CHAPTER 5 Inventory Control Subject to Uncertain Demand

5 -2 The Nature of Uncertainty Suppose that we represent demand as D = Ddeterministic + Drandom If the random component is small compared to the deterministic component, the models of chapter 4 will be accurate. If not, randomness must be explicitly accounted for in the model. In this chapter, assume that demand is a random variable with cumulative probability distribution F(t) and probability density function f(t).

5 -3 The Newsboy Model At the start of each day, a newsboy must decide on the number of papers to purchase. Daily sales cannot be predicted exactly, and are represented by the random variable, D. Costs: co = unit cost of overage cu = unit cost of underage It can be shown that the optimal number of papers to purchase is the fractile of the demand distribution given by F(Q*) = cu / (cu + co). See Figure 5 -4 when demand is normal with μ = 11. 73 and σ = 4. 74, and the critical fractile is 0. 77.

Determination of the Optimal Order Quantity for Newsboy Example 5 -4

5 -5 Lot Size Reorder Point Systems Assumptions u Inventory levels are reviewed continuously (the level of on-hand inventory is known at all times) u Demand is random but the mean and variance of demand are constant. (stationary demand) u There is a positive leadtime, τ. This is the time that elapses from the time an order is placed until it arrives. u The costs are: Set-up each time an order is placed at $K per order u Unit order cost at $c for each unit ordered u Holding at $h per unit held per unit time ( i. e. , per year) u Penalty cost of $p per unit of unsatisfied demand u

5 -6 Describing Demand The response time of the system in this case is the time that elapses from the point an order is placed until it arrives. Hence, the uncertainty that must be protected against is the uncertainty of demand during the lead time. We assume that D represents the demand during the lead time and has probability distribution F(t). Although theory applies to any form of F(t), we assume that it follows a normal distribution for calculation purposes.

5 -7 Decision Variables For the basic EOQ model discussed in Chapter 4, there was only the single decision variable Q. The value of the reorder level, R, was determined by Q. In this case, we treat Q and R as independent decision variables. Essentially, R is chosen to protect against uncertainty of demand during the lead time, and Q is chosen to balance the holding and set-up costs. (Refer to Figure 5 -5)

Changes in Inventory Over Time for Continuous-Review (Q, R) System 5 -8

5 -9 The Cost Function The average annual cost is given by: Interpret n(R) as the expected number of stockouts per cycle given by the loss integral formula. The standardized loss integral values appear in Table A-4. The optimal values of (Q, R) that minimizes G(Q, R) can be shown to be:

5 -10 Solution Procedure The optimal solution procedure requires interating between the two equations for Q and R until convergence occurs (which is generally quite fast). 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 -11 Service Levels in (Q, R) Systems In many circumstances, the penalty cost, p, is difficult to estimate. For this reason, it is common business practice to set inventory levels to meet a specified service objective instead. The two most common service objectives are: 1) Type 1 service: Choose R so that the probability of not stocking out in the lead time is equal to a specified value. 2) Type 2 service. Choose both Q and R so that the proportion of demands satisfied from stock equals a specified value.

5 -12 Computations For type 1 service, if the desired service level is α then one finds R from F(R)= α and Q=EOQ. Type 2 service requires a complex interative solution procedure to find the best Q and R. However, setting Q=EOQ and finding R to satisfy n(R) = (1 -β)Q (which requires Table A-4) will generally give good results.

5 -13 Comparison of Service Objectives Although the calculations are far easier for type 1 service, type 2 service is generally the accepted definition of service. Note that type 1 service might be referred to as lead time service, and type 2 service is generally referred to as the fill rate. Refer to the example in section 5 -5 to see the difference between these objectives in practice (on the next slide).

5 -14 Comparison (continued) Order Cycle 1 2 3 4 5 6 7 8 9 10 Demand 180 75 235 140 180 200 150 90 160 40 Stock-Outs 0 0 45 0 0 10 0 0 For a type 1 service objective there are two cycles out of ten in which a stockout occurs, so the type 1 service level is 80%. For type 2 service, there a total of 1, 450 units demand 55 stockouts (which means that 1, 395 demand are satisfied). This translates to a 96% fill rate.

(s, S) Policies The (Q, R) policy is appropriate when inventory levels are reviewed continuously. In the case of periodic review, a slight alteration of this policy is required. Define two levels, s < S, and let u be the starting inventory at the beginning of a period. Then In general, computing the optimal values of s and S is much more difficult than computing Q and R. 5 -15

5 -16 ABC Analysis ABC analysis is based on the Pareto Curve. Pareto discovered that the distribution of wealth follows an increasing exponential curve. A similar curve describes the distribution of the value of inventory items in a multi-item system. (See Figure 5 -7). The value of a Pareto curve analysis in this context is that one can identify the items accounting for most of the dollar volume of sales. Rough guidelines are that the first 20% of the items account for 80% of the sales, the next 30% of the items account for 15% of the sales, and the last 50% of the items only account for 5% of the sales.

Pareto Curve: Distribution of Inventory by Value 5 -17