Computer implemented system for estimating the demand dependent unit stockout cost of a consumer item at a location of a single peroid inventory system, stockout cost application therefor, and method therefor

[0001] The invention relates to estimating the unit stockout cost of a consumer item at a location of a single period inventory system.

[0002] The following alphabetically listed terms together with their acronyms are employed in the description and claims of this application with respect to the present invention:

Availability A_ij, Optimal Availability A_ij*, and Optimal Availability Matrix A*

[0003] Availability A_ij is an industry term referring to the probability of completely satisfying the demand for an i^th consumer item where i=1, 2, . . . , m at a j^th location where j=1, 2, . . . , n of a single period inventory system without an occurrence of a sellout due to insufficient draw at that location. In mathematical terms, A_ij=F(λ_ij,D_ij) where F is the cumulative probability distribution function (cdf) of demand for the i^th consumer item at the j^th location, and λ_ij and D_ij are its mean demand and draw, respectively. The probability distribution function in the discrete case and the probability density function in the continuous case are both denoted by the letter “f”. The optimal availability A_ij* of an i^th consumer item at a j^th location of a single period inventory system is a function of its unit retail price p_ij, its unit production cost c_ij, its unit return cost g_ij, and its unit stockout cost b_ij and is that availability which maximizes the expected total profit realizable for that i^th j^th item-location of the single period inventory system. A unit return cost g_ij is the difference between the expenses incurred upon returning a unit and its salvage value, and consequently it may assume either a positive or negative value. An optimal availability matrix A* is the matrix of optimal availabilities A_ij* for a single period inventory system, and degenerates to a single so-called optimal common availability A* in the case that the respective values of p_ij, c_ij, g_ij and b_ij are the same for all its i^th j^th item-locations.

Demand X_ij, and Mean Demand λ_ij

[0004] The demand process for a perishable consumer item at a location has a random but non-stationary nature, and therefore cannot be subjected to ensemble inferences based on a single realization. Mean demands λ_ij for a consumer item at a location over time are presupposed to be the outcome of a stochastic process which can be simulated by a forecast model whilst the demand X_ij for an i^th consumer item at a j^th location of a single period inventory system at a future point in time is a random variable with a conditional probability distribution conditioned on its mean demand λ_ij at that point in time.

Distribution Policy

[0005] A distribution policy is the delivered quantities of each i^th consumer item where i=1, 2, . . . , m at each j^th location where j=1, 2, . . . , n of a single period inventory system in accordance with a predetermined business strategy. The most profitable distribution policy for a single period inventory system is realized by an optimal draw matrix D*.

Draw D_ij, Draw Matrix D, and Optimal Draw Matrix D*

[0006] Draw D_ij is an industry term referring to the delivered quantity of an i^th consumer item where i=1, 2, . . . , m at a j^th location where j=1, 2, . . . , n of a single period inventory system. A draw matrix D is the matrix of draws D_ij for all i^th j^th item-locations of a single period inventory system. The optimal draw matrix D* for a single period inventory system is the draw matrix maximizing the expected total profit realizable by a distribution policy therefor.

Returns R(λ_ij,D_ij), and Conditional Expected Returns CER(λ_ij,D_ij)

[0007] Returns R(λ_ij,D_ij) is an industry term referring to the number of unsold units of an i^th consumer item at a j^th location of a single period inventory system, and is given by R(λ_ij,D_ij)=max(D_ij−X_ij,0) where D_ij, X_ij, and λ_ij are its draw, demand, and mean demand, respectively, at that location. The conditional expected returns CER(λ_ij,D_ij) of the i^th j^th item-location of a single period inventory system conditioned on an occurrence of the return of at least one unit of the i^th item at that location is given by:
CER(λ_ij,D_ij)=E[R(λ_ij,D_ij)|X_ij<D_ij]=ER(λ_ij,D_ij)/F(λ_ij,D_ij)
where ER(λ_ij,D_ij) is the expected value of R(λ_ij,D_ij), and F is the cumulative probability distribution function (cdf) of the probability distribution of the demand for the i^th j^th item-location. For a Poisson distribution of demand,
ER(λ_ij,D_ij)=D_ijf(λ_ij,D_ij−1)+(D_ij−λ_ij)F(λ_ij,D_ij−2).
Sales S(λ_ij,D_ij)

[0008] Sales S(λ_ij,D_ij) refers to the quantity of sold items of an i^th consumer item at a j^th location of a single period inventory system as upper bounded by the draw D_ij at that location at each point in time, and is given by S(λ_ij,D_ij)=min(D_ij,X_ij)=D_ij−R(λ_ij,D_ij), where D_ij, X_ij, and λ_ij are its draw, demand, and mean demand, respectively, at that location.

Single Period Inventory Systems

[0009] Single period inventory systems are largely concerned with consumer items having a limited shelf life at the end of which an item loses most, if not all, of its consumer value, and the stock of which at a j^th location is not replenished to prevent an occurrence of a sellout. Such consumer items can include perishable goods, for example, fruit, vegetables, flowers, and the like, and fixed lifetime goods, for example, printed media publications, namely, daily newspapers, weeklies, monthlies, and the like. Two common degenerate problems of single period inventory systems are known in the industry as the so-called “newsvendor” problem i.e. the sale of the same item throughout a multi-location single period inventory system and the so-called “knapsack” problem i.e. the sale of different items at the same location.

Stockout ST(λ_ij,D_ij), and Conditional Expected Stockout EST(λ_ij,D_ij)

[0010] Stockout ST(λ_ij,D_ij) is the quantity of unsatisfied demand for an i^th consumer item at a j^th location of a single period inventory system, and is given by ST(λ_ij,D_ij)=max(X_ij−D_ij,0)=X_ij−S(λ_ij,D_ij) where D_ij, X_ij, and λ_ij are its draw, demand, and mean demand, respectively, at that location. The conditional expected stockout CEST(λ_ij,D_ij) of the i^th j^th item-location of a single period inventory system conditioned on an occurrence of a sellout of the i^th item at the j^th location is given by:
CEST(λ_ij,D_ij)=E[ST(λ_ij,D_ij)|X_ij>D_ij]=EST(λ_ij,D_ij)/(1−F(λ_ij,D_ij))
where EST(λ_ij,D_ij)=λ_ij−ES(λ_ij,D_ij) is the expected value of ST(λ_ij,D_ij), and F is the cumulative probability distribution function (cdf) of the probability distribution of the demand for the i^th j^th item-location.

[0011] One computer implemented approach for calculating a demand forecast involves defining a so-called demand forecast tree capable of being graphically represented by a single top level node with at least two branches directly emanating therefrom, each branch having at least one further node. The demand forecast is computed on the basis of historical sales data typically associated with bottom level nodes of a demand forecast tree by a forecast engine capable of determining a mathematical simulation model for a demand process. One such forecast engine employing statistical seasonal causal time series models of count data is commercially available from Demantra Ltd, Israel, under the name Demantra™ Demand Planner.

[0012] Demand forecast applications include determining an optimal draw matrix D* specifying the number of copies of different consumer items to be delivered to each location of a multi-location single period inventory system to maximize the expected total profit (ETP) realizable by a distribution policy therefor in accordance with the expression:

[0013] $\begin{array}{cc}\begin{array}{c}\mathrm{ETP}=\sum _{ij}{\mathrm{EP}}_{ij}\\ =\sum _{ij}[\left(p_{ij}-c_{ij}\right)D_{ij}-\left(p_{ij}-g_{ij}\right)\mathrm{ER}\left({\lambda }_{ij},D_{ij}\right)-\\ b_{ij}\mathrm{EST}\left({\lambda }_{ij},D_{ij}\right)]\end{array}& \mathrm{Eqn}.\left(1\right)\end{array}$
where p_ij is the unit retail price of an i^th consumer item at a j^th location, c_ij is its unit production cost, g_ij is its unit return cost when unsold, and b_ij is its unit stockout cost. Derived from Eqn. (1), the optimal draw matrix D* for the most profitable distribution policy for a single period inventory system is calculated using optimal availabilities A_ij* where:

[0014] $\begin{array}{cc}{A}_{ij}^{*}=F\left({\lambda }_{ij},D_{ij}^{*}\right)=\frac{p_{ij}-c_{ij}+b_{ij}}{p_{ij}-g_{ij}+b_{ij}}& \mathrm{Eqn}.\left(2\right)\end{array}$

[0015] The unit retail price p_ij of an i^th consumer item at a j^th location, its unit production cost c_ij, and its unit return cost g_ij are tangible costs whilst its unit stockout cost b_ij reflects both tangible and intangible pecuniary considerations due to the lost sale of a unit such as loss of potential profit, customer goodwill, and the like. The unit stockout cost b_ij of an i^th consumer item at a j^th location of a single period inventory system is often estimated to be the unit profit p_ij−c_ij, however, this approach often renders an optimal availability A_ij* incompatible with an industry accepted availability therefor denoted A_ij^I, thereby implying that the unit stockout cost b_ij is, in fact, different than the unit profit p_j−c_j.

[0016] This is now exemplified for a single period inventory system for delivering newspapers for which typical unit cost values are as follows: p_j=US$0.50, b_j=c_j=US$0.25, and g_j=US$0.00 for all locations j. On the assumption that unit stockout cost b_j equals unit profit p_j−c_j, this leads to an optimal common availability A*=66.7% using Eqn. (2). Against this, the industry accepted availability A_ij^I for a single period inventory system for delivering newspaper is 80%±5%. The discrepancy between these availabilities implies that for the same j^th location with mean demand λ=20, the draw D_j thereat would be 21 in accordance with the optimal common availability A*=66.7% and as high as 24 corresponding to A^I=84.3%.

[0017] Broadly speaking, the present invention provides a novel computer implemented system for realistically estimating the demand dependent unit stockout cost b_ij=b(λ_ij) (hereinafter shortened to “unit stockout cost b_ij”) of an i^th consumer item at a j^th location of a single period inventory system as a function of its associated mean demand λ_ij as opposed to the hitherto accepted approach of regarding the unit stockout cost of an item largely as a tangible cost with a fixed value, stockout cost application therefor, and method therefor. The present invention is based on the notion that a realistic estimate of the unit stockout cost b_ij of the i^th consumer item at the j^th location of a single period inventory system is that which substantially balances the cost of its conditional expected stockout at the j^th location with the cost of its conditional expected returns therefrom. It is believed that the realistic estimation of the unit stockout cost b_ij of an i^th consumer item at a j^th location of a single period inventory system in accordance with the present invention will facilitate the determination of a more accurate and profitable distribution policy for a single period inventory system irrespective of whether it is or isn't subject to a constraint.

[0018] The approach for realistically estimating the unit stockout cost b_ij of an i^th consumer item at a j^th location of a single period inventory system in accordance with the present invention can be readily degenerated to estimate the unit stockout cost b_i where i=1, 2, . . . , m in the case of the so-called “knapsack” problem, namely, the sale of different items at the same location, and unit stockout cost b_j where j=1, 2, . . . , n in the case of the so-called “newsvendor” problem, namely, the sale of the same item at different locations.

[0025] FIG. 1 shows an exemplary demand forecast tree 1 having a single top level node (00) with five branches A, B, C, D and E for correspondingly representing the sale of Item I (top level-1 node (10)) at Locations 1 and 2 (bottom level nodes (11) and (21)), Item II (top level-1 node (20)) at Locations 1 and 3 (bottom level nodes (21) and (23)), Item III (top level-1 node (30)) at Location 1, 2 and 3 (bottom level nodes (31), (32) and (33)), Item IV (top level-1 node (40)) also at Locations 1, 2 and 3 (bottom level nodes (41), (42) and (43)); and Item V (top level-1 node (50)) at Location 1 (bottom level node (51)) only. FIG. 2 shows an exemplary table 2 containing historical sales data for Item I at the bottom level nodes (11) and (12). Similar tables exist for the sale of the other items at their respective locations.

[0026] FIG. 3 shows a computer implemented system 3 with a processor 4, memory 6, a user interface 7 including suitable input devices, for example, a keypad, a mouse, and the like, and output means, for example, a screen, a printer, and the like, with possibly other computer components for enabling operation of the system including result analysis. The computer implemented system 3 also includes a database 8 for storing historical time series of sales information of items at locations, a forecast engine 9 for forecasting the mean demand λ_ij for each i^th consumer item at each j^th location of a single period inventory system on the basis of the historical sales data, a unit cost file 11 for storing the unit retail price p_ij, the unit production cost c_ij, and the unit return cost g_ij, of each i^th consumer item at each j^th location of the single period inventory system; and a stockout cost application 12 for estimating the demand dependent unit stockout cost b_ij of an i^th consumer item at a j^th location of a single period inventory system as a function of its unit retail price p_ij, its unit production cost c_ij, its unit return cost g_ij and its mean demand λ_ij. The forecast engine 9 may be implemented and managed as illustrated and described in commonly assigned co-pending U.S. patent application Ser. No. 10/058,830 entitled “Computer Implemented Method and System for Demand Forecast Applications”, the contents are which are incorporated herein by reference. Whilst the present invention is being described in the context of a fully functional computer implemented system, it is capable of being distributed as a program product in a variety of forms, and the present invention applies equally regardless of the particular type of signal bearing media used to carry out distribution. Examples of such media include recordable type media e.g. CD ROM and transmission type media e.g. digital communication links.

[0027] With reference to FIGS. 4 and 5, the present invention will now be exemplified with respect to the single period inventory system at a given location, described hereinabove in the Background of the Invention with the unit cost values p=US$0.50, c=US$0.25, and g=US$0.00, and the assumed Poisson distribution of demand with mean λ_j=20. The stockout cost application 12 calculates the values of the conditional expected stockout CEST(λ,D) and the conditional expected returns CER(λ,D) for an initial value of D which is typically the mean demand λ. In the case of a Poisson distribution, ER(λ,D)=Df(λ,D−1)+(D−λ)F(λ,D−2). The stockout cost application 12 calculates the value of A, and then using the Eqn. (2), calculates the value of the unit stockout cost b_j before calculating the absolute difference Δ between the conditional expected stockout cost bCEST(λ,D), and the conditional expected returns cost 0.25CER(λ,D). The stockout cost application 12 repeats these calculations for increasing values of D in increments of 1 up to 25 as shown in the following table with their respective conditional expected stockout costs and conditional expected returns costs being plotted against availability in FIG. 5:

[0028]

						Δ=|bCEST (λ,D)-
	D	CEST (λ,D)	CER (λ,D)	A	b	0.25 CER (λ,D)|
1	20	4.0	3.778	55.91	0.067	0.7
2	21	3.7	4.178	64.37	0.2017	0.3
3	22	3.5	4.628	72.06	0.3948	0.2
4	23	3.3	5.1348	78.75	0.6764	0.9
5	24	3.1	5.6984	84.32	1.0947	2.0
6	25	3.0	6.322	88.78	1.7285	3.5

[0029] The stockout cost application 12 determines that the absolute difference Δ between the two opposing costs bCEST(λ,D) and (c−g)CER(λ,D) reaches its minimal value 0.2 at D=22 at an optimal availability A*=72.06% which is slightly greater than the availability which would equalize the two opposing costs at their cross over point on FIG. 5. This availability lies approximately midway between the hitherto calculated optimal availability A*=66.7% based on the assumption that stockout cost equals unit profit, and the industry accepted availability A^I=80%±5%, and accordingly, yields the unit stockout cost US$0.3948 which is over 50% greater than the hitherto estimated unit profit US$0.50−US$0.25=US$0.25.

[0030] While the invention has been described with respect to a limited number of embodiments, it will be appreciated that many variations, modifications, and other applications of the invention can be made within the scope of the appended claims.