Order-based material management system

[0002] The invention pertains to the field of manufacturing methods. More particularly, the invention pertains to methods of ordering and managing part and component inventories for services or manufacturing.

[0003] The problem of determining the amount of components and subassemblies that are to be carried in inventory in a factory to meet customer demand has a long history. The complexity of the problem in practice depends on, among other things, the nature of customer demand, supplier lead times, and product structure involved. In most cases, customer demand and supplier lead times are probabilistic. The product structure may require a constant number of units of a list of components per unit of the finished product (“fixed bill-of-materials”) or depend on the customer order (“variable bill-of-materials”). The latter is the case in produce-to-order (engineer-to-order, assembly-to-order) type systems.

[0004] The complexity of the material management problem in the production of products with fixed and variable bills-of-materials (BOM) is significantly different. When a product consists of a known number of units of a fixed list of components (fixed bill-of-materials), there are well established techniques such as the traditional inventory control techniques, Material Requirements Planning (MRP), and their extensions (Nahmias, 1993) to manage component, subassembly, and other material inventories. When a product is built according to a customer order (such as Lucent's 5ESS switch), the set (list) of components/subassemblies as well as the number of units of each component in the set needed for the product may change from one order to another, making the material management problem extremely complex.

[0005] The traditional inventory control techniques, MRP system, and their extensions were originally developed for products with fixed bill-of-materials. They are primarily used for material ordering and management. MRP, for instance, takes either demand forecasts for a finished product or scheduled production quantities for it from a production plan, and determines the numbers of components and subassemblies to be ordered/produced by multiplying the required production quantities by the fixed numbers of units of components and subassemblies specified in the product's fixed BOM. To determine when they should be ordered/produced, it uses the supplier and manufacturing lead times. Some factories also try to use this approach to manage materials for complex products with variable BOM's by first defining building blocks (large subassemblies) that go in to a customer order. The number of units of different building blocks required by an order may vary from order to order as well as blocks' contents (i.e. components and subassemblies in each). With this approach, customer demand is forecasted at building block level. It is then converted in to component level requirements assuming that each building block has a fixed bill-of-material. Composition of a “typical” is obtained from the variable BOM by simply setting the number of units of a component needed per building block equal to a fixed number in order to cover most orders. Hence, this modified MRP approach would not result in effective management of materials.

[0006] None of the existing material management systems links material ordering and inventory control policies with the desired customer service levels in manufacturing products with variable BOM. For instance, the whole order delivery (WOD) performance (or its variants) is one of the critical customer service metrics tracked by factories operating in a manufacture-to-order environment. It is defined as the percentage of customer orders completed within the time interval promised to customers over a period of time (e.g. a week, a month). It is considered to be an indicator of customer satisfaction in terms of timeliness of order completion. As known, material availability is one of the most critical determinants of WOD performance. Without needed materials, an order can not be started, resulting in delays and late completion of it. Specifically, the percentage of the times an arriving order facing no material shortages (referred to as ‘the order fill rate’ in what follows) has a direct relationship with the WOD performance of such a factory. Instead of determining the amount of components, subassemblies, and other materials needed to be ordered and carried in inventory to reach the desired order level fill rate, all existing material management systems determine these quantities for each component/subassembly separately (without taking the order composition and order to order variations), and hope that, if they are high enough, the desired order fill rate will be reached. Thus, these approaches fail to link the material requirements with the performance metrics in a manufacture-to-order environment. The order based material management system described here addresses this problem and links, for the first time, the customer service objectives such as the order fill rate and delivery interval targets with the material requirements in a produce-to-order environment.

[0007] The material management technique of the invention is for managing materials in factories producing highly customized products. It may be used for planning and ordering components/subassemblies and other types of material, timing of these orders, and determining appropriate inventory levels to reach the desired order fill rate targets at factories producing complex products with variable bills-of-materials.

[0008] The Order Based Materials Management (OBMM) method of the invention uses forecasts of actual customer orders (not building block or component level demand forecasts) to determine component stocking levels. There may be several types of orders (an order type may be a particular model with different options or size that can form a complete customer order) and customer demand forecasts for them. Each order type is specified by a variable BOM called an “order profile”. An order profile is represented by a multivariate normal probability distribution (a joint normal probability distribution of two or more variables) with each random variable representing the uncertain number of units of a component or subassembly that may be needed to build a customer order of that type. Thus, the numbers of units of different components/subassemblies needed to build a certain type of order are uncertain (i.e. a variable BOM). The profile of an order type is a joint multivariate distribution of all components/subassemblies (or so called “Type A” items in material management literature) that may be needed for an order of that type. Thus, it also captures the correlation between the numbers of units of different components that may be needed for orders of a particular type.

[0009] Given order level forecasts and order profiles for different order types, the method of the invention uses a desired “order fill rate” (that is, on average, the percentage of the times all components needed for an order will be available in inventory when the order arrives) to determine the needed component/subassembly buffer levels. As explained before, the order fill rate is one of the important factors affecting the Whole Order Delivery (WOD) performance. The time characteristics of final assembly and test operations, and delay time distribution (expedition) when there are material shortages are among the other factors affecting the WOD performance.

[0010] Using i) the desired order fill rate, ii) order profiles, iii) component lead times, iv) forecasts, and v) forecast error variances, OBMM next determines the target number of orders over the lead times plus one review period for which to maintain materials. If materials for this target number of orders are planned for, and ordered and arrive on time, the average order fill rate over time will be the target order fill rate value.

[0011] To determine component level orders and inventory quantities to achieve the desired order fill rate for each order type, this target number and order profile are then combined to obtain a joint multivariate normal distribution of components that may be needed for the target number of orders. The joint distributions for different order types are combined using the statistical properties of these distributions. This allows the invention to take advantage of the component commonalties between order types. Order-up-to periodic inventory policies, also known as the (S, R) system (Silver and Peterson, 1998) for components and subassemblies are then developed from this combined multivariate distribution.

[0012] In summary, the method of the invention starts with the desired order fill rate. Using the desired order fill rate, order level forecasts, and forecast error variances, it first obtains target numbers for different order types. Combining the target numbers with the associated order profiles, an overall multivariate component demand distribution is obtained. And finally, from that multivariate distribution, it determines order-up-levels for components to achieve the desired order fill rate.

[0013] FIG. 1 shows a block diagram of a conceptual single-factory performance model.

[0014] FIG. 2 shows a flowchart of the method of the invention.

[0015] FIG. 3 shows a detail flowchart of the steps from FIG. 2 enclosed in the dotted lines labeled “3”.

[0016] FIG. 1 shows a block diagram of a single-factory, diagramming the interactions of the various functions with the method of the invention (7) and the factory (1).

[0017] Component and subassemblies are ordered either from outside facilities (5) or produced internally (6), and delivered into the “buffer” (2) in the factory (1), from which they are drawn and assembled in the final assembly and test process (3).

[0018] An order is a customer request for a complete functioning product (system). The factory's customers (8) place orders, which they expect to be delivered within the promised delivery interval.

[0019] An important performance metric monitored in this type of systems is the Whole Order Delivery (WOD) performance, generally expressed as the percentage of orders completed on or before the promised delivery time. The factors which affect the WOD performance are:

[0020] (1) The interval (11) between receipt of order and completion;

[0021] (2) The factory capacity (queueing delays);

[0022] (3) Material availability (i.e. order fill rate); and

[0023] (4) The delay, when there is a material shortage.

[0024] If the promised interval is T days, then the WOD Performance (percentage of orders completed on or before T) can be approximated by: $\begin{array}{c}P\left(\begin{array}{c}\mathrm{completing}\\ \mathrm{an}\mathrm{order}\mathrm{on}\mathrm{or}\\ \mathrm{before}T\mathrm{days}\end{array}\right)=P\left(\begin{array}{c}\mathrm{completing}\\ \mathrm{an}\mathrm{order}\mathrm{on}\mathrm{or}\\ \mathrm{before}T\mathrm{days}\end{array}|\begin{array}{c}\mathrm{Order}\mathrm{Start}\\ \mathrm{with}\mathrm{no}\mathrm{Matl}\\ \mathrm{Shortage}\end{array}\right)\times \\ P\left(\begin{array}{c}\mathrm{Order}\mathrm{Start}\\ \mathrm{with}\mathrm{no}\mathrm{Matl}\\ \mathrm{Shortage}\end{array}\right)+\\ P\left(\begin{array}{c}\mathrm{completing}\\ \mathrm{an}\mathrm{order}\mathrm{on}\mathrm{or}\\ \mathrm{before}T\mathrm{days}\end{array}|\begin{array}{c}\mathrm{Order}\mathrm{Start}\\ \mathrm{with}\mathrm{Material}\\ \mathrm{Shortage}\end{array}\right)\times \\ P\left(\begin{array}{c}\mathrm{Order}\mathrm{Start}\\ \mathrm{with}\mathrm{Material}\\ \mathrm{Shortage}\end{array}\right)\end{array}$

[0025] When there is no material shortage, the probability of completing an order on or before T days is determined by the processing time needed and possible delays when there is congestion due to capacity problems. When there are material shortages, however, orders will be delayed until the needed materials become available. This will naturally delay the completion time of the order. Normally, T is chosen so that, in the absence of any delays, an order can be completed within the promised interval. When the facilities are congested (due to heavy load in peak demand periods, for example), T is adjusted to allow time for queuing delays. The only material related factor determining the WOD performance is the order fill rate, that is P(order start with no material shortage). It can be set independently as in the traditional inventory control techniques or in accordance with the WOD target. The Order Based Material Management method of invention uses the order fill rate to determine component stocking levels in a produce-to-order environment.

[0026] There may be several types of orders (an order type may be a particular model with different options or size), and customer demand forecasts for them.

[0027] The method of the invention will be discussed in the context of several examples of increasing complexity, below. The following description, referring to FIG. 2, is presented at this point to explain the method in its general case, before proceeding to the more specific examples.

[0028] Step 20: SET: Order fill rate β is set.

[0029] Step 21: ESTABLISH: Order profiles are established for each type of order. An order profile is a multivariate probability distribution (a joint probability distribution of two or more variables) with each random variable representing the uncertain number of units of a component or subassembly that may be needed to build a customer order of that type. Thus, the numbers of units of different components/subassemblies needed to build an order of a particular type are uncertain (i.e. a “variable BOM”).

[0030] Step 22: REORDER: The lead times for each component/subassembly in the order profile is set in step (22), and the list of lead times is sorted into inverse order from longest to shortest. Counter “I” is set to 1, in preparation for the iteration of the method.

[0031] L_MAXis set to the first (i.e. longest) lead time after the sorting step, above.

[0032] Step 23: FORECAST: The Order Based Materials Management (OBMM) method of the invention uses forecasts (9) of actual customer orders (not building block or component level demand forecasts) to determine component stocking levels, for the lead time plus review (L₁+R) period.

[0033] Step 24: COMPUTE: using i) the order fill rate, ii) component lead times, iii) forecasts, and iv) forecast error variances and their distributions, OBMM next determines the target number of orders over the lead times (plus one review period) for which to maintain materials, and the available quantities of components with lead times greater than the component being considered.

[0034] Step 25: COMPUTE: The component requirements distribution for the target number of orders of each order type are computed.

[0035] Step 26: COMPUTE: The component requirements distribution is computed for all order types, and the order-up-to quantities are computed for all components with shorter lead times than the one being considered, using the available quantities of all components having longer lead times.

[0036] Step 27: ORDER: The component is ordered in an amount to bring the on-hand plus on-order quantity to the order-up-to level. (steps 25 through 27 are iterated for each component)

[0037] Step 28: SET: The index “i” is incremented.

[0038] Step 29: COMPARE: The index is compared against the number of components “k”. If it is less than or equal to k, the method loops back (30) to step 24 If i is greater than k (31), then the method stops (step 32).

[0039] As in any material management system, this method will be executed periodically (daily, weekly, or any time the organization reviews its materials needs and forecasts). In the following this period will be referred to as “R”.

[0040] To facilitate the presentation of the method of the invention, this discussion will first assume that there is only one order type and all components and subassemblies used have the same lead time. The case involving multiple order types, and components with different lead times will be considered later in this section.

[0041] The Target Number of Orders for the lead time plus review period (TNO_t,(R+L)) at time t can be calculated (25) as follows:

[0042] TNO_t,(R+L)=F_t,(R+L)+z(σ_t,(R+L))^0.5

[0043] where:

[0044] F_t,(R+L)is the forecast made at time t of the number of orders (starts) during the next (R+L) periods, where L is the lead time and R is the review period.

[0045] z is a safety factor for the desired order fill rate (fraction of orders filled directly from buffer (2)) of β. In other words, if the component buffer levels are set to meet possible component demand by up to the target number of orders (TNO_t,(R+L)) 100% of the time, then the average percentage of orders filled from the buffer will be β. The safety factor may be calculated by any of the known methods, such as those recommended by Silver (1970) or Johnson, et. al (1995).

[0046] σ_t,(R+L)is the variance of forecast error over the lead time plus review period. For notational convenience, we denote the standard deviation by σ^0.5.

[0047] Next, the component level orders and inventory quantities (step (26), first and second parts) to achieve the desired order fill rate for each order type are determined. For this, the target number of orders and order profile are combined (step (26), third part) to obtain a joint multivariate distribution of components that may be needed for the target number of orders. Order-up-to periodic inventory policies (including safety stocks) for components and subassemblies are then developed (step (27)) from this combined multivariate distribution, and the orders are placed (step (28)) as needed.

[0048] If the preset number of intervals k has not been exceeded (32), the forecasts are revised (step (24)) in accordance with actual orders and performance figures and the system loops back to repeat from step,(24).

[0049] To explain the details of OBMM, consider orders requiring two distinct components, C₁and C₂, in varying quantities. Let X₁and X₂be the number of C₁and C₂, respectively, needed by an order. For this example, assume, as before, that there is just one order type and both components have the same lead time (the more realistic situation of multiple order types and different lead times will be considered, below). Then, the order profile can be described by a random vector and its distribution: $X=\left[\begin{array}{c}{X}_1\\ X_2\end{array}\right]$

[0050] The Order-to-Order Variation is described by a multivariate normal distribution N(μ,V), called the “Order Profile Distribution”. Let:

[0051] E(X₁)=μ₁V(X₁)=σ₁₁

[0052] E(X₂)=μ₂V(X₂)=σ₂₂

[0053] Cov(X₁,X₂)=σ₁₂=σ₂₁

[0054] Then, the mean vector and covariance matrix of the multivariate random vector X=(X₁,X₂)^T(we use superscript T to represent the transpose of a vector or a matrix) are given by: $\mu =\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right],V=\left[\begin{array}{cc}{\sigma }_{11}& {\sigma }_{12}\\ {\sigma }_{21}& {\sigma }_{22}\end{array}\right]$

[0055] Let the profile of order k be given by $X_k=\left[\begin{array}{c}{X}_1^k\\ X_2^k\end{array}\right].$

[0056] Then, TNO orders have a combined profile of: $\sum _{k=1}^{{\mathrm{TNO}}_{t,\left(R+L\right)}}{X}_k=\left[\begin{array}{c}\sum _{k=1}^{{\mathrm{TNO}}_{t,\left(R+L\right)}}{X}_1^k\\ \sum _{k=1}^{{\mathrm{TNO}}_{t,\left(R+L\right)}}{X}_2^k\end{array}\right]=\left[\begin{array}{c}{D}_1\\ D_2\end{array}\right]$

[0057] Then the component demand by TNO_t,(R+L)orders follows a multivariate normal distribution N(μ_TNO,V_TNO) with (Tong, 1990, Theorem 3.3.3., p. 32): ${\mu }_{\mathrm{TNO}}=\sum _{i=1}^{{\mathrm{TNO}}_{t,\left(R+L\right)}}\mu =\left({\mathrm{TNO}}_{t,\left(R+L\right)}\right)\mu$ $\mathrm{and}$ $V_{\mathrm{TNO}}=\sum _{i=1}^{{\mathrm{TNO}}_{t,\left(R+L\right)}}V=\left({\mathrm{TNO}}_{t,\left(R+L\right)}\right)V$

[0058] Thus, if we determine the component order-up-to levels S₁and S₂for C₁and C₂, respectively, to meet 100% of component level demand from TNO orders, the average order fill rate will be β. That is,

[0059] P(D₁≦S₁, D₂≦S₂)=1.0→Order Fill Rate≧β

[0060] But, because this may result in high material buffer levels, we apply the type 1 service concept (Nahmias, 1993, pp.261-262) to determine S₁and S₂using N(μ_TNO, V_TNO) so that P(D₁≦S₁, D₂≦S₂)=α, where α is the service level (i.e probability of having no shortage when TNO orders are received in a cycle). The component order-up-to levels calculated this way will meet component demand from TNO orders α100% of the time.

[0061] Given P(D₁≦S₁, D₂≦S₂)=α, then, a lower bound on the order fill rate is given by: $\left(\begin{array}{c}\mathrm{Effective}\\ \mathrm{Order}\mathrm{Fill}\\ \mathrm{Rate}\end{array}\right)=\mathrm{\alpha \beta }+\left(1-\alpha \right)\left(\begin{array}{c}\mathrm{Guaranteed}\\ \mathrm{Order}\mathrm{Fill}\mathrm{Rate}\\ \mathrm{with}{S}_1\mathrm{and}{S}_2\end{array}\right),$

[0062] where the guaranteed order fill rate with order-up-to levels S₁and S₂is found by decreasing TNO (and recalculating the mean vector and covariance matrix) until P(D₁≦S₁, D₂≦S₂)≧0.9999 . . .

[0063] Thus, there is a tradeoff between α and β.

[0064] The description given above assumed, for simplicity, that there was only one type of order (or, that all orders had the same profile). In reality, there may be a number of different types of orders, each with its own order profile. When there are multiple order types, a target number will be determined for each type. For each order type, its target number and order profile are combined to obtain ajoint multivariate distribution of components that may be needed for the target number of orders of that type. These joint distributions for different order types are combined using the statistical properties of these distributions. This allows us to take advantage of the component commonalties between order types in determining component order-up-to levels.

[0065] It will be understood that the results presented here are valid for any number of 5 order types and components. For purposes of illustration, consider that there are two order types (with different profiles) requiring three different components C₁, C₂, and C₃. Suppose that type 1 orders require components C₁and C₂, while type 2 orders require components C₂and C₃. In this example, C₂is the only component required by both order types. Then, $\mathrm{X1}=\left[\begin{array}{c}{X}_1^1\\ X_2^1\end{array}\right],\mathrm{X2}=\left[\begin{array}{c}{X}_2^2\\ X_3^2\end{array}\right]$

[0066] where X¹₂and X²₂are the random numbers of component C₂needed per order of type 1 and type 2, respectively. Using the order level forecasts, the target number of orders over the lead time plus review period for order type k, TNO^k_t,(R+L), is calculated by:

[0067] TNO^k_t,(R+L)=F^k_t,(R+L)+z_k(σ^k_t,(R+L))^0.5

[0068] where:

[0069] F^k_t,(R+L)is the forecast made at time t of the number of orders (starts) of type k expected during the next (R+L) periods, where L is the lead time and R is the review period.

[0070] z_kis a safety factor for the desired order fill rate (fraction of orders filled directly from buffer (2)) of β. In other words, if the component buffer levels are set to meet possible component demand by up to the target number of orders (TNO^k_t,(R+L)) 100% of the time, then the average percentage of type k orders filled from the buffer will be β. The safety factor may be calculated by any of the known methods such as those recommended by Silver (1970) or Johnson, et. al (1995).

[0071] σ^k_t,(R+L)is the variance of forecast error for order type k over the lead time plus review period.

[0072] Using a multinomial distribution with a mean vector μ^kand a covariance matrix V^kto represent the profile of a type k order, we obtain a multinormal distribution for the component demand by TNO^k_t,(R+L)orders of type k with a mean vector: ${\mu }_{\mathrm{TNO}}^k=\sum _{i=1}^{{\mathrm{TNO}}_{t,\left(R+L\right)}^k}{\mu }^k=\left({\mathrm{TNO}}_{t,\left(R+L\right)}^k\right){\mu }^k$

[0073] and a covariance matrix $V_{\mathrm{TNO}}^k=\sum _{i=1}^{{\mathrm{TNO}}_{t,\left(R+L\right)}^k}{V}^k=\left({\mathrm{TNO}}_{t,\left(R+L\right)}^k\right)V^k$

[0074] Let the component demand vectors for TNO¹_t,(R+L)orders of type 1, and TNO²_t,(R+L)orders of type 2 be $\left[\begin{array}{c}{D}_1^1\\ D_2^1\end{array}\right],\left[\begin{array}{c}{D}_2^2\\ D_3^2\end{array}\right],$

[0075] respectively. In order to take advantage of component commonality, one needs the distribution of: $\left[\begin{array}{c}{D}_1^1\\ D_2^1+D_2^2\\ D_3^2\end{array}\right]$

[0076] where (D¹₂+D²₂) is the combined demand for component C₂.

[0077] According to Hausman, et. al, (1997, proof of Theorem 1), if X and Y are two independent multinormal variates with means μ_x, μ_yand covariances V_x, V_y, then the augmented variate (X,Y)^Tis a multinomial variate with mean (μ_x, μ_y)^Tand covariance $V=\left[\begin{array}{cc}{V}_x& 0\\ 0& V_y\end{array}\right].$

[0078] Using this procedure, the augmented distribution of (D¹₁, D¹₂, D²₂, D²₃)^Tis obtained as a multinormal distribution with parameters: $\mu =\left[\begin{array}{c}{\mu }_1^1\\ {\mu }_2^1\\ {\mu }_2^2\\ {\mu }_3^2\end{array}\right],V=\left[\begin{array}{cccc}{\sigma }_{11}^1& {\sigma }_{12}^1& 0& 0\\ {\sigma }_{21}^1& {\sigma }_{22}^1& 0& 0\\ 0& 0& {\sigma }_{22}^2& {\sigma }_{23}^2\\ 0& 0& {\sigma }_{32}^2& {\sigma }_{33}^2\end{array}\right]$

[0079] According to Tong, 1990, (Theorem 3.3.3, p.32), if X follows a multinormal distribution with a mean vector μ and a covariance matrix V, then Y=CX+b, where C is any given m×n real matrix with rank n≧m, and b is any m×1 real vector, follows a multinormal distribution with a mean vector (Cμ+b) and covariance matrix CVC^T. By applying this result with b=0, and with $C=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right],$

[0080] parameters of the distribution of (D¹₁, D¹₂, D²₂, D²₃)^Tare obtained as: $\mu =\left[\begin{array}{c}{\mu }_1^1\\ {\mu }_2^1+{\mu }_2^2\\ {\mu }_3^2\end{array}\right],V=\left[\begin{array}{ccc}{\sigma }_{11}^1& {\sigma }_{12}^1& 0\\ {\sigma }_{21}^1& \left({\sigma }_{22}^1+{\sigma }_{22}^2\right)& {\sigma }_{23}^2\\ 0& {\sigma }_{32}^2& {\sigma }_{33}^2\end{array}\right]$

[0081] The Order-up-to levels for components C₁, C₂and C₃are determined using this distribution and P(D₁≦S₁,D₂≦S₂,D₃≦S₃)=α as described before.

[0082] A lower bound on the order fill rate can be calculated using the approach described before by decreasing TNO¹_t,(R+L), and TNO²_t,(R+L)both until P(D₁≦S₁, D₂≦S₂, D₃≦S₃)≧0.9999 . . . .

[0083] In actual practice, the lead time for each component may be different. Consider the previous case with orders requiring three components C₁, C₂, and C₃with unequal lead times L₁, L₂, and L₃, respectively. Without losing generality, we assume L₁>L₂>L₃(equality of lead times are discussed below). If all order-up-to levels are determined for the longest lead time, L₁, this will cause excessive stock levels (both cycle and safety) for C₂and C₃(which have the shorter lead times of L₂and L₃), due to early material arrival and coverage over a longer period than needed. Also, order quantities for C₂and C₃will be determined using inaccurate forecasts (i.e. farther in to the future then L₂, and L₃periods).

[0084] To eliminate these inefficiencies, OBMM recursively recalculates order-up-to points (steps (24)-(32), FIG. 2). This recursion is explained in more detail below, and shown in the flowchart in FIG. 3 (which details the steps in FIG. 2 enclosed in the dotted lines labeled “3”).

[0085] Note that FIG. 3 shows the recursion in a general sense, for n order types and k components, while the discussion below follows the recursion process for the two order type and three component type example above. For the purpose of this discussion, it is assumed that the lead times of the three components are different, and the components have been arranged in decreasing order of lead times, so that L₁>L₂>L₃. It will be understood by one skilled in the art how the specific example below translates into the general procedure of the flowchart of FIG. 3.

[0086] In the example discussed above, the recursive process done as follows: At each review point, update on-order and on-hand quantities of C₁, C₂, and C₃. Determine the order-up-to levels and order quantities as follows:

[0087] (enter from step 23, with i=1)

[0088] Step 40, Iteration 1:

[0089] Consider component C₁.

[0090] Determine TNO¹, and TNO²for (L₁+R) periods

[0091] Step 41, Iteration 1:

[0092] Since there are no components with a longer lead time than L₁, by definition, this step is not executed in this iteration.

[0093] Step 42, Iteration 1:

[0094] Determine the mean vector μ, and covariance matrix V for the combined demand distribution of components C₁, C₂, and C₃for TNO¹, and TNO²orders

[0095] Step 43, Iteration 1:

[0096] Combine the mean vector μ, and covariance matrix V for TNO¹and TNO², for the combined demand distribution for components C₁, C₂, and C₃.

[0097] Step 44, Iteration 1:

[0098] Calculate the order-up-to points S₁, S₂and S₃from P(D₁≦S₁,D₂≦S₂,D₃≦S₃)=α

[0099] Since i<2, we do not use the available quantities or order-up-to levels.

[0100] Step 45, Iteration 1:

[0101] Check if it is possible to reach α.

[0102] If it is, proceed to step 46.

[0103] If it is not,

[0104] Step 50: Reduce TNO¹and TNO²for (L₁+R) periods, and loop back to step 42.

[0105] Step 46, Iteration 1:

[0106] Check if order-up-to point S₁is less than or equal to the maximum consumption capacity. If it is not (i.e. cannot get as many C₁as needed), (step 51) set S₁to the maximum consumption capacity.

[0107] Step 47, Iteration 1:

[0108] Place an order for C₁for Max{0, S₁−(on-hand inventory)−(on-order quantity)}.

[0109] Step 48, Iteration 1:

[0110] S₂and S₃will be used for planning purposes only. Do not use S₂or S₃for ordering, as these will be recalculated.

[0111] (from FIG. 2, perform step 28 (i=2) and step 30−compare i(=2) to k(=3) and loop back)

[0112] Step 40, Iteration 2:

[0113] Consider component C₂.

[0114] Recalculate TNO¹, and TNO²over (R+L₂) periods.

[0115] Step 41, Iteration 2:

[0116] Determine for component C₁, available quantities over (L₂+R):

[0117] H₁=(on hand inventory)+(arrivals expected over the next L₂+R periods)

[0118] Step 42, Iteration 2:

[0119] Determine the mean vector μ, and covariance matrix V for the[combined] demand distribution of components C₁, C₂, and C₃for TNO¹, and TNO2 orders

[0120] Step 43, Iteration 2:

[0121] Combine the mean vector μ, and covariance matrix V for TNO¹and TNO², for the combined demand distribution for components C₁, C₂, and C₃

[0122] Step 44, Iteration 2:

[0123] Calculate the order-up-to points S₂and S₃from P(D₁≦S₁,D₂≦S₂,D₃≦S₃)=α, using the available quantity H₁for order-up-to level S₁

[0124] Step 45, Iteration 2:

[0125] Check if it is possible to reach α.

[0126] If it is, proceed to step 46.

[0127] If it is not,

[0128] Step 50: Reduce TNO¹and TNO²for (L₂+R) periods, and loop back to step 42.

[0129] Step 46, Iteration 2:

[0130] Check if order-up-to point S₂is less than or equal to the maximum consumption capacity. If it is not (i.e. cannot get as many C₂as needed), (step 51) set S₂to the maximum consumption capacity.

[0131] Step 47, Iteration 2:

[0132] Place an order for C₂for Max{0, S₂−(on-hand inventory)−(on-order quantity)}.

[0133] Step 48, Iteration 2:

[0134] S₃will be used for planning purposes only. Do not use S₃for ordering, as it will be recalculated.

[0135] (from FIG. 2, perform step 28 (i=3) and step 29−compare i(=3) to k(=3) and loop back)

[0136] Step 40, Iteration 3:

[0137] Consider Component C₃

[0138] Recalculate TNO¹, and TNO²over (R+L₃) periods.

[0139] Step 41, Iteration 3:

[0140] Determine for components C₁and C₂, available quantities over (L₃+R):

[0141] H₁=(on hand inventory)+(arrivals expected over the next L₃+R periods)

[0142] H₂=(on hand inventory)+(arrivals expected over the next L₃+R periods)

[0143] Step 42, Iteration 3:

[0144] Determine the mean vector μ, and covariance matrix V for the [combined] demand distribution of components C₁, C₂, and C₃for TNO¹, and TNO²orders

[0145] Step 43, Iteration 3:

[0146] Combine the mean vector μ, and covariance matrix V for TNO¹and TNO², for the combined demand distribution for components C₁, C₂, and C₃

[0147] Step 44, Iteration 3:

[0148] Calculate the order-up-to point S₃from P(D₁≦S₁,D₂≦S₂,D₃≦S₃)=α, using the available quantities H₁and H₂for order-up-to levels S₁and S₂

[0149] Step 45, Iteration 3:

[0150] Check if it is possible to reach α.

[0151] If it is, proceed to step 46.

[0152] If it is not,

[0153] Step 50: Reduce TNO¹and TNO²for (L₃+R) periods, and loop back to step 42.

[0154] Step 46, Iteration 3:

[0155] Check if order-up-to point S₃is less than or equal to the maximum consumption capacity. If it is not (i.e. cannot get as many C₃as needed), (step 51) set S₃to the maximum consumption capacity.

[0156] Step 47, Iteration 3:

[0157] Place an order for C₃for Max{0, S₃−(on hand inventory)−(on order quantity)}.

[0158] Step 48, Iteration 3: (no i+1−this step not executed)

[0159] (from FIG. 2, perform step 28 (i=4) and step 29−compare i(=4) to k(=3) and stop)

[0160] In determining the component order-upto levels in this fashion, two situations may require redetermination of α, both of which are incorporated in the discussion above:

[0161] First, as one calculates order-up-to levels for components with shorter lead times, reaching a service level of α with the expected deliveries of the components with longer lead times (quantities denoted by H_iabove) may not be possible (step 44). When this is observed for a component, order-up-to levels for all components have to be recalculated by lowering the service level incrementally until a feasible level is reached (step 50).

[0162] In other situations, there may be a limit on the number of components that can be consumed even when the facility is run at the maximum capacity level. If the order-up-to level for a component exceeds this level (step 46), this means the desired order fill rate is not realistic due to capacity shortage. In this case, the achievable order fill rate is obtained by setting the order-up-to point to the maximum capacity (step 51).

[0163] Accordingly, it is to be understood that the embodiments of the invention herein described are merely illustrative of the application of the principles of the invention. Reference herein to details of the illustrated embodiments are not intended to limit the scope of the claims, which themselves recite those features regarded as essential to the invention.

[0164] Hausman, W. H., H. L. Lee, and A. X. Zhang, “Joint Demand Fulfillment Probability in a Multi-Item Inventory System with Independent Order-Up-To Policies,” European Journal of Operational Research,101-114, 1997.

[0165] Johnson, E. M., H. L. Lee, T. Davis, and R. Hall, “Expressions for Item Fill Rates in Periodic Inventory Systems,” Naval Research Logistics,57-80, 1995.

[0166] Nahmias, S., Production and Operations Analysis,2^nded., Irwin Publ., 1993.

[0167] Silver, E. A., “A Modified Formula for Calculating Customer Service Under Continuous Inventory Review,” AIIE Transactions,241-245, 1970.

[0168] Silver, E. A., D. E. Pyke, and R. Peterson, Inventory Management and Production Scheduling,3^rded., John Wiley and Sons, 1998.

[0169] Tong, Y. L., The Multivariate Normal Distribution,Springer-Verlag, New York, 1990.