FZ, Z, and EZ Note: Poisson Distribution Single parameter P > d P Z > t Z dz Normal random variate representing lead time demand average lead time demand in units S saety stock in units Z standard normal variate - standard deviation o lead time demand Poisson distribution is not symmetrical with respect to mean when mean is small - Skewed to the right t 5 Department o echanical Engineering

6 Normal approximation to Poisson is usually adequate when mean is 1 or greater Negative Exponential Distribution Single parameter SAFETY STOCK ESTIATION: KNOWN STOCKOUT COST AND LEAD TIE Stockout cost is expressed as backorder cost per unit, backorder cost per outage, lost sales cost per unit or lost sales per outage ackorder cost per outage is a ixed amount and can occur at most once during a replenishment cycle Assumed a variable demand and constant lead-time ackorder Case: Stockout Cost per unit No lost sales Notations reorder points in units S saety stock in units H holding cost per unit o inventory per year A backordering cost per unit R average annual demand in units lot size or order quantity in units lead-time demand in units a random variable average lead time demand in units probability density unction o lead time demand TC s expected annual saety stock cost s 0 d d 0 0 d E > d Expected annual saety stock cost holding cost stockout cost AR TC s SH d 6 Department o echanical Engineering

7 AR H d dtc s To get optimum value o, 0 d LEINIZ S RULE G x k x h x g x, y dy k x dg x dx h x g x, y dk x dh x dy g x, k x g x, h x x dx dx d b d d d P > dtc s d P > RA H H AR [ P > ] 0 This ormula can be applied to both discrete and continuous probability distribution o lead-time demand When discrete distributions are used, the exact optimum stockout probability is usually unattainable When the optimum stockout probability cannot be attained, the next lower attainable stockout probability is selected Problem What is the optimal reorder point or the inventory problem speciied below? The lead time demand distribution is given in table below. 7 Department o echanical Engineering

8 R 1800 units per year C Rs 300 per order F 15 % P Rs 0 per unit A Rs 10 per unit backordered Also, calculate the expected stockout quantity, saety stock and expected annual saety stock cost. Determine the standard deviation and mean o the distribution. Consider this mean and standard deviation as the parameters o normal distribution and determine reorder level and saety stock. ackorder Case: Stockout Cost Per Outage G backorder cost per outage R TC s SH G d For Normal distribution For discrete distribution GR P H dtc s 0 d Z H GR > When the optimum reorder point lies between two integer values select the integer with the lower Lost sales case: Stockout cost per Unit All stockouts are lost and not recovered Saety stock is zero whenever Average number o annual cycles is R E > Expected stockout quantity per cycle is E> R Lead time demand Probability Department o echanical Engineering

9 Department o echanical Engineering 9 d S 0 0 d d d > d E s d AR SH TC d H AR H H AR H P d dtc s > 0 Lost Sales Case: Stockout Cost Per Outage s d GR SH TC d GR d F H H GR H F P d dtc s > 1 0 SAFETY STOCK ESTIATION: KNOWN STOCKOUT COST AND VARIALE LEAD TIE Constant Demand and Variable Lead Time Solution technique similar to variable demand, constant lead time case Demand distribution during lead time is obtained by multiplying the constant demand by probability distribution or the lead time I lead time ollows normal L ZD DL Z Where D constant demand rate per period standard deviation o demand during lead time

10 L standard deviation o lead time L average lead time in periods Variable Demand and Variable Lead Time Joint probability distribution JPD is required Range o the JPD is rom the level indicated by the product o the smallest demand and shortest lead time to level indicated by the product o largest demand and largest lead time JPD is used to analyse the appropriate stockout cost situation as previously discussed When demand and lead time distributions are independent DL L D D L Where L average lead time length in periods D average demand per period D standard deviation o demand distribution L standard deviation o lead time distribution standard deviation o demand during lead time When demand and lead time distributions are not independent DL L D D L With variable demand and variable lead time, the solution procedures are the same as those previously discussed SAFETY STOCK ESTIATION: SERVICE LEVELS When stockout costs are not known or eels very uneasy about estimating them, it is customary or management to set service levels or which reorder points can be ascertained. A service level indicates the ability to meet customer demands rom stock, or in some other timely manner For make-to-stock or order-to-stock environment, service implies illing demand rom inventory and better service involves excess inventory investment For make-to-order or assemble-to-order, service implies providing in time and better service involves excess capacity As the service level approaches 100 %, the investment in saety stock oten increases drastically The principle o diminishing return applicable here D L 10 Department o echanical Engineering

11 While perect customer service might not be attainable, lead time reduction and just-in-time approaches can substantially improve the customer service A production line where ailure to provide needed parts can bring the line to halt and in such environment RP system is suitable than a ixed order size system The service level takes on dierent meaning, depending upon how it is stated as a decision criterion Service per order cycle Service per units demanded Saety stocks under dierent service concepts will be dierent Service per Order Cycle It indicates the probability o not running out o stock during the replenishment lead time It is not concerned with how large the shortage is, but with how oten it can occur Service level per order cycle SLc is deined as the raction o replenishment cycles without depletion o stock number o cycles with a stockout SL c 1- total number o order cycles 1- P > P> is the probability o a stockout during the lead time or the stockout probability per order cycle. It represents the probability o at least one stockout during the lead time or the raction o lead time periods during which the demand will exceed the reorder point Service per order cycle does not indicate how requently stockout will occur over a given time period or all products This is because the order cycle will vary rom product to product ore requently stock is replenished, the greater the number o expected stockout cycles This indicates that service per order cycle does not allow or uniorm treatment o dierent products It also does not indicates the percentage o demand that will be satisied Service per Units Demanded It does indicate the percentage o demand that will be satisied and allow uniorm treatment o dierent products 11 Department o echanical Engineering

12 number o units stockouts SL u 1 - total number o units demanded E > 1 - or backorder E > 1 - E > or Imputed Stockout Costs or the given Service Level Service level really does impute a stockout cost lost sales For a given service level the imputed stockout cost can be calculated rom previously developed optimum ormulations or the probability o a stockout Imputed stockout cost is a convenient way to determine i the value chosen or the service level is appropriate FIXED ORDER INTERVAL SYSTE: PROAILISTIC ODELING Inventory position is monitored at discrete point in time Once an order is placed at time t, another order can not be placed until t T, and the second order will not be illed until the lead time period has elapsed, at t T L Thus saety stock protection is needed or the lead-time L plus the order interval T In the ixed order size system, saety stock is needed only or the lead-time period, because the inventory position is monitored with each transaction In the ixed order size system, a higher than normal demand causes a shorter time between orders whereas in the ixed order interval system, the result would be a larger order size Predetermined inventory level E E S RT LR Where S is the saety stock considering demand variation during T L period RT LR is the average demand during T L period 1 Department o echanical Engineering

13 Realistic inventory model or P system Predetermined inventory level E UANTITY Inventory level o expected demand during lead-time and saety stock Saety stock level L L L T T TIE Stockout Fig: P-System: Inventory variation with time Saety stock is determined based on the inventory variation during lead time and review period. That is, or saety stock calculation, the demand variation during the TL period is considered. The order interval T T C HR 0 EOI in years Determination o Saety Stock when stockout cost known Approach is same that o -system is a random variable shows the demand distribution or T L period 13 Department o echanical Engineering

14 Table. Formula or determining saety stock Stockout case Stockout cost per unit Stockout cost per outage ackorder P > E HT A E HT G Lost sale HT P E > E A HT 1 P > E G HT Determination o Saety Stock when service level known Service level per order interval SL c number o order intervals with a stockout SL c 1- total number o order intervals 1- P > E Service levels per units demanded SL u A worked out Problem number o units stockouts SL u 1 - total number o units demanded E > E 1 - TR Daily demand or hamburger buns at a city centre production plant is as ollows: Daily Demand Probability The plant can produce several other products using the same acility. The lie o the product is 5 days. Consider 5 days as planning horizon and -system o inventory control, determine the batch size. The ollowing data is also available or the above problem. Carrying cost Rs 1 per unit day Setup cost Rs 450 per setup What should be the reorder point with a lead-time o 1 day and the service level per order cycle o 85 %? What is the saety stock? What is the stockout cost per unit imputed by this service level? Assume that the daily demand is normally distributed with parameters that can be determined rom the daily demand distribution given above. Determine the reorder point and saety stock i the above service level is applicable. 14 Department o echanical Engineering

15 I P-system o inventory control is used, what are the review period, predetermined inventory level and saety stock? Compare the total cost o expected saety stock o both the system o inventory controls. 1. Given data: Daily demand distribution Daily demand Probability Total 1.0 Service level per order cycle SL c 85 %. Lead time 1 day Planning horizon 5 days Carrying cost H Rs. 1 per unit per day. Set up cost C Rs. 450 per set up. atch size RC H Where, R average demand in the planning horizon average demand per day Number o days in a planning horizon Average demand per day Units. Now R Units Reorder Point: SL c 1- P> P > P > Units. 15 Department o echanical Engineering

16 Daily demand Probability Cumulative probability Stock out probability P > 1- cumulative probability From the above table, Reorder point, or 0.15 stockout probability is 550 units. Reorder point S S Thereore, Saety stock, S Units. Stock out cost per unit A imputed by given service level : H P > AR A 450 A 9.0 Rupees per unit. I the daily demand is normally distributed with parameters that can be determined rom the daily demand distribution given above: ean, 490 Units, Variance, n i i 1 P Standard normal deviate, Z From statistical tables, Z or P> 0.15, is Units Department o echanical Engineering

17 Reorder point S S Thereore, Saety stock, S Units. I P system o inventory control is used: Economic order interval, T o days 1.35 days 1 day approximately C RH horizon o the planning Service level per order cycle SL c 85 %. SL c 1- P>E P >E P >E 0.15 For inding, maximum inventory level E, we need demand distribution or T o Lead time period. T o Lead time 11 days. Preparation o demand distribution or days First day Second day Demand Probability Demand Probability Total demand probability Department o echanical Engineering

18 Demand distribution or days Demand Calculation o probability Probability Cumulative probability Total 1.0 Stock out probability P >E 1- cumulative probability From the above table, aximum inventory level, E or 0.15 stock out probability is 1050 Units. aximum inventory level E S Units. Thereore, Saety stock, S E Units. 18 Department o echanical Engineering

19 Comparison o total cost o maintaining expected saety stock in and P systems, i stock out cost is Rs. 6 per unit: System: max R TC SH A P s P-System: s Rupees. 1 max T o E 1 TC SH A E P Rupees. 19 Department o echanical Engineering