5-1 ISE 315 – Production Planning, Design and Control Chapter 5 – Inventory Control Subject to Unknown Demand McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

5-2 In this chapter Inventory Control with uncertain demand  Demand is a random variable with a given CDF  If demand is a random variable, cost is also a random variable (it is a function of demand) Models – Newsboy model (one period model, one-time decision) – Lot size – reorder point systems (continuous review systems, namely (Q, R) ) – Periodic review systems (namely (s, S) )

5-3 The Nature of Uncertainty Suppose that we represent demand D as D = D deterministic + D random If the random component is small compared to the deterministic component – the models of Chapter 4 will be accurate. – if not, randomness must be accounted explicitly in the model. In this chapter, assume that demand is a random variable with cumulative distribution function F(x) probability density function f(x)

5-4 Timing Decisions Intermittent-Time Decisions Continuous Decisions One-Time Decision Continuous Review Systems Periodic Review Systems EOQ, EPQ (Q, R) System Two Bins (s, S) System Optional Replenishment (S, T) System Structure of timing decisions Newsboy

Newsboy Model one period, one time decision model Demand is random At the end of the period, unsold items perish or become obsolete It could apply to a person who purchases newspapers at the beginning of the day -> sells a random amount -> must discard/scrap any leftovers We need to determine the appropriate order quantity

Lot size – reorder point sytems Also called (Q, R) model Demand occurs randomly, possibly in batches Inventory is monitored continuously – Inventory level is known at all times POLICY: When the inventory level reaches reorder point R, an order of size Q is placed After a lead-time (during which a stockout might occur) the order is received – The problem is to determine appropriate values of Q and R.

(s, S) model Extension of Newsboy model – unsold items can be used in the future periods. Demand occurs randomly Inventory is reviewed periodically (period length is constant) POLICY: At the time of review, – if the inventory level is less than s order up to S The problem is to determine best values of s and S Harder to develop and find optimal levels

Newsboy model: an example A manufacturer of Christmas lights Demand is unpredictable and occurs just prior to Christmas The decision of how many sets of lights to produce must be made prior to the holiday season If the production is not enough, some demand will be lost The cost of holding unsold inventory until next year is too high so that storage is not an attractive option Therefore, any unsold sets of lights are sold after Christmas at a steep discount.

5-9 Appropriate production quantity To choose an appropriate production quantity, the important pieces of information to consider are: (1)probability distribution of demand (2)unit costs of producing more than or less than the actual demand

5-10 Christmas Light Example Suppose that a set of lights costs $1 and sells for $2 Any unsold set on Christmas will be discounted to $0.5. Demand fits the Normal Distribution with mean 10,000 units and 1,000 units standard deviation. What is the optimal order quantity?

5-11 Notation It can be shown that the optimal number of newspapers to purchase is the fractile of the demand distribution given by F(Q*) = c u / (c u + c o )

5-12 Calculation of the costs Underage cost = lost profit (selling price – purchase cost) + loss of goodwill Overage cost = unit cost (purchase cost –scrap value/amount paid back) + cost of disposal The overage cost is c o = 1 ‑ 0.5 = $0.5 The underage cost is c u = 2 ‑ 1 = $1

5-13 Christmas Light Example The symmetry (i.e., bell shape) of the normal distribution implies that it is equally likely for demand to be above or below 10,000 units. – If demand is below 10,000 units, the firm will lose c o = $0.5 per unit – If demand is above 10,000 units, the firm will lose c u = $1 per unit – Clearly, shortages are worse than overages This suggests that perhaps the firm should produce more than 10,000 units. But, how much more?

5-14 Probability of not selling this extra unit at the end of the period = P(D < Q) The Concept of Marginal Analysis one more unit Marginal analysis: Finding the expected impact of ordering one more unit. Probability of selling this extra unit = P(D ≥ Q)  Q

5-15 Marginal Analysis Continued P(X>Q) (C u applies ) Single Period Inventory Model Marginal Analysis: E (overage cost of last sale) + E (underage cost of last sale) = 0

5-16 Marginal Analysis Continued Expected overage cost of the extra unit is c o P(Demand < Q) = c o F(Q) Expected underage cost of the extra unit is - c u P(Demand > Q) = - c u (1-F(Q)) At Q *, c o F(Q * ) - c u (1-F(Q * )) = 0 Probability of satisfying the demand during the period, F(Q*), is known as critical ratio To calculate Q * we must use cumulative distribution function of demand

5-17 Amount of Overage and Underage We order Q units Demand is D units Overage = Underage =

5-18 Expected overage and underage

5-19 The expected cost Note that is not a function of x. Note that in cost calculation, the purchase cost (purchase cost * Q) is ignored since included in the overage and underage costs. Which minimizes this cost?

5-20 Optimization We obtain the first derivative. Leibnitz's rule:

5-21 Optimization Applying the Leibnitz’s rule

5-22 Optimal Order level for Newsboy To minimize expected overage plus shortage cost, we have to choose a production or order quantity Q* that satisfies: F(x) increases in x, so that anything that increases the right-hand side of the equation, results in a larger Q*. Increasing c u will increase Q*, Increasing c o will decrease Q*

5-23 Christmas Light Example Demand is assumed to be normally distributed z represents the standard normal variable Ф represents the cumulative distribution function of the standard normal distribution

5-24 Determination of the Optimal Order Quantity for Newsboy Example

5-25 Basics Insights In an environment of uncertain demand, the appropriate production/order quantity depends on – distribution of demand – the relation of overage and underage costs

Lot Size-Reorder Point (Q, R) Systems Assumptions – Inventory levels are reviewed continuously – Demand is random and stationary. We cannot know the exact demand, expected value of demand of any length of horizon is constant, – There is a positive lead time, τ Two decisions – Order quantity, Q – Reorder point, R

5-27 Related parameters

The (Q, R) Policy When the inventory level reaches (or goes below) R, an order of size Q is placed. After a lead-time of τ during which a stock out might occur, the order is received. The problem is to determine appropriate values of Q and R

5-29 Basic mechanics of the (Q, R) policy R Q time Inventory level Order placed Order received τ: Lead time

5-30 Example The manager of a maintenance department must stock spare parts to facilitate equipment repairs Demand for parts is a function of machine breakdowns and random The cost incurred in placing a purchase order (for parts obtained from an outside supplier) and the costs associated with setting up the production facility (for parts produced internally) are significant enough to make one-at-a-time replenishment impractical The maintenance manager must determine not only how much stock to carry but also how many to produce/order at a time (as in the EOQ and newsboy models)

5-31 Reorder Point, R ) Safety Stock (SS) Time Inventory Level SS R Expected Demand P(Stockout) Probability Lead Time Place order Receive order When to Order?

5-32 Uncertain Demand Time On-hand inventory Order received On Hand Order placed Order placed Order received IP R Cycle 1Cycle 2Cycle 3 τ1τ1 τ 2 τ 3 Order placed Order received Order received

5-33 Expected Inventory (Assumptions) Time I(t) Slope - Q SS

5-34 Expected cost function Total cost = Holding + ordering + shortage Average Holding Cost: Average Ordering Cost:

5-35 Expected average annual cost D: random variable represents demand during lead time Expected number of shortages per cycle: n(R) is called the Loss Function Expected Penalty Cost : Loss Function values for Normal distribution in Table A-4)

5-36 Expected average annual cost

5-37 Cost Minimization Expected Cost Function: Partial Derivatives: (1) (2)

5-38 Cost Minimization Partial Derivatives: (2)

5-39 Optimal replenishment quantity Simultaneously solve two equations to find optimal replenishment quantity Q*, and reorder point R*

5-40 Think and Discuss Insights from the model 1.Cycle stock increase as replenishment frequency decrease 2.Safety stock provide a buffer against stock out

5-41 Think and Discuss Suppose you are stocking parts purchased from vendors in a warehouse. How could you use a (Q, R) model to determine whether a vendor of a part with a higher price but a shorter lead time is offering a good deal? What other factors should you consider in deciding to change vendors?

5-42 Solution Procedure The optimal solution procedure requires iterating between the two equations for Q and R until convergence occurs A cost effective approximation is to set Q=EOQ and find R from the second equation. A slightly better approximation is to set Q = max(EOQ,σ) where σ is the standard deviation of lead time demand when demand variance is high)

5-43 Finding Q and R, iteratively 1. Compute Q = EOQ. 2. Substitute Q in to Equation (2) and compute R. 3. Use R to compute n(R) in Equation (1). 4. Solve for Q in Equation (1). 5. Go back to Step 2, continue until convergence.

5-44 Example Demand is Normally distributed with mean of 40 per week and a weekly variance of 8 The ordering cost is $50 Lead time is two weeks Shortages cost an estimated $5 per unit short to expedite orders to appease customers The holding cost is $ per week Find (Q,R)

5-45 Demand is per week. Lead time is two weeks long. Thus, during the lead time:  Mean demand is 2(40) = 80  Variance is (2*8) = 16  Demand observed in one week is independent from demand observed in any other week:  E(demand over 2 weeks) = E (2*demand over week 1) = 2 E(demand in a single week) = 2 μ = 80 Standard deviation over 2 weeks is = 4 Solution

5-46 Solution Iteration 1: From the standard normal table:

5-47 Solution Iteration 2: This is the unit normal loss expression. Table A - 4 gives values.

5-48 Solution Iteration 2:

5-49 Example A company observes the expected demand of air filters as 2600 per year. They order it from a supplier. $10 to place an order Unit cost is $25 per filter Inventory carry cost is $2/unit per year Shortage cost is $5 Lead time is 2 weeks Assume demand during lead time follows a uniform distribution from 0 to 200 Find the optimal Q and R values

5-50 Solution Partial derivative outcomes:

5-51 Solution From Uniform U(0,200) distribution:

5-52 Solution Iteration 1: F(R) 2000 R

5-53 Solution Iteration 2:

5-54 Solution R didn’t change => CONVERGENCE (Q*,R*) = (163.78,194.96) I(t) Slope -  With lead time equal to 2 weeks: SS = R –  = (2/52)=159