Discrete Time Horizon Research Articles

Multi-Product Inventory-Routing Problem in the Supermarket Distribution Industry

Abstract The Inventory Routing Problem (IRP) is an integrated logistic problem arising in several industries (e.g. petrochemical, grocery, soft drink and automotive). A vendor decides the optimal delivery strategy for a set of customers, taking into account their inventory policies and avoiding product stock-out in a finite and discrete time horizon. Delivery strategy includes the time and size of deliveries in order to minimize the total delivery cost. Most commonly studied are IRP real cases where a single homogeneous product with deterministic but time-varying demand is delivered over a finite time horizon. This paper is focused on an efficient methodology for industrial problems where a vendor resupplies a set of customers of heterogeneous products (as in the supermarket distribution industry). In this context, the paper reports on an effort facing the inventory routing problem for multi-category products per customer in conjunction with different inventory policies per category. The paper is motivated by real applications arising in the food engineering field. For instance, industries dealing with food’s distribution to stores located in a given geographic area. The planning strategy is formulated as a linear model. The core of the decision problem consists in determining both the delivery route and the corresponding day of activation along the time horizon. A decomposition of the problem into two phases has been proposed. A suitable penalty cost modeled by simulating the possibility of having an early or delayed product delivery on the delivery day returned from the inventory model (e.g. Economic Order Quantity) is the key feature of the first phase. In the second phase, deliveries are scheduled on a daily basis by taking into account the time windows associated to each customer. This is accomplished by using a constructive heuristic algorithm for the vehicle routing problem with time windows. Computational results on some realistic instances are presented and discussed.

An efficient algorithm based on eigenfunction expansions for some optimal timing problems in finance

This paper considers the optimal switching problem and the optimal multiple stopping problem for one-dimensional Markov processes in a finite horizon discrete time framework. We develop a dynamic programming procedure to solve these problems and provide easy-to-verify conditions to characterize connectedness of switching and exercise regions. When the transition or Feynman–Kac semigroup of the Markov process has discrete spectrum, we develop an efficient algorithm based on eigenfunction expansions that explicitly solves the dynamic programming problem. We also prove that the algorithm converges exponentially in the series truncation level. Our method is applicable to a rich family of Markov processes which are widely used in financial applications, including many diffusions as well as jump–diffusions and pure jump processes that are constructed from diffusion through time change. In particular, many of these processes are often used to model mean-reversion. We illustrate the versatility of our method by considering three applications: valuation of combination shipping carriers, interest-rate chooser flexible caps and commodity swing options. Numerical examples show that our method is highly efficient and has significant computational advantages over standard numerical PDE methods that are typically used to solve such problems.

Superhedging of American options on an incomplete market with discrete time and finite horizon

We establish an existence criterion for the decomposition that generalizes a wellknown uniform Doob decomposition to a set of equivalent probability measures. Based on this criterion, we obtain necessary and sufficient existence conditions for a minimal superhedging (with respect to any measure out of the set of equivalent measures) American option portfolio on an incomplete frictionless market with a finite number of risky assets, discrete time, and finite horizon. We give a sample construction of such a portfolio for an American option with an arbitrary bounded dynamical contingent claim on an incomplete market with one risky asset.

An Efficient Algorithm Based on Eigenfunction Expansions for Some Optimal Timing Problems in Finance

This paper considers the optimal switching problem and the optimal multiple stopping problem for one-dimensional Markov processes in a finite horizon discrete time framework. We develop a dynamic programming procedure to solve these problems and provide easy-to-verify conditions to characterize connectedness of switching and exercise regions. When the transition or Feynman-Kac semigroup of the Markov process has discrete spectrum, we develop an efficient algorithm based on eigenfunction expansions that explicitly solves the dynamic programming problem. We also prove that the algorithm converges exponentially in the series truncation level. Our method is applicable to a rich family of Markov processes which are widely used in financial applications, including many diffusions as well as jump-diffusions and pure jump processes that are constructed from diffusion through time change. In particular, many of these processes are often used to model mean-reversion. We illustrate the versatility of our method by considering three applications: valuation of combination shipping carriers, interest-rate chooser flexible caps and commodity swing options. Numerical examples show that our method is highly efficient and has significant computational advantages over standard numerical PDE methods that are typically used to solve such problems.

Brak arbitrażu na rynkach z proporcjonalnymi kosztami transakcji

The aim of the paper was to present in a clear and unified way various results concerning the absence of arbitrage in the modelling of financial markets with proportional transaction costs. The absence of weak and strict arbitrage opportunities criteria in a finite time horizon discrete time market model are given. Sufficient conditions for the absence of simple arbitrage (i.e. arbitrage over simple investment strategies) in a continuous time market model are presented. Special attention is devoted to transactions without short selling.

Preventive maintenance scheduling of multi-component systems with interval costs

We introduce the preventive maintenance scheduling problem with interval costs (PMSPIC), which is to schedule preventive maintenance (PM) of the components of a system over a finite and discretized time horizon, given a common set-up cost and component costs dependent on the lengths of the maintenance intervals. We present a 0-1 integer linear programming (0-1 ILP) model for the PMSPIC; the model is identical to that presented by Joneja (1990) for the joint replenishment problem within inventory management. We study this model from a polyhedral and exact solutions’ point of view, as opposed to previously studied heuristics (e.g. Boctor, Laporte, & Renaud, 2004; Federgruen & Tzur, 1994; Levi, Roundy, & Shmoys, 2006; Joneja, 1990). We show that most of the integrality constraints can be relaxed and that the linear inequality constraints define facets of the convex hull of the feasible set. We further relate the PMSPIC to the opportunistic replacement problem, for which detailed polyhedral studies were performed by Almgren et al. (2012a). The PMSPIC can be used as a building block to model several types of maintenance planning problems possessing deterioration costs. By a careful modeling of these costs, a polyhedrally sound 0-1 ILP model is used to find optimal solutions to realistic-sized multi-component maintenance planning problems. The PMSPIC is thus easily extended by side-constraints or to multiple tiers, which is demonstrated through three applications; these are chosen to span several levels of unmodeled randomness requiring fundamentally different maintenance policies, which are all handled by variations of our basic model.Our first application considers rail grinding. Rail cracks increase with increasing intervals between grinding occasions, implying that more grinding passes must be performed—thus generating higher costs. We optimize the grinding schedule for a set of track sections presuming a deterministic model for crack growth; hence, no corrective maintenance (CM) will occur between the grinding occasions scheduled. The second application concerns two approaches for scheduling component replacements in aircraft engines. The first approach is bi-objective, simultaneously minimizing the cost for the scheduled PM and the probability of unexpected stops. In the second approach the sum of costs for PM and expected CM—without rescheduling—is minimized. When rescheduling is allowed, the 0-1 ILP model is used as a policy by re-optimizing the schedule at a component failure, which then constitutes an opportunity for PM. The policy manages the trade-off between costs for PM and unplanned CM and is evaluated in a simulation of the engine. The third application considers components’ replacement in wind mills in a wind farm, extending the PMSPIC to comprise multiple tiers with joint set-up costs. Due to the large number of components unexpected stops occur frequently, thus calling for a dynamic rescheduling, which is evaluated through a simulation of the system. In each of the three applications, the use of the 0-1 ILP model is compared with age or constant-interval policies; the maintenance costs are reduced by up to 16% as compared with the respective best simple policy. The results are strongest for the first two applications, possessing low levels of unmodeled randomness.

Formulations for an inventory routing problem

AbstractIn this paper, we present and compare formulations for the inventory routing problem (IRP) where the demand of customers has to be served, over a discrete time horizon, by capacitated vehicles starting and ending their routes at a depot. The objective of the IRP is the minimization of the sum of inventory and transportation costs. The formulations include known and new mathematical programming formulations. Valid inequalities are also presented. The formulations are tested on a large set of benchmark instances. One of the most significant conclusions is that the formulations that use vehicle‐indexed variables are superior to the more compact, aggregate formulations.

A Multiperiod Maximal Covering Location Model for the Optimal Location of Intersection Safety Cameras on an Urban Traffic Network

In this paper we propose a multiperiod optimization model based on the maximal covering location problem in order to support safety policies within urban areas. In particular, we focus on the field of car accidents control, by considering the problem of the optimal location of intersection safety cameras (ISC) on an urban traffic network to maximize road control and reduce the number and the impact of car accidents. The effectiveness of accidents prevention programs can be increased by changing periodically the position of the available ISCs on a given time horizon. To this aim, we propose a novel multiperiod maximal covering location approach designed to maximize the overall coverage on the whole discretized time horizon. The results of the application of this methodology on a real dataset concerning road accidents occurred on a portion of the urban traffic network of the city of Rome are presented and discussed.

Estimating Vehicles in the Dilemma Zone and Application to Fixed-Time Coordinated Signal Optimization

Most traffic accidents are related to driving behaviors. Dangerous driving behaviors may occur if a vehicle is trapped in the dilemma zone. Traditional methods for calculating the number of vehicles in the dilemma zone (NVDZ) are usually based on field data collection and postanalysis and require a great deal of time and human resources. This paper proposes an analytical model for estimating NVDZ on the basis of signal timing, arterial geometry, traffic demand, and driving characteristics. Through application of Robertson's platoon dispersion model, the proposed model calculates the flow rate in the dilemma zone area at yellow onset as well as queue lengths under a discrete time horizon. A VISSIM-based microscopic simulation is then calibrated to validate the NVDZ calculation. The mathematical framework for signal optimization is implemented. Most of the current signal optimization methodologies do not consider safety measures such as NVDZ to be an objective. In this paper, delay and NVDZ are formulated as a multi-objective optimization problem addressing efficiency and safety together. Examples show that delay and NVDZ are competing objectives and cannot be optimized at the same time. Finally, an economic model is applied to quantify both delay and NVDZ in monetary values. The optimal solution considering both factors reduces NVDZ by more than 20% with a less than 2% increase in delay compared with minimizing only delay. The total cost is reduced.

An Algorithm for Infinite Horizon Lot Sizing with Deterministic Demand

We analyze an infinite horizon discrete time inventory model with deterministic but non-stationary demand for a single product at a single stage. There is a finite cycle of vectors of characteristics of the environment (demand, fixed ordering cost, variable procurement cost, holding cost) which is repeated after a finite number of periods. Future cost is discounted. In general, minimization of the sum of discounted total cost over the cycle does not give the minimum of the sum of discounted total cost over the infinite horizon. We construct an algorithm for computing of an optimal strategy over the infinite horizon. It is based on a forward in time dynamic programming recursion.

Equilibria Immune to Deviations by Coalitions in Infinite Horizon Non-Cooperative Games

Infinite horizon discrete time non-cooperative games with observable actions of players and discounting of future single period payoffs are a suitable tool for analyzing emergence and sustainability of cooperation between all players because they do not contain the last period. A subgame perfect equilibrium is a standard solution concept for them. It requires only immunity to unilateral deviations in any subgame. It does not address immunity to deviations by coalitions. In particular, it does not rule out cooperation based on punishments of unilateral deviations that the grand coalition would like to forgive. We first briefly review concepts of renegotiation-proofness that rule out such forgiveness. Then we discuss the concept of strong perfect equilibrium that requires immunity to all deviations by all coalitions in all subgames. In games with only one level of players (e.g. members of the population engaged in the same type of competitive activity), it fails to exist when the Pareto efficient frontier of the set of single period payoff vectors has no sufficiently large flat portion. In such a case, it is not possible to punish unilateral deviations in a weakly Pareto efficient way. In games with two levels of players (e.g. members of two populations with symbiotic relationship, while activities within each population are competitive), it is possible to overcome this problem. The sum of benefits of all players during a punishment can be the same as when nobody is punished but its distribution between the two populations can be altered in favor of the punishers.

The multi-player nonzero-sum Dynkin game in discrete time

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ( $$N \ge 2$$ ) with stopping times as strategies (or pure strategies). The payoff depends on the set of players that stop at the termination stage (where the termination stage is the minimal stage in which at least one player stops). We prove existence of a Nash equilibrium point for the game provided that, for each player $$\pi _i$$ and each nonempty subset $$S$$ of players that does not contain $$\pi _i$$ , the payoff if $$S$$ stops at a given time is at least the payoff if $$S$$ and $$\pi _i$$ stop at that time.

On the convergence of optimal solutions in infinite horizon discrete time models

We analyse the dynamic properties of a one-sector optimal growth model and then extend this analysis to the optimal growth model in which each generation cares about its own parent generation, its own consumption and the utility of its successor.

Mixed Integer Formulations for a Short Sea Fuel Oil Distribution Problem

We consider a short sea fuel oil distribution problem occurring in the archipelago at Cape Verde. Here, an oil company is responsible for the routing and scheduling of ships between the islands such that the demand for various fuel oil products is satisfied during the planning horizon. Inventory management considerations are taken into account at the demand side but not at the supply side. The ports have restricted opening hours each day, so multiple time windows are considered. In contrast to the ships in many other studies within ship routing and scheduling, ships here spend considerable time in the ports compared to at sea. Hence, the time in port is modeled in detail by incorporating both a variable (un)loading time and a setup time for loading different products in the same ports. A mathematical model of the problem is presented and it includes a combined continuous and discrete time horizon because of the multiple time windows and a daily varying consumption rate of the various products in the different ports. We discuss several strategies to improve the proposed model, such as tightening bounds, using extended formulations, and including valid inequalities. The computational study shows that the real problem can be solved to optimality within reasonable time by the use of improved formulations based on a combination of such strategies.

Stochastic Dominance-Constrained Markov Decision Processes

We are interested in risk constraints for infinite horizon discrete time Markov decision processes (MDPs). Starting with average reward MDPs, we show that increasing concave stochastic dominance constraints on the empirical distribution of reward lead to linear constraints on occupation measures. An optimal policy for the resulting class of dominance-constrained MDPs is obtained by solving a linear program. We compute the dual of this linear program to obtain average dynamic programming optimality equations that reflect the dominance constraint. In particular, a new pricing term appears in the optimality equations corresponding to the dominance constraint. We show that many types of stochastic orders can be used in place of the increasing concave stochastic order. We also carry out a parallel development for discounted reward MDPs with stochastic dominance constraints. A portfolio optimization example is used to motivate the paper.

Robust Inventory Routing Under Demand Uncertainty

This paper introduces a robust inventory routing problem where a supplier distributes a single product to multiple customers facing dynamic uncertain demands over a finite discrete time horizon. The probability distribution of the uncertain demand at each customer is not fully specified. The only available information is that these demands are independent and symmetric random variables that can take some value from their support interval. The supplier is responsible for the inventory management of its customers, has sufficient inventory to replenish the customers, and distributes the product using a capacitated vehicle. Backlogging of the demand at customers is allowed. The problem is to determine the delivery quantities as well as the times and routes to the customers, while ensuring feasibility regardless of the realized demands, and minimizing the total cost composed of transportation, inventory holding, and shortage costs. Using a robust optimization approach, we propose two robust mixed integer programming (MIP) formulations for the problem. We also propose a new MIP formulation for the deterministic (nominal) case of the problem. We implement these formulations within a branch-and-cut algorithm and report results on a set of instances adapted from the literature.

항공사 이산형 동적가격 결정 및 좌석통제 문제

Revenue management problems originated in the 1970's in the context of the airline industry have been successfully introduced in airline industries. It has started on the capacity control by booking classes for available seats, and has been recognized as a powerful tool to maximize the total revenue. Changing customer behavior and airline market environments, however, has required a new mechanism for improving the revenue. Dynamic pricing is one of innovative tools which is to adjust prices according to the market status. In this paper, we consider a dynamic pricing and seat control problem for discrete time horizon. The problem can be modeled as a stochastic programming problem. Applying the linear approximation technique and given the price set for each time, we suggest a mixed Integer Programming model to solve our problem efficiently. From the simulation results, we can find our model makes good performance and can be expanded to other comprehensive problems.

A genetic algorithm for dynamic inbound ordering and outbound dispatching problem with delivery time windows

This article considers an inbound ordering and outbound dispatching problem for a single product in a third-party warehouse, where the demands are dynamic over a discrete and finite time horizon, and moreover, each demand has a time window in which it must be satisfied. Replenishing orders are shipped in containers and the freight cost is proportional to the number of containers used. The problem is classified into two cases, i.e. non-split demand case and split demand case, and a mathematical model for each case is presented. An in-depth analysis of the models shows that they are very complicated and difficult to find optimal solutions as the problem size becomes large. Therefore, genetic algorithm (GA) based heuristic approaches are designed to solve the problems in a reasonable time. To validate and evaluate the algorithms, finally, some computational experiments are conducted.

MIP formulations and heuristics for two-level production-transportation problems

A two-level supply chain with multiple items, production sites and client areas and a discrete time horizon is considered. After introducing different mixed integer programming formulations, including an initial formulation that is small but provides weak bounds and a multi-commodity extended formulation that provides much improved bounds but is of large size, we develop a hybrid heuristic that uses the strong formulation to provide a good dual bound and suggest certain variable fixing, and the initial formulation restricted by the variable fixing to then provide the heuristic solution. For different classes of medium-sized instances, we show that the hybrid heuristic provides solutions of a guaranteed quality that are as good or better than those provided by the MIP optimizer with a considerably larger run time.

A study on the state set invariance for a class of discrete-time linear control systems

This paper considers the state set invariance for a class of linear control systems over the discrete time horizon. Given a state set and a control set, the author proposes the methodology to design a state feedback controller and its corresponding maximum invariant subset of the state set. The corresponding invariant subset means that with the controller any state starting from the subset remains in the subset at all times. Fixing the controller, the necessary and sufficient condition for the maximum invariant subset to be the state set itself is first derived. For the systems not satisfying the condition, the author characterizes the corresponding maximum invariant subset based on the spectral analysis of matrices. Numerical examples for different cases are given to illustrate the proposed computation techniques.