Deterministic hierarchical substitution inventory models

In this paper, we consider a deterministic nested substitution problem where there are multiple products which can be substituted one for the other, if necessary, at a certain cost. We consider the case when there are n products, and product j can substitute products j + 1,…,n at certain costs. The trade-off is the cost of storing products (for example, customised products) at a higher inventory holding stage versus the cost of transferring downwards from a lower inventory holding cost (generic product) stage. The standard approach to solving the problem yields an intractable formulation, but by reformulating the problem to determine the optimal run-out times, we are able to determine the optimal order and substitution quantities. Numerical examples showing the effect of various system parameters on the optimal order and substitution policy are also presented.     

A Stochastic Petri Net Approach for Inventory Rationing in Multi-Echelon Supply Chains

Manufacturing supply chains are considered as discrete event dynamical systems (DEDS) where coordination of material and information flows is essential to satisfy customer orders and to improve the bottomline of the constituent organizations. A critical problem that is often faced by distribution centres that hold finished good inventory is that of inventory rationing. Inventory rationing is a useful strategy to tackle the problem of conflicting objectives i.e., minimizing inventory costs (holding and backorder) on the one hand and achieving the desired customer service levels (CSLs) on the other. The focus of this paper is to formulate Generalized Stochastic Petri net models to address the inventory rationing problem in the context of multi-echelon make-to-stock distribution chains, where the goods flow through multiple echelons, typically from product manufacturers all the way up-to the retail outlets. The statistical inventory control (SIC) policies modeled by the GSPN are (R, s, S) and a variant that we propose, (R^∗, s, S). We compare the performance of the model under two rationing settings. The first setting considers a case without cooperation, where the individual local stockpoints maximize their own performance. The second setting considers a case with cooperation, where the local stockpoints cooperate with each other to maximize the overall system performance. We provide a methodology to approximately determine the optimal rational fractions with different weights assigned to expected backorder and holding cost components (b/h). We present some interesting results obtained after rigorous numerical experimentation on the model.     

Analysis of optimal opportunistic replenishment policies for inventory systems by using a (s,S) model with a maximum issue quantity restriction

The analysis of optimal inventory replenishment policies for items having lumpy demand patterns is difficult, and has not been studied extensively although these items constitute an appreciable portion of inventory populations in parts and supplies types of stockholdings. This paper studies the control of an inventory item when the demand is lumpy. A continuous review (s,S) policy with a maximum issue quantity restriction and with the possibility of opportunistic replenishment is proposed to avoid the stock of these items being depleted unduly when all the customer orders are satisfied from the available inventory and to reduce ordering cost by coordinating inventory replenishments. The nature of the customer demands is approximated by a compound Poisson distribution. When a customer order arrives, if the order size is greater than the maximum issue quantity w, the order is satisfied by placing a special replenishment order rather than from the available stock directly. In addition, if the current inventory position is equal to or below a critical level A when such an order arrives, an opportunistic replenishment order which combines the special replenishment order and the regular replenishment order will be placed, in order to satisfy the customer's demand and to bring the inventory position to S. In this paper, the properties of the cost function of such an inventory system with respect to the control parameters s, S and A are analysed in detail. An algorithm is developed to determine the global optimal values of the control parameters. Indeed, the incorporation of the maximum issue quantity and opportunistic replenishment into the (s,S) policy reduces the total operating cost of the inventory system.     

The dimension of the Cartesian product of partial orders

If P and Q are partial orders, then the dimension of the cartesian product P × Q does not exceed the sum of the dimensions of P and Q. There are several known sufficient conditions for this bound to be attained. On the other hand, the only known lower bound for the dimension of a cartesian product is the trivial inequality dim (P × Q)⩾ max{dim P, dim Q}. In particular, if P has dimension n, we know only that n⩽dim(P×P)⩽2n. In this paper, we show that for each n⩾3, the crown S_n⁰ is an n-dimensional partial order for which dim(S_n⁰×S_n⁰)=2n−2. No example for which dim(P×Q<dimP+dimQ−2 is known.     

On the value of customer information for an independent supplier in a continuous review inventory system

We consider the inventory control problem of an independent supplier in a continuous review system. The supplier faces demand from a single customer who in turn faces Poisson demand and follows a continuous review (R, Q) policy. If no information about the inventory levels at the customer is available, reviews and ordering are usually carried out by the supplier only at points in time when a customer demand occurs. It is common to apply an installation stock reorder point policy. However, as the demand faced by the supplier is not Markovian, this policy can be improved by allowing placement of orders at any point in time. We develop a time delay policy for the supplier, wherein the supplier waits until time t after occurrence of the customer demand to place his next order. If the next customer demand occurs before this time delay, then the supplier places an order immediately. We develop an algorithm to determine the optimal time delay policy. We then evaluate the value of information about the customer’s inventory level. Our numerical study shows that if the supplier were to use the optimal time delay policy instead of the installation stock policy then the value of the customer’s inventory information is not very significant.     

Optimal integrated production and inventory control of an assemble-to-order system with multiple non-unitary demand classes

We study a pure assemble-to-order system subject to multiple demand classes where customer orders arrive according to a compound Poisson process. The finished product is assembled from m different components that are produced on m distinct production facilities in a make-to-stock fashion. We show that the optimal production policy of each component is a state-dependent base-stock policy and the optimal inventory allocation policy is a multi-level state-dependent rationing policy. Using numerical experimentation, we first study the system behavior as a function of order size variability and order size. We show that the optimal average cost rate is more sensitive to order size variability than to order size. We also compare the optimal policy to the first-come first-serve policy and show that there is great benefit to inventory rationing. We also propose two simple heuristics and show that these can effectively mimic the optimal policy which is generally much more difficult to determine and, especially, to implement.     

Spines and Embeddings of n-Manifolds

Every compact, connected PL manifold Mⁿ, with MⁿØ, collapsesto a codimension-one subpolyhedron Q^n–1, called a spineof Mⁿ. The purpose of this paper is to prove that, if Q^n–1is appropriately chosen, one can reconstruct Mⁿ from Q^n–1,after taking the Cartesian product with an interval I=[0, 1].     

On the mixed Cauchy problem with data on singular conics

We consider a problem of mixed Cauchy type for certain holomorphicpartial differential operators with the principal part Q_2p(D)essentially being the (complex) Laplace operator to a power,^p. We provide inital data on a singular conic divisor givenby P = 0, where P is a homogeneous polynomial of degree 2p.We show that this problem is uniquely solvable if the polynomialP is elliptic, in a certain sense, with respect to the principalpart Q_2p(D).     

On an (<Emphasis Type="Italic">s,Q</Emphasis>) Production Policy for an Integrated Inventory Model for a Single Product with Linearly Increasing Demand

We analyse an (s, Q) production policy for an inventory system consisting of a single finished product and the raw materials used for manufacturing it, and where the demand rate of the product increases linearly with time. We formulate a mathematical programming model with the objective of minimizing total inventory cost per unit time. The problem of grouping raw materials optimally so that common replenishment periods may be used is considered. Solution procedures are developed, and numerical examples are presented.     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.     

Stochastic inventory system with lead time flexibility: offered by a manufacturer/transporter

This paper studies lead time flexibility in a two-stage continuous review supply chain in which the retailer uses the (R, Q) inventory system: when his inventory position reaches R, the retailer places orders with size Q to the manufacturer, who uses a transportation provider to deliver them with different lead time options. According to the contract, the manufacturer is able to expedite or postpone the delivery if the retailer makes such a request. Hence, the retailer has the flexibility to modify the lead time by using the most up-to-date demand information. The optimal lead time policy is found to be a threshold-type policy. The sensitivity analysis also shows that R is much more sensitive to the change of lead time than Q, and thus, the paper is primarily focused on finding optimal R. We also provide a cost approximation which yields unimodal cost in R. Furthermore, we analyze the order crossing problem and derive an upper bound for the probability of order crossing. Finally, we conduct an extensive sensitivity analysis to illustrate the effects of lead time flexibility on supply chain performance and discuss the managerial insights.     

A dynamic inventory model with supplier selection in a serial supply chain structure

Considering the inherent connection between supplier selection and inventory management in supply chain networks, this article presents a multi-period inventory lot-sizing model for a single product in a serial supply chain, where raw materials are purchased from multiple suppliers at the first stage and external demand occurs at the last stage. The demand is known and may change from period to period. The stages of this production–distribution serial structure correspond to inventory locations. The first two stages stand for storage areas for raw materials and finished products in a manufacturing facility, and the remaining stages symbolize distribution centers or warehouses that take the product closer to customers. The problem is modeled as a time-expanded transshipment network, which is defined by the nodes and arcs that can be reached by feasible material flows. A mixed integer nonlinear programming model is developed to determine an optimal inventory policy that coordinates the transfer of materials between consecutive stages of the supply chain from period to period while properly placing purchasing orders to selected suppliers and satisfying customer demand on time. The proposed model minimizes the total variable cost, including purchasing, production, inventory, and transportation costs. The model can be linearized for certain types of cost structures. In addition, two continuous and concave approximations of the transportation cost function are provided to simplify the model and reduce its computational time.     

Modeling and analysis for determining optimal suppliers under stochastic lead times

The policy of simultaneously splitting replenishment orders among several suppliers has received considerable attention in the last few years and continues to attract the attention of researchers. In this paper, we develop a mathematical model which considers multiple-supplier single-item inventory systems. The item acquisition lead times of suppliers are random variables. Backorder is allowed and shortage cost is charged based on not only per unit in shortage but also per time unit. Continuous review (s,Q)$(s\text{,}Q)$ policy has been assumed. When the inventory level depletes to a reorder level, the total order is split among n suppliers. Since the suppliers have different characteristics, the quantity ordered to different suppliers may be different. The problem is to determine the reorder level and quantity ordered to each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost, and shortage cost, is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared with the models in the literature. According to our extensive experiments, the model developed in this paper is the best model in the literature which considers order splitting for n-supplier inventory systems since it is the nearest model to the real inventory system.     

Order splitting in continuous review (Q,r) inventory models

A number of recent articles in the literature have argued the case, when lead time is variable, for splitting a replenishment order for Q between n suppliers by comparing this with the alternative of placing a single order for Q on one supplier. The split order compares favourably on the grounds that the arrival of the first component of a split order cannot be later than the arrival of an order from any one specified supplier. This note argues that an alternative comparison could be made with a policy of ordering Q/n from a single supplier (n times as often). It makes this comparison in the context of a continuous review (Q, r) inventory model but does so not by comparing aggregate costs but by fixing Q and the customer stock service level and comparing the average stock — an approach which is more appropriate to how many companies manage inventory in practice. We consider Poisson and deterministic demand processes, a general lead time distribution and both lost sales and backorder models.     

Supplier selection in make-to-order environment with risks

The problem of allocation of orders for parts among part suppliers in a customer driven supply chain with operational risk is formulated as a stochastic single- or bi-objective mixed integer program. Given a set of customer orders for products, the decision maker needs to decide from which supplier to purchase parts required for each customer order to minimize total cost and to mitigate the impact of delay risk. The selection of suppliers and the allocation of orders is based on price and quality of purchased parts and reliability of on time delivery. To control the risk of delayed supplies, the two popular percentile measures of risk are applied: value-at-risk and conditional value-at-risk. The proposed approach is capable of optimizing the supply portfolio by calculating value-at-risk of cost per part and minimizing mean worst-case cost per part simultaneously. Numerical examples are presented and some computational results are reported.     

Optimal order-limit policies for an (r, Q) inventory system

The reorder-point-reorder-quantity policies referred to as (r,Q) policies are widely used in industry and extensively studiedin the inventory literature. It should be noted, however, thatshortages often occur in practice even when the optimal reorderpoint and reorder quantity are achieved. In this paper, we presenta mathematical model to control a shortage period by an emergencyorder. The problem is the determination of the best timing todeliver items after a shortage occurs. By applying the conceptof repair-limit replacement policy in the context of maintenancetheory, the optimal order time limits based on three kinds ofcost criteria are derived in the framework of a simple (r, Q)inventory system. Finally, some examples of the inventory controlmodel with a stochastic lead time are given to explain the optimalorder-limit policies.     

Stock Rationing in a Continuous Review Two-Echelon Inventory Model

In this paper we consider a 1-warehouse, N-retailer inventory system where demand occurs at all locations. We introduce an inventory model which allows us to set different service levels for retailers and direct customer demand at the warehouse. For each retailer a critical level is defined, such that a retailer replenishment order is delivered from warehouse stock if and only if the stock level exceeds this critical level. It is assumed that retailer replenishment orders, which are not satisfied from warehouse stock, are delivered directly from the outside supplier, instead of being backlogged. We present an analytical upper bound on the total cost of the system, and develop a heuristic method to optimize the policy parameters. Numerical experiments indicate that our technique provides a very close approximation of the exact cost. Also, we show that differentiating among the retailers and direct customer demand can yield significant cost reductions.     

In–Out Algorithm for assortment planning under a ranking-based consumer choice model

We consider a retailer’s assortment planning problem under a ranking-based consumer choice model. The retailer incurs a fixed carrying cost per product offered, a substitution penalty cost for each customer who does not purchase his first choice, and a penalty cost on lost sales. We develop an effective In–Out Algorithm to identify the optimal solution. The extensive numerical study shows that the algorithm performs well, and is more than 10,000 times faster than enumeration on problems with 20 products.