Stochastic Inventory Routing Problem Research Articles

Scenario-based Distributionally Robust Optimization for the Stochastic Inventory Routing Problem

We consider a class of the inventory routing problem in a discrete and finite time horizon, where the demand for homogeneous products at retail stores is uncertain and varies across different scenarios. The supplier is required to determine the times to visit retailers, the replenishment quantities to each retailer, and the routing of a vehicle so as to minimize the sum of stockout, holding, and transportation costs. We propose a scenario-based distributionally robust optimization framework to tackle this problem. We transform the distributionally robust optimization model into a mixed-integer problem, which can be solved efficiently by our proposed algorithm. We adopt a warm-start procedure that utilizes the solution to the nominal model in our methodological framework. Then we apply a Tabu search algorithm, integrated with column generation, to solve a set-partitioning-like integer linear programming model so that a better route set can be identified. By doing so, a large-scale scenario-based distributionally robust optimization model can be solved. We conduct a case study of a fuel company and construct realistic instances to demonstrate the performance of our proposed method. Computational results suggest that the model taking into account various scenarios is more effective when random demands can be classified; the model with a linear decision rule outperforms a non-adaptive model; and the model with the route set identified by an improved algorithm can deliver a better solution than the original route set.

The stochastic inventory routing problem on electric roads

In this work we introduce a green inventory routing problem termed the Stochastic Inventory Routing Problem on Electric Roads (S-IRP-ER), in which a hybrid vehicle navigates a road network with charging opportunities in some road sections, to cover the non-stationary stochastic demand of a single product for a set of retailers in the network. We model the problem using isochrone graphs to represent the real road network. In an isochrone graph nodes are located such that the time to travel any arc is constant all over the network. This allows for tracking the battery level of the vehicle serving retailers, as it charges and discharges continuously while travelling. We formulate a mathematical programming heuristics and prove its effectiveness. We use our model on a realistic instance of the problem, showcasing the different strategies that a vehicle may follow depending on fuel costs in relation to the costs of electricity.

Optimizing food logistics through a stochastic inventory routing problem under energy, waste and workforce concerns

The trend towards sustainable operations management makes it inevitable for companies to carry out their operations by considering environmental and social externalities. This tendency has implications also in the food logistics industry. This study addresses a single-period closed Inventory Routing Problem under environmental and social sustainability concerns in daily food logistics systems. In particular, the study focuses on reducing CO2 emissions in a refrigerated transportation system, collecting and disposing of waste, and offering employees more enticing work schedules, that have not been simultaneously addressed in the literature. The problem has been mathematically defined as a Mixed Integer Linear Programming model. A solution approach based on a clustering algorithm has been proposed to solve large-sized cases. The applicability of the proposed decision support models and the potential practical benefits obtained from their use are shown by performing numerical analyses on an industrial case and a set of larger instances. The results show that simultaneously respecting workforce constraints, waste collection/disposal, and demand uncertainty provide improved economic, environmental and social outputs for food logistics companies. Due to workforce constraints, the delivery time is shortened by 5.5 h, which allow the manufacturing to start later. Respecting waste collection and disposal as well as demand uncertainty enables cost reductions of %40.9 and %6, respectively.

A sampling-based matheuristic for the continuous-time stochastic inventory routing problem with time-windows

This article considers a continuous-time variant of the stochastic inventory routing problem. In most of the articles present in the literature related to inventory routing, customers reveal their demands at the end of each period, which is when the inventory level is calculated. In the variant of the problem at hand, the demand that each customer experiences on each period, results in a continuous decrease of the inventory for the customer during the period. This characteristic strongly affects the quantities that can be delivered to each customer and, if the deliveries are not sufficient or arrive too late, they can cause some stock-out situations within the periods. In the inventory routing problem, the vendor manages all the replenishment decisions of the vendees and, therefore, creates delivery plans for a planning horizon, aiming to reduce the routing, consistency, inventory and lost-sales costs. We formulate the problem as a two-stage mathematical program. Customers experience a continuous stochastic demand within each period and present information about inventory level and capacity at each period, as well as the time windows in which deliveries can take place. To solve this problem, we develop a matheuristic solution approach based on an adaptive large neighborhood search algorithm. In addition, we evaluate the impact of applying recourse actions to deal with expected lost sales during the planning horizon. We compare the performance of the algorithm with adapted variants of the multiple scenario approach and the branch and regret algorithms from the literature. Samples based on the stochastic demands are considered in the algorithms, to find robust solutions that can minimize the objective function. We evaluate the solutions by means of a sample average estimator procedure, and we compare the efficiency of the algorithms as well as the impact of different levels of stochasticity.

Stochastic inventory routing with time-based shipment consolidation

Inspired by the retail industry, we consider a stochastic inventory routing problem where retailers are replenished from a central warehouse using a time-based shipment consolidation policy. Such a time-based dispatching policy, where retailers facing stochastic demand are repetitively replenished at fixed times, is essential in practice. It allows for easy incorporation with dependent up- and downstream planning problems such as personal staffing and warehouse operations, and has become a standard part of transportation contracts. We provide a new chance-constrained model that determines an optimal clustering of retailers in groups, their associated routing and shipment interval, and each retailers’ optimal inventory level. A newly developed branch-price-and-cut algorithm solves our model to optimality. Its efficiency comes from a tailored labeling algorithm for solving the pricing problem that relies, among others, on an optimality pruning criterion based on the approximate solution of a 0,1-knapsack problem. Computational experiments show that our exact method can solve instances of up to 60 retailers to optimality. Besides, we accommodate practitioners by providing a fast heuristic that provides excellent solutions with an optimality gap of less than 1%. Finally, we show that incorporating uncertainty already in the planning process is essential for stochastic inventory routing with time-based shipment consolidation, as it results in overall cost-savings of 7.7% compared to the current state-of-the-art.

A P-Robustness Approach for the Stochastic Inventory Routing Problem

Context: Approaches to logistics solutions through mathematical optimization are widely studied in the literature given their importance for business operations and their computational complexity. In this way, studying the uncertainty associated to operations is a key factor in modeling and decision-making. Method: A stochastic mathematical model is proposed for the Inventory Routing Problem (IRP), considering scenarios with variation in the demands. To obtain a suitable approach, a p-robustness approach and the reformulation of the classical IRP are presented. Results: The performed experiments show the benefits of including uncertainty through a p-robust approach when they are analyzed within an instance of the IRP. Moreover, given the selected modeling, the benefits of combining the approaches can be analyzed. Conclusions: The development of stochastic approaches for decision-making applied to the IRP allow analysts to handle uncertainty and also reduce the complexity of decision when combining different types of problems (Routing + Inventory) in the same model.

Scenario-Based Distributionally Robust Optimization for the Stochastic Inventory Routing Problem

Scenario-Based Distributionally Robust Optimization for the Stochastic Inventory Routing Problem

Solving inventory routing with transshipment and substitution under dynamic and stochastic demands using genetic algorithm and deep reinforcement learning

In this paper, we investigate a two-level supply chain consisting of a company which manufactures a set of products and distributes them via its central warehouse to a set of customers. The problem is modelled as a dynamic and stochastic inventory routing problem (DSIRP) that considers two flexible instruments of transshipment and substitution to mitigate shortages at the customer level. A new resolution approach, based on the hybridisation of mathematical modelling, Genetic Algorithm and Deep Reinforcement Learning is proposed to handle the combinatorial complexity of the problem at hand. Tested on the 150 most commonly used benchmark instances for single-vehicle-product DSIRP, results show that the proposed algorithm outperforms the current best results in the literature for medium and large instances. Moreover, 450 additional instances for multi-products DSIRP are generated. Different demand distributions are examined in these experiments, namely, Normal distribution, Poisson distribution for demand occurrence, combined with demands of constant size; Stuttering Poisson distribution and Negative Binomial distribution. In terms of managerial insights, results show the advantages of promoting inventory sharing and substitutions on the overall supply chain performance.

Stochastic inventory-routing problem with lateral transshipment for perishable product

PurposeThis study aims to design a supply chain network in an uncertain environment while exists two options for distribution of the perishable product and production lot-sizing is concerned.Design/methodology/approachOwing to the complexity of the mathematical model, a solution approach based on a Lagrangian relaxation (LR) heuristic is developed which provides good-quality upper and lower bounds.FindingsThe model output is discussed through various examples. The introduction of some enhancements and using some heuristics results in better outputs in the solution procedure.Practical implicationsThis paper covers the modeling of some real-world problems in which demand is uncertain and managers face making some concurrent decisions related to supply chain management, transportation and logistics and inventory control issues. Furthermore, considering the perishability of product in modeling makes the problem more practically significant as these days there are many supply chains handling dairy and other fresh products.Originality/valueConsidering uncertainty, production, transshipment and perishable product in the inventory-routing problem makes a new variant that has not yet been studied. The proposed novel solution is based on the LR approach that is enhanced by some heuristics and some valid inequalities that make it different from the current version of the LR used by other studies.

Inventory routing under stochastic supply and demand

It is well known that the integrated optimization of multiple and inter-related decisions in a supply chain can bring important benefits to companies. In this spirit, the inventory routing problem focuses on jointly optimizing inventory replenishment and vehicle routing decisions in a distribution context. In practice, the presence of uncertainty often further complicates the problem. We address a stochastic inventory routing problem under the consideration that both the product supply and the customer demands are uncertain. We propose a two-stage stochastic programming formulation, where routing decisions are made in the first stage, while delivery quantities, inventory levels and specific recourse actions are determined in the second stage. In this context, we analyze different recourse mechanisms such as lost sales, backlogging and an additional source for the product in a capacity reservation contract setting. We provide managerial insights from the results of computational experiments using instances based on a benchmark test set. In particular, we study the response mechanisms of the optimal solutions for different levels of uncertainty and cost configurations. Furthermore, we observe that supply and demand uncertainty have different effects on the value of taking the uncertainty into account. We also study the effect of incorporating a service level. Finally, we propose a heuristic solution method which is based on the progressive hedging algorithm and provides high-quality solutions within reasonable running times for problems with a large number of scenarios.

Waste collection inventory routing with non-stationary stochastic demands

We solve a rich routing problem inspired from practice, in which a heterogeneous fixed fleet is used for collecting recyclable waste from large containers over a finite planning horizon. Each container is equipped with a sensor that communicates its level at the start of the day. Given a history of observations, a forecasting model is used to estimate the expected demands and a forecasting error representing the level of uncertainty. The problem falls under the framework of the stochastic inventory routing problem and our main contribution is the modeling of the dynamic probability-based cost of container overflows and route failures over the planning horizon. We cast the problem as a mixed integer non-linear program and, to solve it, we develop an adaptive large neighborhood search algorithm that integrates a purpose-designed forecasting model, tested and validated on real data. We demonstrate the strength of our modeling approach on a set of rich inventory routing instances derived from real data coming from the canton of Geneva, Switzerland. Our approach significantly outperforms alternative deterministic policies in its ability to limit the occurrence of container overflows for the same routing cost. Finally, we show the benefit of a rolling horizon solution and derive lower and upper bounds on its cost.

Distributionally robust inventory routing problem to maximize the service level under limited budget

This paper studies a stochastic inventory routing problem with alternative handling modules and limited capital budget, under partial distributional information (i.e., the mean and covariance matrix of customer demands). The objective is to maximize the service level, i.e., the probability of jointly ensuring no stockout and respecting the warehouse capacities for all customers at the end of each period. A novel distributionally robust chance constrained formulation is proposed. The sample average approximation method and a model-based hierarchical approach based on problem analysis are developed. Computational results show that the latter approach is more efficient. We also draw some managerial insights.

Integrated Analysis of Inventory Management and Transportation Systems for the Single-Period Problem

In a two-stage supply chain system, vendor managed inventory policies (VMI) is an integrating decisions between a supplier and his customers.The supplier assumes the responsibility of maintaining inventory at its customers and ensuring that they will not run out of stock at any moment.The objective of this paper is to optimize the inventory and the transportation costs of the customers for a two-stage supply chain. 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.This paper considers the single-period stochastic inventory routing problem (SP-SIRP) system. The problem formulated as a linear mixed-integer stochastic program for the solutions of the SP-SIRP problem. An illustrative example for the SP-SIRP is presented to explain the proposed model. The finding shows that the vehicle capacity is optimized efficiently with the average capacity utilization for all tours is 100%.

Robustness of inventory replenishment and customer selection policies for the dynamic and stochastic inventory-routing problem

When inventory management, distribution and routing decisions are determined simultaneously, implementing a vendor-managed inventory strategy, a difficult combinatorial optimization problem must be solved to determine which customers to visit, how much to replenish, and how to route the vehicles around them. This is known as the inventory-routing problem. We analyze a distribution system with one depot, one vehicle and many customers under the most commonly used inventory policy, namely the (s,S), for different values of s. In this paper we propose three different customer selection methods: big orders first, lowest storage first, and equal quantity discount. Each of these policies will select a different subset of customers to be replenished in each period. The selected customers must then be visited by a vehicle in order to deliver a commodity to satisfy the customers' demands. The system was analyzed using public benchmark instances of different sizes regarding the number of customers involved. We compare the quality and the robustness of our algorithms and detailed computational experiments show that our methods can significantly improve upon existing solutions from the literature.

A stochastic inventory routing problem for infectious medical waste collection

Abstract–We consider the problem of designing a logistic system to organize adequate collection of infectious medical waste. Waste materials are produced by patients in self‐treatment, stored at pharmacies and picked up by local authorities for disposal. The problem is formulated as a collector‐managed inventory routing problem, encompassing stochastic aspects, which are common in such problems. Social objectives, specifically the satisfaction of pharmacists and local authorities, as well as the minimization of public health risks, are considered for the real‐world‐motivated inventory routing problem. To optimize the planning process for a predefined time horizon, we take advantage of radio frequency identification technology, which allows an improvement of the planning process for pharmacists as well as local authorities. We develop two solution approaches to optimize the determination of visit dates and corresponding vehicle routes. We design a sampling method and an approach based on an adaptive large neighborhood search algorithm to treat the stochastic problem. The suggested approaches are tested on real‐world data from the region of Provence‐Alpes‐Côte d'Azur, in France. To evaluate the performance of the proposed solution methods, the results for different waste collection scenarios are analyzed and compared. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 63(1), 82–95 2014

A stochastic inventory routing problem with stock-out

In this paper, we study an inventory routing problem in which a supplier has to serve a set of retailers. For each retailer, a maximum inventory level is defined and a stochastic demand has to be satisfied over a given time horizon. An order-up-to level policy is applied to each retailer, i.e. the quantity sent to each retailer is such that its inventory level reaches the maximum level whenever the retailer is served. An inventory cost is applied to any positive inventory level, while a penalty cost is charged and the excess demand is not backlogged whenever the inventory level is negative. The problem is to determine a shipping strategy that minimizes the expected total cost, given by the sum of the expected total inventory and penalty cost at the retailers and of the expected routing cost. A hybrid rollout algorithm is proposed for the solution of the problem and its performance is evaluated on a large set of randomly generated problem instances.

Large scale stochastic inventory routing problems with split delivery and service level constraints

A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, which determines delivery volumes to the customers that the depot serves in each period, and vehicle routes to deliver the volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer’s delivery in each period can be split and satisfied by multiple vehicle routes if necessary. This paper considers SIRPSD under the multi-criteria of the total inventory and transportation costs, and the service levels of customers. The total inventory and transportation cost is considered as the objective of the problem to minimize, while the service levels of the warehouses and the customers are satisfied by some imposed constraints and can be adjusted according to practical requests. In order to tackle the SIRPSD with notorious computational complexity, we first propose an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We then develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model found by the approach is used to construct a near optimal solution of the SIRPSD. Randomly generated instances of the problem with up to 200 customers and 5 periods and about 400 thousands decision variables where half of them are integer are examined by numerical experiments. Our approach can obtain high quality near optimal solutions within a reasonable amount of computation time on an ordinary PC.

Scenario Tree-Based Heuristics for Stochastic Inventory-Routing Problems

In vendor-managed inventory replenishment, the vendor decides when to make deliveries to customers, how much to deliver, and how to combine shipments using the available vehicles. This gives rise to the inventory-routing problem in which the goal is to coordinate inventory replenishment and transportation to minimize costs. The problem tackled in this paper is the stochastic inventory-routing problem, where stochastic demands are specified through general discrete distributions. The problem is formulated as a discounted infinite-horizon Markov decision problem. Heuristics based on finite scenario trees are developed. Computational results confirm the efficiency of these heuristics.

A coordinated approach to hedge the risks in stochastic inventory-routing problem

The inventory routing problem (IRP) studied in this research involves repeated delivery of products from a depot to a set of retailers that face stochastic demands over a long period. The main objective in the IRP is to design the set of routes and delivery quantities that minimize transportation cost while controlling inventory costs. Traditional IRP focuses on risk-neutral decision makers, i.e., characterizing replenishment policies that maximize expected total net present value, or equivalently, minimize expected total cost over a planning horizon. In this research, for incorporating risk aversion, a hedge-based stochastic inventory-routing system (HSIRS) integrated with Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and Forward Option Pricing (FOP)model based on Black-Scholes model, from hedge point of view, is proposed to solve the multi-product multi-period inventory routing problem with stochastic demand. Computational results demonstrate the importance of this approach not only to risk-averse decision makers, but also to maximize the net present value at an acceptable service level. As a result, an optimal portfolio ( R, s, S) system of product group can be generated to maximize the net present value under an acceptable service level in a given planning horizon. Meanwhile, the target group needed to be served and the relative transportation policy also can be determined accordingly based on the time required to be served as a priori partition to minimize the average transportation costs; hence, the routing assignment problem can be successfully optimized through a Predicting Particle Swarm Optimization algorithm.

Using scenario trees and progressive hedging for stochastic inventory routing problems

The Stochastic Inventory Routing Problem is a challenging problem, combining inventory management and vehicle routing, as well as including stochastic customer demands. The problem can be described by a discounted, infinite horizon Markov Decision Problem, but it has been showed that this can be effectively approximated by solving a finite scenario tree based problem at each epoch. In this paper the use of the Progressive Hedging Algorithm for solving these scenario tree based problems is examined. The Progressive Hedging Algorithm can be suitable for large-scale problems, by giving an effective decomposition, but is not trivially implemented for non-convex problems. Attempting to improve the solution process, the standard algorithm is extended with locking mechanisms, dynamic multiple penalty parameters, and heuristic intermediate solutions. Extensive computational results are reported, giving further insights into the use of scenario trees as approximations of Markov Decision Problem formulations of the Stochastic Inventory Routing Problem.