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The inventory routing problem (IRP) and the production routing problem (PRP) are two difficult problems arising in the planning of integrated supply chains. These problems are solved in an attempt to jointly optimize production, inventory, distribution, and routing decisions. Although several studies have proposed exact algorithms to solve the single-vehicle problems, the multivehicle aspect is often neglected because of its complexity. We introduce multivehicle PRP and IRP formulations, with and without a vehicle index, to solve the problems under both the maximum level (ML) and order-up-to level (OU) inventory replenishment policies. The vehicle index formulations are further improved using symmetry breaking constraints; the nonvehicle index formulations are strengthened by several cuts. A heuristic based on an adaptive large neighborhood search technique is also developed to determine initial solutions, and branch-and-cut algorithms are proposed to solve the different formulations. The results show that the vehicle index formulations are superior in finding optimal solutions, whereas the nonvehicle index formulations are generally better at providing good lower bounds on larger instances. IRP and PRP instances with up to 35 customers, three periods, and three vehicles can be solved to optimality within two hours for the ML policy. By using parallel computing, the algorithms could solve the instances for the same policy with up to 45 and 50 customers, three periods, and three vehicles for the IRP and PRP, respectively. For the more difficult IRP (PRP) under the OU policy, the algorithms could handle instances with up to 30 customers, three (six) periods, and three vehicles on a single core machine, and up to 45 (35) customers, three (six) periods, and three vehicles on a multicore machine.

Bike Sharing Systems (BSSs) are widely adopted in major cities of the world due to concerns associated with extensive private vehicle usage, namely, increased carbon emissions, traffic congestion and usage of nonrenewable resources. In a BSS, base stations are strategically placed throughout a city and each station is stocked with a pre-determined number of bikes at the beginning of the day. Customers hire the bikes from one station and return them at another station. Due to unpredictable movements of customers hiring bikes, there is either congestion (more than required) or starvation (fewer than required) of bikes at base stations. Existing data has shown that congestion/starvation is a common phenomenon that leads to a large number of unsatisfied customers resulting in a significant loss in customer demand. In order to tackle this problem, we propose an optimisation formulation to reposition bikes using vehicles while also considering the routes for vehicles and future expected demand. Furthermore, we contribute two approaches that rely on decomposability in the problem (bike repositioning and vehicle routing) and aggregation of base stations to reduce the computation time significantly. Finally, we demonstrate the utility of our approach by comparing against two benchmark approaches on two real-world data sets of bike sharing systems. These approaches are evaluated using a simulation where the movements of customers are generated from real-world data sets.

O perational problems arising in the planning of integrated supply chains have been increasingly studied in the past decade. Among these, the production routing problem (PRP) is a difficult problem that aims to jointly optimize production, inventory, distribution, and routing decisions in order to satisfy the dynamic demand of customers and minimize the overall system cost. This paper introduces an optimization-based adaptive large neighborhood search heuristic for the PRP. In this heuristic, binary variables representing setup and routing decisions are handled by an enumeration scheme and upper-level search operators, respectively, and continuous variables associated with production, inventory, and shipment quantities are set by solving a network flow subproblem. Extensive computational experiments have been performed on benchmark instances from the literature. The results show that our algorithm generally outperforms existing heuristics for the PRP and can produce high-quality solutions in short computing times.

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The production routing problem (PRP) is a generalization of the inventory routing problem and concerns the production and distribution of a single product from a production plant to multiple customers using capacitated vehicles in a discreteand finite-time horizon. In this study, we consider the stochastic PRP with demand uncertainty in two-stage and multistage decision processes. The decisions in the first stage include production setups and customer visit schedules, while the production and delivery quantities are determined in the subsequent stages. We introduce formulations for the two problems, which can be solved by a branch-and-cut algorithm. To handle a large number of scenarios, we propose a Benders decomposition approach, which is implemented in a single branch-and-bound tree and enhanced through lower-bound lifting inequalities, scenario group cuts, and Pareto-optimal cuts. For the multistage problem, we also use a warm start procedure that relies on the solution of the simpler two-stage problem. Finally, we exploit the reoptimization capabilities of Benders decomposition in a sample average approximation method for the two-stage problem and in a rollout algorithm for the multistage problem. Computational experiments show that instances of realistic size can be solved to optimality for the two-stage and multistage problems, and that Benders decomposition provides significant speedups compared to a classical branch-and-cut algorithm.

We consider the vehicle routing problem with deadlines under travel time uncertainty in the contexts of stochastic and robust optimization. The problem is defined on a directed graph where a fleet of vehicles is required to visit a given set of nodes and deadlines are imposed at a subset of nodes. In the stochastic vehicle routing problem with deadlines (SVRP-D), the probability distribution of the travel times is assumed to be known and the problem is solved to minimize the sum of probability of deadline violations. In the robust vehicle routing problem with deadlines (RVRP-D), however, the exact probability distribution is unknown but it belongs to a certain family of distributions. The objective of the problem is to optimize a performance measure, called lateness index, which represents the risk of violating the deadlines. Although novel mathematical frameworks have been proposed to solve these problems, the size of problem that those approaches can handle is relatively small. Our focus in this paper is the computational aspects of the two solution schemes. We introduce formulations that can be applied for the problems with multiple capacitated vehicles and discuss the extensions to the cases of incorporating service times and soft time windows. Furthermore, we develop an algorithm based on a branch-and-cut framework to solve the problems. The experiments show that these approaches provide substantial improvements in computational efficiency compared to the approaches in the literature.

Material Requirements Planning (MRP), a core component of enterprise resource planning (ERP) systems, is widely used by manufacturers to determine the production lot sizes of components. These lot sizes are typically computed based on deterministic and dynamic demand assumptions, while safety stocks, which hedge against demand uncertainty, are determined independently based on different assumptions. As the lot sizes and safety stocks are not determined simultaneously, sub‐optimal decisions are often used in practice. The critical impact of inventories and service levels in manufacturing motivates the study of stochastic optimization methods for MRP. In this study, we investigate stochastic optimization methods for MRP systems under demand uncertainty. A two‐stage and a multi‐stage model are proposed to deal with the static‐static and static‐dynamic decision frameworks, respectively. We first derive structural properties of the two‐stage and multi‐stage models to provide insights on the differences between the plans created with these two models. As multi‐stage stochastic programs are not convenient in real‐world applications, several practical enhancements are proposed. First, to address scalability issues, we employ heuristics in combination with advanced sampling methods. Second, to allow real‐time static‐dynamic decisions, we derive a policy from the solution of the multi‐stage model. Third, to deal with the dynamic‐dynamic decision framework, we employ a rolling horizon implementation. The effectiveness and performance of stochastic optimization for MRP are validated by numerical experiments, which demonstrate that the stochastic optimization approaches have the potential to generate significant cost savings compared to traditional methods for production planning and safety stocks determination.