Probabilistic Inventory Models 1. CONTINUOUS REVIEW MODELS 1. 1 “Probabilitized” EOQ Model Some practitioners have sought to adapt the deterministic EOQ model to reflect the probabilistic nature of demand by using an approximation that superimposes a constant buffer stock on the inventory level throughout the entire planning horizon. The size of the buffer is determined such that the probability of running out of stock during lead time (the period between placing and receiving an order) does not exceed a prespecified value. LetL = Lead time between placing and receiving an order [pic] = Random variable representing demand during lead time [pic] = Average demand during lead time [pic]= Standard deviation of demand during lead time B = Buffer stock size a = Maximum allowable probability of running out of stock during lead time The main assumption of the model is that the demand,[pic] ,during lead time L is normally distributed with mean [pic] and standard deviation [pic]-that is, N([pic], [pic]) FIGURE 14.

1 Buffer stock imposed on the classical EOQ model pic] Figure 14. 1 depicts the relationship between the buffer stock, B, and the parameters of the deterministic EOQ model that include the lead time L, the average demand during lead time, [pic] , and the EOQ, y*. Note that L must equal the effective lead time. The probability statement used to determine B can be written as P {[pic] [pic] B +[pic]} [pic] [pic] We can convert [pic] into a standard N (O, 1) random variable by using the following substitution [pic] Thus, we have [pic] Figure 14. 2 defines [pic] such that [pic]Hence, the buffer size must satisfy [pic] The demand during the lead time L usually is described by a probability density function per unit time (e. g. , per day or week), from which the distribution of the demand during L can be determined.

Given that the demand per unit time is normal with mean D and standard deviation [pic], the mean and standard deviation, [pic] and [pic], of demand during lead time, L, are computed as [pic] The formula for [pic] requires L to be (rounded to) an integer value. [pic] Example EOQ = 1000 units.If the daily demand is normal with mean D = 100 lights and standard deviation [pic] = 10 lights-that is, N (100, 10)-determine the buffer size so that the probability of running out of stock is below a = . 05.

We know, the effective lead time is L = 2 days. Thus, [pic] Given [pic]= 1. 645, the buffer size is computed as B [pic] 14. 14 * 1.

645 [pic] 23 neon lights Thus, the optimal inventory policy with buffer B calls for ordering 1000 units whenever the inventory level drops to 223 (= B + [pic]= 23 + 2 * 100) units. 1. 2 Probabilistic EOQ ModelThere is no reason to believe that the “probabilitized” EOQ model will produce an optimal inventory policy. The fact that pertinent information regarding the probabilistic nature of demand is initially ignored, only to be “revived” in a totally independent manner at a later stage of the calculations, is sufficient to refute optimality.

To remedy the situation, a more accurate model is presented in which the probabilistic nature of the demand is included directly in the formulation of the model. Unlike the case in Section 1. , the new model allows shortage of demand, as Figure 14. 3 demonstrates. The policy calls for ordering the quantity y whenever the inventory drops to level R. As in the deterministic case, the reorder level R is a function of the lead time between placing and receiving an order. The optimal values of y and R are determined by minimizing the expected cost per unit time that includes the sum of the setup, holding, and shortage costs. [pic] The model has three assumptions.

1. Unfilled demand during lead time is backlogged. 2. No more than one outstanding order is allowed. 3.The distribution of demand during lead time remains stationary (unchanged) with time. To develop the total cost function per unit time, let f(x) = pdf of demand, x, during lead time D = Expected demand per unit time h = Holding cost per inventory unit per unit time p = Shortage cost per inventory unit K = Setup cost per order Based on these definitions, the elements of the cost function are now determined. 1.

Setup cost. The approximate number of orders per unit time is[pic], so that the setup cost per unit time is approximately[pic]. 2.

Expected holding cost. The average inventory is [pic]The formula is based on the average of the beginning and ending expected inventories of a cycle, y + E {R – x} and E {R – x}, respectively. As an approximation the expression ignores the case where R – E {x} may be negative. The expected holding cost per unit time thus equals hI. 3.

Expected shortage cost. Shortage occurs when x > R. Thus, the expected shortage quantity per cycle is [pic] Because p is assumed to be proportional to the shortage quantity only, the expected shortage cost per cycle is pS, and, based on [pic] cycles per unit time, the shortage cost per unit time is[pic].The resulting total cost function per unit time is [pic] The solutions for optimal y* and R* are determined from [pic] We thus get [pic] Because y* and R* cannot be determined in closed forms from (1) and (2), a numeric algorithm, developed by Hadley and Whitin (1963, pp.

169-174), is used to find the solutions. The algorithm converges in a finite number of iterations, provided a feasible solution exists. For R = 0, (1) and (2) above yield [pic] If [pic], unique optimal values of y and R exist. The solution procedure recognizes that the smallest value of y* is[pic], which is achieved when S = 0.The steps of the algorithm are Step 0. Use the initial solution[pic], and let Ro = O. Set i = 1, and go to step i. Step i.

Use [pic] to determine [pic] from Equation (2). If [pic] [pic] [pic] stop; the optimal solution is y* =[pic], and R* =[pic]. Otherwise, use [pic] in Equation (1) to compute[pic]. Set i = i + 1, and repeat step i.

Example Electro uses resin in its manufacturing process at the rate of 1000 gallons per month. It costs Electro $100 to place an order for a new shipment. The holding cost per gallon per month is $2, and the shortage cost per gallon is $10.Historical data show that the demand during lead time is uniform over the range (0, 100) gallons. Determine the optimal ordering policy for Electro. Using the symbols of the model, we have D = 1000 gallons per month K = $100 per order h = $2 per gallon per month p = $10 per gallon f (x) = [pic], 0[pic]x[pic]100 E{x} = 50 gallons First, we need to check whether the problem has a feasible solution. Using the equations for [pic] and [pic] get [pic] Because[pic][pic][pic], a unique solution exists for y* and R*.

The expression for S is computed as [pic]Using S in Equations (1) and (2), we obtain [pic] The last equation yields [pic] We now use Equations (3) and (4) to determine the solution. Iteration 1 [pic] Iteration 2 [pic] Hence, [pic] Iteration 3 [pic] Thus, [pic] Because [pic] and[pic], the optimum is R* [pic] 93. 611 gallons, y* [pic] 319. 44 gallons. File excelContRev. xls can be used to determine the solution to any desired degree of accuracy.

The optimal inventory policy calls for ordering approximately 320 gallons whenever the inventory level drops to 94 gallons.