# Murat Kaya, Sabancı Üniversitesi 1 MS 401 Production and Service Systems Operations Spring 2009-2010 Inventory Control – IV Multiperiod Probabilistic Demand:

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Murat Kaya, Sabancı Üniversitesi 1 MS 401 Production and Service Systems Operations Spring 2009-2010 Inventory Control – IV Multiperiod Probabilistic Demand: (Q,R) Model Slide Set #6

Murat Kaya, Sabancı Üniversitesi 2 Lot Size – Reorder Point Systems Infinite horizon model (similar to the EOQ model) Continuous review –the level of on-hand inventory is know at all times Demand is random and stationary –the mean rate of demand is know to be λ Replenishment lead time τ is known and fixed Setup cost K per order Holding cost h per unit hold per unit time Unit order cost c per item Stock out cost p per unit of unsatisfied demand –shortage cost, penalty cost

Murat Kaya, Sabancı Üniversitesi 3 Problem Setup Relevant random variable: Demand during lead time, D –this is what we guard against –let μ=E(D) and σ: standard deviation of D – Decision variables –Q: Lot size (order quantity) –R: Reorder level Policy: When the on-hand inventory drops to R, place an order of Q units which will arrive in τ units of time

Murat Kaya, Sabancı Üniversitesi 4 Actual On-hand Inventory Plot Copyright © 2001 by The McGraw-Hill Companies, Inc

Murat Kaya, Sabancı Üniversitesi 5 Expected Inventory Level R R- μ +Q R- μ Safety Stock Reorder Point Order Arrival Slope= -λ Cycle, length T=Q/ λ Order Arrival

Murat Kaya, Sabancı Üniversitesi 6 The Objective The objective is to find the (Q,R) values that minimize the expected annual average cost function Three cost components –holding cost –setup cost –penalty cost

Murat Kaya, Sabancı Üniversitesi 7 The Holding Cost The expected inventory level varies linearly between –R-μ+Q and (where μ is the mean of the lead time demand D) –R-μ (safety stock) Average holding cost per unit time This is only an approximation –in reality, the true inventory level may be negative in which case no holding charge should apply

Murat Kaya, Sabancı Üniversitesi 8 The Setup Cost Number of setups per unit time = number of cycles per unit time = 1/T = λ/Q Average setup cost per unit time = K λ / Q

Murat Kaya, Sabancı Üniversitesi 9 The Penalty Cost Stock-outs occur if demand during leadtime (D) is greater than the reorder level (R) Expected number of stock-outs (shortages) in a cycle: Expected number of stock-outs per unit time:

Murat Kaya, Sabancı Üniversitesi 10 The Cost Function & Solution The optimal solution (Q, R) is found by iteratively solving the following two equations: Objective: Minimize the expected average annual cost of holding, setup and shortages

Murat Kaya, Sabancı Üniversitesi 11 1.Select Q 0 =EOQ 2.Using equation (2) find R 0. Let i=1 3.Using equation (1) find Q i Using equation (2) find R i. If the differences Q i -Q i-1 and R i -R i-1 are small, stop; otherwise let i=i+1 and continue step 3. The Algorithm to Find (Q*, R*) Equation (1)Equation (2)

Murat Kaya, Sabancı Üniversitesi 12 Normally Distributed Demand Case Note that n(R) is not easy to calculate in general If demand is normally distributed, we have where L(z) is the standardiz normal loss function –tabulated in Table A-4 of the Appendix of Nahmias

Murat Kaya, Sabancı Üniversitesi 13 (Q, R) Policy: Example Harvey’s Specialty Shop sells is a popular mustard. The mustard costs Harvey \$10 a jar. Replenishment lead time is 6 months. Harvey uses a 20 percent annual interest rate and estimates the loss of goodwill cost as \$25 per jar in case of stockout. Bookkeeping expenses for placing an order is \$50. During the six-month replenishment period, Harvey estimates that he sells an average of 100 jars, and the standard deviation of demand is 25. Assume that demand is described by a normal distribution. How should Harvey control the replenishment of the mustard?

Murat Kaya, Sabancı Üniversitesi 14 (Q, R) Policy: Example First, note that λ=200 (the annual demand)

Murat Kaya, Sabancı Üniversitesi 15 (Q, R) Policy: Example

Murat Kaya, Sabancı Üniversitesi 16 (Q, R) Policy: Example (cont.) Determine the following for the example stated above: –the safety stock –the average annual holding, setup and penalty costs associated with the inventory control of the mustard –the average time between placement of orders –the proportion of order cycles in which no stock out occurs –the proportion of demands that are not met

Murat Kaya, Sabancı Üniversitesi 17 Service Levels

Murat Kaya, Sabancı Üniversitesi 18 Managers often have difficulty in determining the stock out cost p because it includes intangible components such as the loss of goodwill cost A common substitute is the service level There are number of different definitions of service level, all related to the probability that the demand is met We will discuss two commonly used service levels: –type-1 Service (  ) –type-2 Service (  ) Service Levels in (Q, R) Systems

Murat Kaya, Sabancı Üniversitesi 19 Type-1 Service Level (  ): The probability of not stocking out during lead time Equivalent to Computation of (Q, R) values: –determine R value that satisfies F(R) =  –set Q =EOQ A Type-1 Service objective is appropriate when a shortage has the same consequence independent of its time or amount Type 1 Service Level

Murat Kaya, Sabancı Üniversitesi 20 Type-2 Service Level (  ): The proportion of demand met from on-hand stock (fill rate) –i.e. Expected number of stock outs per cycle: n(R) Expected demand during a cycle: Q –why? Therefore, n(R)/Q is the proportion of demand NOT met from on hand stock in each cycle Hence, R is determined from n(R)/Q=1-  Q is set as the EOQ –not optimal but a good approximation Type 2 Service Level

Murat Kaya, Sabancı Üniversitesi 21 Scenario 1Scenario 2 Order Cycle DemandStocks Order Cycle DemandStocks 11,00099011,000 22,0001,99022,000 34,0003,99034,0000 42,0001,99042,000 51,00099051,000 Total10,0009,950Total10,0006,000 Type 1 vs. Type 2 Service Levels

Murat Kaya, Sabancı Üniversitesi 22 Type 1 vs. Type 2 Service Levels Type-2 (fill rate) assumes as if there is a cost of disappointing each customer –and, the customers are treated as equal Type-1 (in-stock rate) assumes as if there is a high cost of disappointing even one customer –no credit for partial fulfillment in a given period

Murat Kaya, Sabancı Üniversitesi 23 Recall the Harvey’s Specialty Shop Suppose Harvey feels uncomfortable with the stock-out cost concept and decides to use a service level of 98% Type-1:  =0.98. From Table A-4 (Nahmias) we obtain z=2.05 hence Type-2:  =0.98. In this case again, use EOQ as the Q. We need to solve n(R)=EOQ*(1-  ) to determine R n(R)=100*(1-0.98)=2=L(z)*  hence Therefore R=100+25*1.02 = 126.

Murat Kaya, Sabancı Üniversitesi 24 Important Notes Type-1 and type-2 service levels are used as a substitute to “penalty cost” –hence, if either of the service levels is specified, we do not need a penalty cost to solve the problem Type-1 service: Q*=EOQ. Hence, no iteration to find Q* Type-2 service: EOQ is an approximation to Q*. We assume that the approximation is good enough, hence we will not iterate to find Q* –ignore the section in p.257 of Nahmias

Murat Kaya, Sabancı Üniversitesi 25 Scaling of Lead Time Demand: Example Weekly demand for a certain type of automotive spark plug is normally distributed with mean 34 and standard deviation 12. Procurement lead time is six weeks. Determine the lead time demand distribution. The demand over the lead time is normally distributed with –mean: μ=6*34=204 –standard deviation: 12* sqrt(6)=29.39 Caution!

Murat Kaya, Sabancı Üniversitesi “Inventory Position” Concept We checked “on hand inventory” level when making the ordering decision When lead times are long, this can cause problems We might need to place a second order before an order arrives For such cases, checking the “inventory position” is better: Inventory position = Inventory on hand + (inventory on order) Inventory position never falls before R 26

Murat Kaya, Sabancı Üniversitesi Periodic Ordering Systems: (s,S) Policy (Q,R) model assumes that –the inventory level is monitored continuously –and we can place an order anytime we need In practice, –the firm might know its inventory position only at certain time intervals (say, each Friday evening, after an inventory count) –or the firm might order only at certain times (due to supplier’s rules etc.) In such cases, it is not possible to use the (Q,R) model The correct model to use: (s,S) model 27

Murat Kaya, Sabancı Üniversitesi Periodic Ordering Systems: (s,S) Policy (s,S) policy: –Check your inventory position every T time units –If the position is less than s, order (S-s) units –If the position is at least s, do not order Note that the order quantities might be different at each order Determining the optimal (s,S) values is difficult. A good approximation: –compute the (Q,R) values as if using a (Q,R) model –set s=R, S=Q+R 28

Murat Kaya, Sabancı Üniversitesi 29 Multiproduct Systems: ABC Analysis

Murat Kaya, Sabancı Üniversitesi 30 There is a trade-off between the cost of controlling the system and the potential benefits from that control Typically –the top 20% of the items account for the 80% of the annual dollar value of sales –the next 30% for the next 15% value of sales –remaining 50% for 5% value of sales To use ABC analysis: 1.select criterion for ranking (for example, annual sales value) 2.rank items on basis of criterion 3.calculate cumulative percentages 4.set up classes around break points Multiproduct Systems: A-B-C Analysis

Murat Kaya, Sabancı Üniversitesi 31 The Distribution of Inventory by Value Copyright © 2001 by The McGraw-Hill Companies, Inc

Murat Kaya, Sabancı Üniversitesi 32 Example A-B-C Analysis

Murat Kaya, Sabancı Üniversitesi 33 Example A-B-C Analysis: Sorted by \$ Usage

Murat Kaya, Sabancı Üniversitesi 34 Example A-B-C Analysis: Cumulatives Graph

Murat Kaya, Sabancı Üniversitesi 35 Multiproduct Systems: Exchange Curves

Murat Kaya, Sabancı Üniversitesi Exchange Curve Example Consider a deterministic system consisting of n products with varying demand rates λ i and item values c i Assume that EOQ is used for each item’s replenishment The same K and I value for all items We may treat K/I as an aggregate policy parameter –If K/I is large, lot sizes and average investment in inventory will be large –If K/I is small, the number of annual replenishments will be high An exchange curve allows us to see this trade-off easily Also, we do not need to know the “exact” values of K and I together 36

Murat Kaya, Sabancı Üniversitesi 37 Exchange Curve Example The number of annual replenishments: The multiplication of these two expressions is equal to: The average value of on-hand inventory: The exchange curve makes it easy to see the trade-offs

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