A single supplier–single retailer system with an order-up-to level inventory policy

We consider a two-level vendor-managed system in which external demand occurs only at a retailer and a supplier replenishes the retailer employing an order-up-to S policy over T periods. We present an O(T³) algorithm to coordinate the system when S is known. We also show that S can be optimized in O(aT³) time for an input parameter a.     

Ensuring service levels in routing problems with time windows and stochastic travel times

In the stochastic variant of the vehicle routing problem with time windows, known as the SVRPTW, travel times are assumed to be stochastic. In our chance-constrained approach to the problem, restrictions are placed on the probability that individual time window constraints are violated, while the objective remains based on traditional routing costs. In this paper, we propose a way to offer this probability, or service level, for all customers. Our approach carefully considers how to compute the start-service time and arrival time distributions for each customer. These distributions are used to create a feasibility check that can be “plugged” into any algorithm for the VRPTW and thus be used to solve large problems fairly quickly. Our computational experiments show how the solutions change for some well-known data sets across different levels of customer service, two travel time distributions, and several parameter settings.     

Optimal inventory threshold for a dynamic service make‐to‐stock system with strategic customers

We consider a make‐to‐stock production system with one product type, dynamic service policy, and delay‐sensitive customers. To balance the waiting cost of customers and holding cost of products, a dynamic production policy is adopted. If there is no customer waiting in the system, instead of shutting down, the system operates at a low production rate until a certain threshold of inventory is reached. If the inventory is empty and a new customer emerges, the system switches to a high production rate where the switching time is assumed to be exponentially distributed. Potential customers arrive according to the Poisson process. They are strategic in the sense that they make decisions on whether to stay for product or leave without purchase on the basis of on their utility value and the system information on whether the number of products is observable to customers or not. The strategic behavior is explored, and a Stackelberg game between production manager and customers is formulated where the former is the game leader. We find that the optimal inventory threshold minimizing the cost function can be obtained by a search algorithm. Numerical results demonstrate that the expected cost function in an observable case is not greater than that in an unobservable case. If a customer's delay sensitivity is relatively small, these two cases are entirely identical. With increasing of delay sensitivity, the optimal inventory threshold might be positive or zero, and hence, a demarcation line is depicted to determine when a make‐to‐stock policy is advantageous to the manager.     

An integrated inventory model with controllable lead time and distribution‐free demand

The impact of lead time reduction on an integrated periodic review inventory system comprising a single vendor and multiple buyers with a step crashing cost function and service‐level constraints is studied. The probability distribution of demand during the protection period for each buyer is unknown, but the mean and the variance are given. Each production lot of the vendor can be delivered in a number of shipments to all buyers. A minimax distribution‐free procedure with Lagrange multipliers is applied to determining the lead time, the common shipment cycle time, the target levels of replenishments and the number of shipments per production cycle so that the expected total system cost is minimized. Numerical experiments along with sensitivity analysis were performed to illustrate the effects of parameters on the decision and the total system cost. Copyright © 2009 John Wiley & Sons, Ltd.     

Determination of the optimal ordering policy for the retailer with limited capitals when a supplier offers 2 levels of trade credit

In this article, we consider and investigate the cases when the retailer's capitals are restricted and when the supplier offers another kind of 2‐level trade credit. This means that the supplier offers 2‐level trade credit for the retailer to settle the account and the retailer's capitals are restricted, so the retailer decides to pay off the unpaid balance as follows: Firstly, the retailer decides to pay off the unpaid balance at the end of the first credit period if the retailer can pay off all accounts and, in addition, the retailer can use the sales revenue to earn interest throughout the replenishment cycle time. Secondly, the retailer decides to pay off all accounts either after the end of the first credit period, but before the second credit period, or after the second credit period if the retailer cannot pay off the unpaid balance at the end of the first credit period. Additionally, the delay will incur interest charges on the unpaid and overdue balance due to the difference between the interest earned and the interest charged. Consequently, the main purpose of this article is to characterize the optimal solution processes and (in accordance with the functional behavior of the cost function) to search for the optimal replenishment cycle time. Finally, numerical examples are given to illustrate the theoretical results which are proven in this article by means of mathematical solution procedures.     

Lot‐sizing policies for deterioration items under two‐level trade credit with partial trade credit to credit‐risk retailer and limited storage capacity

The main purpose of this article is to investigate the optimal wholesaler's replenishment decisions for deterioration items under two levels of the trade credit policy and two storage facilities in order to reflect the supply chain management situation within the economic order quantity framework. In this study, each of the following assumptions have been made: (1) The own warehouse with limited capacity always is not sufficient to store the order quantity, so that a rented warehouse is needed to store the excess units over the capacity of the own warehouse; (2) The wholesaler always obtains the partial trade credit, which is independent of the order quantity offered by the supplier, but the wholesaler offers the full trade credit to the retailer; (3) The wholesaler must take a loan to pay his or her supplier the partial payment immediately when the order is received and then pay off the loan with the entire revenue. Under these three conditions, the wholesaler can obtain the least costs. Furthermore, this study models the wholesaler's optimal replenishment decisions under the aforementioned conditions in the supply chain management. Two theorems are developed to efficiently determine the optimal replenishment decisions for the wholesaler. Finally, numerical examples are given to illustrate the theorems that are proven in this study, and the sensitivity analysis with respect to the major parameters in this study is performed. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic inventory and pricing policy in a capacitated stochastic inventory system with fixed ordering cost

We study a dynamic inventory and pricing optimization problem in a periodic review inventory system with setup cost and finite ordering capacity in each period. We show that the optimal inventory control is characterized by an (s,s^′,p) policy in four regions of the starting inventory level.     

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.     

Monotonicity,uniqueness, and stability of traveling waves in a nonlocal reaction‐diffusion system with delay

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction‐diffusion system with delay arising from the spread of an epidemic by oral‐faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c ?c _?. Then we show that the traveling wave fronts with speed c >c _? are exponentially asymptotically stable.