On multi-item inventory

The paper gives a new approach towards a two––item inventory model for deteriorating items with a linear stock––dependent demand rate. In fact, for the first time, the interacting terms showing the mutual increase in the demand of one commodity due to the presence of the other is accommodated in the model. Again, from the linear demand rate, it follows that more is the inventory, more is the demand. So a control parameter is introduced, such that it maintains the continuous supply to the inventory. Next an objective function is formed to calculate the net profit with respect to all possible profits and all possible loss (taken with negative sign). The paper obtains a necessary criterion for the steady state optimal control problem for optimizing the objective function subjected to the constraints given by the ordinary differential equations of the inventory. It also considers a particular choice of parameters satisfying the above necessary conditions. Under this choice, the optimal values of control parameters are calculated; also the optimal amount of inventories is found out. Finally, with respect to these optimal values of control parameters and those of the optimal inventories, the optimal value of the objective function is determined.Next another choice of parameters is considered for which the aforesaid necessary conditions do not hold. Obviously, in that case the steady state solution is non-optimal. In such a case a suboptimal problem is considered corresponding to the more profitable inventory. It is shown that such suboptimal steady state solution fails to exist in this case.     

Financing newsvendor inventory

If the cost of borrowing is not too high, the capital-constrained newsvendor borrows funds to procure an amount that is less than would be ideal. The lender charges an interest rate that decreases in the newsvendor’s equity. Furthermore, we derived a non-linear loan schedule that coordinates the channel.     

Optimal policy for brownian inventory models with general convex inventory cost

We study an inventory system in which products are ordered from outside to meet demands, and the cumulative demand is governed by a Brownian motion. Excessive demand is backlogged. We suppose that the shortage and holding costs associated with the inventory are given by a general convex function. The product ordering from outside incurs a linear ordering cost and a setup fee. There is a constant leadtime when placing an order. The optimal policy is established so as to minimize the discounted cost including the inventory cost and ordering cost.     

A stochastic inventory model

Summary In some cases arising in certain industries or military installations not only the demand for a particular commodity is a stochastic variable but its supply as well. In these cases it is convenient to consider the inventory level resulting from the interaction of supply and demand as a third stochastic variable. The variation of the inventory level in time can then be considered as a stochastic process. If this process is ergodic, the total inventory cost over a certain timeT may be represented as a function of the mean inventory level. This mean level can then be manipulated in such a way as to minimize the total inventory cost.

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Zusammenfassung Es kommt vor, daß in gewissen Industriezweigen sowohl der Verbrauch, als auch die Anlieferung eines bestimmten Gutes stochastische Variable sind. In solchen Fällen ist es zweckmäßig, wenn man die aus der Zusammenwirkung von Verbrauch und Anlieferung resultierende Vorratsmenge als eine dritte stochastische Variable einführt. Man kann dann die Oszillationen der Vorratsmenge in der Zeit als einen stochastischen Prozeß auffassen. Falls dieser Prozeß ergodisch ist, können die gesamten Vorratshaltungskosten für eine bestimmte ZeitT dargestellt werden als eine Funktion der mittleren Vorratsmenge. Diese mittlere Vorratsmenge kann dann so bestimmt werden, daß sie die gesamten Vorratshaltungskosten minimalisiert.

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SHAPE Air Defence Technical Centre. Formerly with Tidewater Oil Company, Los Angeles, California, where the present problem was originally investigated.

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Vorgel. v.:J. Nitsche.     

A comparison of different inventory policies of the one-period multi-location inventory system

In this paper we compare different policies that must meet some demand requirement in a multi-center one-period inventory system. More specifically, this paper concerns a multi-location, one-period inventory problem with linear holding costs and a probability of out-of-stock constraint at each location. Such systems are common in the public sector. Three different policies are considered: (a) A decentralized policy, (b) a centralized policy without priorities, and (c) a centralized priority policy. The superiority of policy (c) over policies (a) and (b) is discussed and demonstrated for some specific examples. It is shown that this superiority is very sensitive to the size of the system.     

Non-stationary inventory position processes

This paper investigates single-product non-stationary inventory problems associated with non-stationary demand processes, backorders allowed, stochastically independent inter-review-period demands, and review periods not necessarily of equal length. Sufficient conditions are obtained for the convergence in distribution of inventory positions in 〈R, r, T〉 inventory systems; in particular, the uniform long-run distribution is found robust in 〈nQ, r, T〉 systems for a large class of demand processes. It is also shown that in 〈R, r, T〉 systems the subsequences of cyclic non-stationary inventory position process associated with cyclic behavior of demand patterns converge in distribution.     

Dynamic inventory targets revisited

This paper is about periodic review inventory systems which are controlled by replenishing the inventory up to a target level which is set based on a demand forecast generated from past demands. It is shown that the replenishments will fluctuate more than final demand. Furthermore the effectiveness of a given safety stock is significantly reduced, indeed even with a very high target the maximum achievable service level can be quite limited. The paper is based on an actual spare parts inventory control system used in a domestic appliance company.     

Approximations of inventory models

Summary A general procedure is presented for constructing approximations of discrete review single product dynamic inventory models. Bounds are derived for the approximations and compared with the ones ofHinderer [1978],Whitt [1978] for the approximation of a general dynamic program. Good order-policies are constructed. Finally, conditions are given under which the sequence of bounds associated with a sequence of approximating models converges to zero.

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Zusammenfassung Es wird ein allgemeines Verfahren zur Approximation periodisch inspizierter dynamischer Ein-Produkt-Lagerhaltungsmodelle betrachtet. Dazu werden Fehlerschranken gegeben. Diese werden verglichen mit denen vonHinderer [1978],Whitt [1978] für ein allgemeines dynamisches Entscheidungsmodell. Außerdem werden gute Bestellpolitiken konstruiert und hinreichende Bedingungen gegeben, unter denen die Fehlerschranken einer Folge von approximierenden Modellen gegen Null konvergiert.

     

Managing decentralized inventory and transshipment

Any decentralized retail or wholesale system of competing entities requires a benefit sharing arrangement when competing entities collaborate after their demands are realized. For instance, consider a distribution system similar to the observed behavior of independent car dealerships. If a dealership does not have in stock the car requested by a customer, it might consider acquiring it from a competing dealer. Such behavior raises questions about procurement strategies that achieve system optimal (first-best) outcomes. In this paper, we examine the existence and uniqueness of pure strategy Nash equilibrium (PSNE) for a decentralized system that adopts a transfer payment approach proposed by Anupindi et al. (Manuf. Serv. Oper. Manag. 4(3):349–368, 2001). In particular, we state a set of conditions on cost parameters and distributions that guarantee uniqueness of PSNE and discuss its consequences. We also examine a situation with incomplete information and expand the scope of the earlier models by relaxing the assumption of satisfying local demand first. That is, we allow the retailers to transship their inventory regardless of the local demand status if such transshipment increases retailer’s profit, and observe that this model extension does not affect our results relative to the more restrictive case. In short, our results provide important insights, clarifications, and strategic limitations regarding collaborations in decentralized distribution system.     

Customer differentiated end-of-life inventory problem

This paper deals with the service parts end-of-life inventory problem in a circumstance that demands for service parts are differentiated. Customer differentiation might be due to criticality of the demand or based on various service contracts. In both cases, we model the problem as a finite horizon stochastic dynamic program and characterize the structure of the optimal inventory policy. We show that when customers are differentiated based on the demand criticality then the optimal structure consists of time and state dependent threshold levels for inventory rationing. In case of differentiation based on service contracts, we show that in addition to rationing thresholds we also need contract extension thresholds by which the system decides whether to offer an extension to an expiring contract or not. By numerical experiments in both cases, we identify the value of incorporating such decisions in service parts end-of-life inventory management with customer differentiation. Moreover, we show that these decisions not only result in cost efficiency but also decrease the risk of part obsolescence drastically.     

Lot sizing with inventory gains

The lot sizing problem with inventory gains generalizes the classical lot sizing problem to one in which stock is not conserved. Instances of this problem can be polynomially transformed into instances of the classical problem. The implications for problems involving different production capacity limitations, backlogging and multilevel production are discussed.     

Robust multi-echelon multi-period inventory control

We consider the problem of minimizing the overall cost of a supply chain, over a possible long horizon, under demand uncertainly which is known only crudely. Under such circumstances, the method of choice is Robust Optimization, in particular the Affinely Adjustable Robust Counterpart (AARC) method which leads to tractable deterministic optimization problems. The latter is due to a recent re-parametrization technique for discrete time linear control systems. In this paper we model, analyze and test an extension of the AARC method known as the Globalized Robust Counterpart (GRC) in order to control inventories in serial supply chains. A simulation study demonstrates the merit of the methods employed here, in particular, it shows that a good control law that minimizes cost achieves simultaneously good control of the bullwhip effect.     

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.     

Decomposition in multi-item inventory control

This paper considers a Lagrangian decomposition approach to a stochastic demand multi-item inventory control problem with a single resource constraint. The work is a generalization of existing decomposition methods.Three decomposition methods are proposed, and bounds on the loss of optimality for each are given in terms of the Lagrange multiplier used. One method allows the calculation of the complete decision rule in advance of the realization of the states, but is expected to perform worse than the other two methods. The second and third method allow the determination of decisions as an optimization problem as the states are realized. Since, in any problem with many states, only a small proportion will actually be realized even in a large time-horizon problem, there may be some advantage in taking this approach.     

Plant location with minimum inventory

We present an integer programming model for plant location with inventory costs. The linear programming relaxation has been solved by Dantzig-Wolfe decomposition. In this case the subproblems reduce to the minimum cut problem. We have used subgradient optimization to accelerate the convergence of the D-W algorithm. We present our experience with problems arising in the design of a distribution network for computer spare parts. In most cases, from a fractional solution we were able to derive integer solutions within 4% of optimality.