Retailer’s inventory policies for a one time only manufacturer trade deal of uncertain duration

This paper models a retailer’s response to temporary manufacturer’s trade deals characterized by a time interval of random length, but with the ending date known before its occurrence. Uncertainty is handled through a reordering point, which serves as a trigger mechanism for a new special order and is activated at the discount termination date. The model generates ordering policies, applicable to any probability distribution and is shown to yield well-known deterministic optimal policies as a limiting case.     

The inventory model for trade credit in economic ordering policies of deteriorating items in a supply chain system

In 1994, professors Jaggi and Aggarwal presented the economic ordering policies of deteriorating items in the presence of trade credit using a discounted cash-flows (DCF) approach. This paper discusses the same problem as that of Jaggi and Aggarwal and indicates that some approximations to the optimal cycle times proposed by Jaggi and Aggarwal are inappropriate sometimes. A theorem is derived out to find the optimal cycle time. With that theorem, a simple algorithm is developed to locate the optimal cycle time.     

A sample-path approach to the optimality of echelon order-up-to policies in serial inventory systems

We present a new proof of the optimality of echelon order-up-to policies in serial inventory systems, first proved by Clark and Scarf. Our proof is based on a sample-path analysis as opposed to the original proof based on dynamic programming induction.     

Fair transfer price and inventory holding policies in two-enterprise supply chains

A key issue in supply chain optimisation involving multiple enterprises is the determination of policies that optimise the performance of the supply chain as a whole while ensuring adequate rewards for each participant.In this paper, we present a mathematical programming formulation for fair, optimised profit distribution between echelons in a general multi-enterprise supply chain. The proposed formulation is based on an approach applying the Nash bargaining solution for finding optimal multi-partner profit levels subject to given minimum echelon profit requirements.The overall problem is first formulated as a mixed integer non-linear programming (MINLP) model. A spatial and binary variable branch-and-bound algorithm is then applied to the above problem based on exact and approximate linearisations of the bilinear terms involved in the model, while at each node of the search tree, a mixed integer linear programming (MILP) problem is solved. The solution comprises inter-firm transfer prices, production and inventory levels, flows of products between echelons, and sales profiles.The applicability of the proposed approach is demonstrated by a number of illustrative examples based on industrial processes.     

Simulation-based evolution of resupply and routing policies in rich vendor-managed inventory scenarios

Vendor managed inventory combines inventory management and transportation. Compared to classical inventory management approaches, this strategy offers various degrees of freedom for the vendor while providing a certain service quality level for the customers. To capture the characteristics of rich real-world scenarios, our problem formulation consists of multiple customers, many products and stochastic product usages. Additionally, we also consider mixed formulations, where only a certain part of the customers is switched to a vendor managed inventory to allow a stepwise transition. We show that resupply and routing policies can be evolved autonomously for those scenarios using a simulation-based optimization approach. By combining inventory management and routing, the resulting policies aim to minimize costs and to maximize resource usage while maintaining a given service level. In order to validate our approach, we perform case studies and apply the evolved rules on a large-scale vendor managed inventory scenario for supermarkets. Furthermore, we show that our methodology can be used to perform a sensitivity analysis by considering the influence of exogenous and endogenous factors on the decision process, if a customer base should be transitioned to a vendor managed inventory.     

Optimal retailer's ordering policies in the EOQ model under trade credit financing

The main purpose of this note is to modify the assumption of the trade credit policy in previously published results to reflect the real-life situations. All previously published models implicitly assumed that the supplier would offer the retailer a delay period, but the retailer would not offer the trade credit period to his/her customer. In most business transactions, this assumption is debatable. In this note, we assume that the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer's replenishment model. Furthermore, we assume that the retailer's trade credit period offered by supplier M is not shorter than the customer's trade credit period offered by retailer N(M?N). Under these conditions, we model the retailer's inventory system as a cost minimization problem to determine the retailer's optimal ordering policies. Then a theorem is developed to determine efficiently the optimal ordering policies for the retailer. We deduce some previously published results of other researchers as special cases. Finally, numerical examples are given to illustrate the theorem obtained in this note.     

Integrated inventory models considering the two-level trade credit policy and a price-negotiation scheme

This paper develops the integrated inventory models with permissible delay in payment, in which customers’ demand is sensitive to the buyer’s price. The models consider the two-level trade credit policy in the vendor–buyer and buyer–customer relationships in supply chain management. A simple recursive solution procedure is proposed for the integrated models to determine the buyer’s optimal pricing and production/order strategy. Although the total profit from the buyer and vendor increases together, the buyer’s share lessens. To compensate the buyer’s loss due to the cooperative relationship, a negotiation system is presented in order to allocate the profit increase to the vendor and buyer to determine the pricing and production/order strategy. A numerical example and sensitivity analysis are provided to illustrate the proposed model. The results indicate that the total profit from the buyer and vendor together can increase, although a price discount is given to the buyer in the proposed models.     

Optimal ordering policies in inventory systems with random demand and random deal offerings

In this paper, we determine the optimal order policies for a firm facing random demand and random deal offerings. In a periodic review setting, a firm may first place an order at the regular price. Later in the period, if a price promotion is offered by the supplier (with a certain probability), the firm may decide to place another order. We consider two models in the paper. In the first model, the firm does not share the cost savings (due to the promotion offered by the supplier) with its own customers, i.e. its demand distribution remains fixed. In the second model, the cost savings are shared with the final customers. As a result, the demand distribution shifts to the right. For both the models, in a dynamic finite-horizon problem, the order policy structure is divided into three regions and is as follows. If the initial inventory level for the firm exceeds a certain threshold level, it is optimal not to order anything. If it is in the medium range, it is optimal to wait for the promotion and order only if it is offered. The order quantity when the promotion is offered has an ‘order up to’ policy structure. Finally, if the inventory level is below another threshold, it is optimal to place an order at the regular price, and to place a second order if the promotion is offered. The low initial inventory level makes it risky to just wait for the promotion to be offered. The sum of the order quantities in this case has an ‘order up to’ structure. Finally, we model the supplier's problem as a Stackelberg game and discuss the motivation for the supplier to offer a promotion for the case of uniform demand distribution for the firm. In the first model (when the firm does not share the cost savings with its customers), we show that it is rarely optimal for the supplier to offer a promotion. In the second model, the supplier may offer a promotion depending on the price elasticity of the product.     

On the interaction between retrials and sizing of call centers

This paper models a call center as a Markovian queue with multiple servers, where customer impatience, and retrials are modeled explicitly. The model is analyzed as a continuous time Markov chain. The retrial phenomenon is explored numerically using a real example, to demonstrate the magnitude it can take and to understand its sensitivity to various system parameters. The model is then used to assess the impact of disregarding existing retrials in the staffing of a call center. It is shown that ignoring retrials can lead to under-staffing or over-staffing with respect to the optimal, depending on the forecasting assumptions being made.     

On the interaction between indirect cost allocations and the firm's objectives

The import of cost allocation procedures are through their ex ante important decision making. Hence, it is important that the allocation issue be placed squarely within the context of those firm's objectives which gave rise to the need for the specific allocation. To that effect, this paper focuses the debate on the identification of the indirect cost allocation method that is best suited to the specific reasons for requiring the cost information. First, it is shown that all existing allocation schemes (i) may be expressed in a common equation, flexible enough to be adapted to whatever decision-making purpose the firm desires; and (ii) fulfil the individual rationality conditions of game theory. Then, in light of the controversy as to whether the US Defense Department indirectly subsidizes the commercial side of its suppliers' operations, necessary and sufficient conditions are provided for allocations which do and do not subsidize. Non-subsidized allocations are shown to belong to the core. Subsidized allocations occur when the players (divisions') rational objectives are superseded by higher priority coordinating objectives of non-players (the firm).     

New and efficient algorithms for transfer prices and inventory holding policies in two-enterprise supply chains

We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582–599, 2002). Existing global optimization methods to solve this problem are computationally expensive. We propose a continuous approach based on difference of convex functions (DC) programming and DC Algorithms (DCA) for solving this combinatorial optimization problem. The problem is first reformulated as a DC program via an exact penalty technique. Afterward, DCA, an efficient local approach in non-convex programming framework, is investigated to solve the resulting problem. For globally solving this problem, we investigate a combined DCA-Branch and Bound algorithm. DCA is applied to get lower bounds while upper bounds are computed from a relaxation problem. The numerical results on several test problems show that the proposed algorithms are efficient: DCA provides a good integer solution in a short CPU time although it works on a continuous domain, and the combined DCA-Branch and Bound finds an \(\epsilon \) -optimal solution for large-scale problems in a reasonable time.     

On the demand pooling anomaly in inventory theory

The demand pooling anomaly of inventory theory of type F amounts to a kind of restricted order relation between the individual demands (assumed to be independent) and their average. In this paper, we present some sufficient conditions for the type F anomaly not to occur for two i.i.d. demands; furthermore we provide an asymptotic result showing whether this anomaly occurs for large n for a class of distributions containing all distributions with finite mean.     

The optimal pricing and ordering policy for an integrated inventory model when trade credit linked to order quantity

In traditional inventory models, it is implicitly assumed that the buyer must pay for the purchased items as soon as they have been received. However, in many practical situations, the vendor is willing to provide the buyer with a permissible delay period when the buyer’s order quantity exceeds a given threshold. Therefore, to incorporate the concept of vendor–buyer integration and order-size-dependent trade credit, we present a stylized model to determine the optimal strategy for an integrated vendor–buyer inventory system under the condition of trade credit linked to the order quantity, where the demand rate is considered to be a decreasing function of the retail price. By analyzing the total channel profit function, we developed some useful results to characterize the optimal solution and provide an iterative algorithm to find the retail price, buyer’s order quantity, and the numbers of shipment per production run from the vendor to the buyer. Numerical examples and sensitivity analysis are given to illustrate the theoretical results, and some managerial insights are also obtained.     

Uniform distribution of inventory positions in two-echelon periodic review systems with batch-ordering policies and interdependent demands

We consider a two-level inventory system in which there are one supplier and multiple retailers. The retailers face stochastic, interdependent customer demands. Each location employs a periodic-review (R,nQ), or lot-size reorder point, inventory policy. We show that each location's inventory positions are stationary and the stationary distribution is uniform and independent of any other's.     

Trade-off between inventory and repair capacity in spare part networks

The availability of repairable technical systems depends on the availability of (repairable) spare parts, to be influenced by (1) inventory levels and (2) repair capacity. In this paper, we present a procedure for simultaneous optimisation of these two factors. Our method is based on a modification of the well-known VARI-METRIC procedure for determining near-optimal spare part inventory levels and results for multi-class, multi-server queuing systems representing repair shops. The modification is required to avoid non-convexity problems in the optimisation procedure. To include part-time and overtime working, we allow for a non-integer repair capacity. To this end, we develop a simple approximation for queuing systems with a non-integer number of servers. Our computational experiments show that the near-optimal utilisation rate of the repair servers is usually high (0.80–0.98) and depends mainly on the relative price of the servers compared with inventory items. Further, the size of the repair shop (the minimal number of servers required for a stable system) plays its part. We also show that our optimisation procedure is robust for the choice of the step size for the server capacity.     

On the optimality of inventory models with deteriorating items for demand and on-hand inventory dependent production rate

In this paper we present two flexible production lot size inventorymodels for deteriorating items in which the production rateat any instant depends on the demand and the on-hand inventorylevel at that instant. Each of demand and deterioration rateare general continuous functions of time. For one model, shortagesare allowed but are partially backordered. Also, all cost componentsare affected by inflation and the time value of money. For eachmodel, a closed form of the per unit time total relevant costis derived and sufficient conditions that minimize this totalcost are built. Then, mathematical methods are used to showthat, under certain conditions, each of the underlying inventorysystem can attain a unique global optimal solution.     

Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy

In 2007, Huang proposed the optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy, in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides its customer a permissible delay period N (with N < M). In this paper, we extend his EPQ model to complement the shortcoming of his model. In addition, we relax the dispensable assumptions of N < M and others. We then establish an appropriate EPQ model to the problem, and develop the proper theoretical results to obtain the optimal solution. Finally, a numerical example is used to illustrate the proposed model and its optimal solution.     

On weak interaction between a ground state and a non-trapping potential

We show that ground states of the NLS moving at nonzero speed are asymptotically stable if they either stay far from the potential, or the potential is small, or the ground state has large speed. We search an effective Hamiltonian using the Birkhoff normal forms argument in [11], treating the potential as a perturbation. The so-called Fermi Golden Rule, which is used to describe the decay to 0 of the internal discrete modes of the ground state, is similar to that in [12]. The continuous modes dispersion requires the theory in [40] on charge transfer models.