Production Routing Problem Research Articles

A MILP-based approach to address the production and distribution planning of large industrial gas supply chains

This paper focuses on addressing the Production Routing Problem (PRP) of industrial gas supply chains (SC). The work introduces a Mixed-Integer Linear Programming (MILP) based approach to effectively tackle this challenging problem. The proposed framework aims to increase company profits by optimizing both production and distribution activities simultaneously. For the distribution part, a route generation algorithm is employed to initially generate a set of feasible routes, ensuring that only routes with practical significance are considered. Subsequently, the mathematical model is developed considering only the set of routes generated. The formulation includes decisions associated with plants production levels, plant selection to supply the different consumers, clients to be visited per trip and day, quantity to be delivered and routes to be used. As part of the problem, it is considered that trucks can make several trips per day, as well as trips of several days. To take into account the last feature, the approach accounts for the explicit effect of the lead time on production and distribution decisions. In this way, the work allows for different delivery times for customers included in the same route, according to their inventories and usage levels. The model considers fleet unavailability until they return to the plants after finishing a trip. Finally, two case studies, one motivating and the other industrial size, are presented and solved. The results obtained demonstrate the effectiveness of the proposed approach to solve the case studies in reasonable CPU times.

A novel robust decomposition algorithm for a profit-oriented production routing problem with backordering, uncertain prices, and service level constraints

AbstractThe Production Routing Problem (PRP) seeks optimal production and distribution planning that minimises costs and fulfils customer orders. Yet, existing literature often overlooks the potential impact on profitability. Achieving optimal profit does not necessarily imply meeting all customer orders. The cost-to-profit ratio should be considered when serving customer orders, as there are circumstances where it might be more profitable to cancel or backorder certain orders. Thus, this paper proposes, for the first time, a novel extension of PRP that maximises profit where demand is price-sensitive and allows order cancellation and backorders under service level targets. From on-field observations, price is inherently subject to uncertainty; thus, we propose a robust mathematical model for the problem that optimises the worst-case profit. To solve the problem, the paper proposes a decomposition algorithm that splits the problem into a master problem and a set of subproblems, enhanced by valid inequalities and warming up lower bounds to alleviate the model complexity. Through a series of computational tests, we prove the ability of the proposed algorithm to tighten the optimality gaps and alleviate computational time. An additional economic study is conducted to investigate how parameter variation affects profit and how sensitive it is to service level targets.

Production routing problem in shared manufacturing: Robust chance-constrained formulation and multi-diversification based matheuristic

Building on the concept of the sharing economy, which capitalizes on the advancements in information and manufacturing technologies, the shared manufacturing paradigm has tremendous potential for enhancing the performance of production–distribution systems and has become prevalent in the today’s manufacturing industry. Despite the extensive research on the production routing problem, which integrates decisions of production, inventory, and distribution routing, there is a notable absence of investigation within the context of shared manufacturing. This paper aims to bridge the above-mentioned research gap and develop valuable decision-support tools for managers, assisting efficient management of production–distribution systems in shared manufacturing environments. To achieve this, this paper introduces a general model for the production routing problem with shared manufacturing resources. The problem is formulated as a robust chance-constrained model to address uncertainties arising in customer demand and the demand and supply of shared manufacturing resources. Subsequently, a computationally tractable formulation is derived, for which a matheuristic that incorporates three diversification techniques is proposed to solve. Extensive computational experiments are conducted to validate the feasibility and effectiveness of both the model and the matheuristic, as well as to evaluate the benefits of manufacturing resource sharing. The experimental results demonstrate that our matheuristic outperforms state-of-the-art solution approaches. Moreover, manufacturing resource sharing can lead to significant reductions in operational costs and improvements in service levels. Finally, the advantages are found to increase as manufacturing resource unit purchasing costs decrease, demand fluctuations decrease, and production interruption severity increases.

A progressive hedging-based matheuristic for the stochastic production routing problem with adaptive routing

The production routing problem (PRP) arises in the context where a manufacturing facility manages its production schedule, the delivery of goods to customers by a fleet of vehicles, and the inventory levels both at the plant and at the customers. The presence of uncertainty often complicates the planning process. In particular, production and distribution costs may significantly increase if demand uncertainty is ignored in the planning phase. Nevertheless, only a few studies have considered demand uncertainty in the PRP. In this article, we propose a two-stage stochastic programming approach for a one-to-many PRP with a single product and demand uncertainty. Unlike previous studies in the literature, we consider the case where routing decisions are made in the second stage after customer demands become known. This offers more flexibility, which can decrease transportation costs by preventing unnecessary customer visits. In addition to the static–dynamic case, in which setup decisions are made first and production quantities are decided in the second stage, we also consider the static–static setting where both sets of decisions must be made prior to the demand realization. A progressive hedging algorithm combined with a matheuristic is developed to solve the problem. The role of the progressive hedging algorithm is to decompose the stochastic problem into more tractable scenario-specific subproblems and lead the first-stage variables towards convergence by modifying their Lagrangean multipliers. However, solving the subproblems remains challenging since they include routing decisions, and we thus propose a matheuristic to exploit the structural characteristics of the subproblems. First, a Traveling Salesman Problem (TSP) is solved to find the optimal tour for all customers regardless of demand and capacity. Utilizing the sequence obtained from the TSP, we then solve a restricted PRP while taking into account the other constraints of the original problem. Finally, for each period and scenario, a vehicle routing problem is solved to improve the quality of the solutions. Computational experiments are conducted to analyze the algorithm’s efficiency and to assess the benefits that can be achieved by handling routing in the second stage.

Physical internet enabled sustainable and resilient production-routing problem with modular capacity

PurposeModular capacity units enable rapid reconfiguration, providing tactical flexibility to efficiently meet customer demand during disruptions and ensuring sustainability. Moreover, the Physical Internet (PI) enhances the potential of modular capacity in addressing efficiency, sustainability, and resilience challenges. To evaluate the sustainability and resilience advantages of the PI-enabled reconfigurable modular system (PI-M system), this paper studies a PI-enabled sustainable and resilient production-routing problem with modular capacity.Design/methodology/approachWe develop a multi-objective optimization model to assess the sustainability and resilience benefits of combining PI and modular capacity in a chemical industry case study. A hybrid solution approach, combining the augmented e-constraint method, construction heuristic, and hybrid adaptive large neighborhood search, is developed.FindingsThe experimental results reveal that the proposed solution approach is capable of obtaining better solutions than the Gurobi and the existing heuristic in a shorter running time. Moreover, compared with the traditional system, the PI only and traditional with modular capacity systems, PI-M system has significant advantages in both sustainability and resilience.Originality/valueTo the best of our knowledge, this study is the first to integrate the PI and modular capacity and investigate sustainability and resilience in the production-routing problem.

Simultaneous Pickup-and-Delivery Production-Routing Problem in closed-loop supply chain with remanufacturing and disassembly consideration

In a context of Closed-Loop Supply Chain, Production-Routing Problem with Pickup and Delivery is presented. In its basic form, Production-Routing Problem (PRP) attempts to solve jointly Dynamical Lot-Sizing and Inventory-Routing Problems. In this paper, a novel PRP model seeking to minimize the total cost of manufacturing, remanufacturing, disassembly, inventory, and routing by taking into account the remanufacturing and disassembly processes of End-of-Life returned products is provided. Novel hybrid heuristics based on Two-Phase Iterative and Relax-and-Fix heuristics were developed and outperform Branch-and-Cut algorithm for large size instances with a small vehicle capacity. Main insights derived from sensitivity analyses are (i) managers can use the model to trade off the cost saving resulting from the integration of forward and reverse flows, on the one hand, and any additional cost incurred by the organizational changes related to the integration, on the other hand, (ii) a high level of remanufacturing rate for returned products is not necessarily profitable, and managers should develop a return policy that leads to the appropriate level of remanufacturing rate, and (iii) investing in expanding the remanufacturing or the disassembly capacities can significantly reduce the total cost until a threshold level, beyond which any further investment is not beneficial.

A sample average approximation-based heuristic for the stochastic production routing problem

AbstractThe Production Routing Problem under demand uncertainty is an integrated problem containing production, inventory, and distribution decisions. At the planning level, the aim is to meet retailers demand, when only the demand distribution is known in advance, while minimizing the corresponding costs. In this study, a two-stage formulation is presented in which the routing can be adjusted at short notice. In the first stage, only production decisions are made, while delivery and inventory quantities and retailer visit schedules are determined in the second stage. To handle a large number of scenarios, two solution methods based on Sample Average Approximation are introduced. Furthermore, the impact of the routing quality is explored by applying a simple heuristic and an effective metaheuristic on the routing part. It is shown that, on average, the simple heuristic within an adjustable Sample Average Approximation approach provides better objective function values than the metaheuristic within a non-adjustable approach. Also all solution approaches outperform an expected value based approach in terms of runtime and objective function value.

Split demand and deliveries in an integrated three-level lot sizing and replenishment problem

We address a three-level lot sizing and replenishment problem (3LSRP), an extension of the production routing problem. We consider one production plant, with limited capacity, that produces items over a discrete and finite planning horizon. The items are sent from the plant to the warehouses using direct capacitated shipments, and routes are designed to deliver the goods from the warehouses to the retailers. The objective is to minimize the sum of all operational costs. We contribute to the literature on integrated problems by introducing the concept of demand splitting which allows the demand from a specific period to be satisfied by deliveries over multiple periods. Our second contribution lies in the development of the two heuristics that we propose: a top-down approach and a bottom-up approach. The production decisions and the transportation decisions between the warehouses and the retailers are the leading decisions in the top-down and the bottom-up approach, respectively. We compare them to a branch-and-cut algorithm that we also developed. We run computational experiments to assess the performance of each heuristic. We analyze the impact of allowing splitting possibilities. The results show that the top-down approach obtains better performance in terms of cost, except when we allow demand splitting only. The bottom-up approach leads to a greater use of delivery splitting. Results also show that we obtain large gains thanks to the splitting possibilities.

Designing a multi-period dynamic electric vehicle production-routing problem in a supply chain considering energy consumption

The coordinated decision-making approach for considering sequential activities of the supply chain results in additional benefits by optimizing production, inventory, and distribution operations. Accordingly, this paper proposes a mixed integer linear mathematical model to optimize a multi-period production routing problem utilizing electric vehicles. The proposed model optimizes the total cost associated with fixed and variable production, holding inventory, and routing, including the fixed cost of utilizing electric vehicles and travel time. However, mileage limitation is one of the main restrictions of utilizing electric vehicles in performing deliveries which is strongly affected by the consumed energy. Although optimization of the routes of vehicles can facilitate using them, considering the variation of travel speed network links during different times of the day because of traffic conditions can obviously affect the required energy to perform the assigned deliveries. To the best of our knowledge, this paper is the first to study simultaneous multi-period dynamic production routing problems using a set of heterogenous electric vehicles whose travel time of links can vary by dividing each production period into several hourly time intervals to capture different traffic conditions. Finally, a series of capable and hybrid metaheuristic algorithms are designed and implemented to solve this problem in a real-case dimension, and all proposed algorithms are compared.

A Three-Front Parallel Branch-and-Cut Algorithm for Production and Inventory Routing Problems

Production and inventory routing problems consider a single-product supply chain operating under a vendor-managed inventory system. A plant creates a production plan and vehicle routes over a planning horizon to replenish its customers at minimum cost. In this paper, we present two- and three-index formulations, implement a branch-and-cut algorithm based on each formulation, and introduce a local search matheuristic-based algorithm to solve the problem. In order to combine all benefits of each algorithm, we design a parallel framework to integrate all three fronts, called the three-front parallel branch-and-cut algorithm (3FP-B&C). We assess the performance of our method on well-known benchmark instances of the inventory routing problem (IRP) and the production routing problem (PRP). The results show that our 3FP-B&C outperforms by far other approaches from the literature. For the 956 feasible small-size IRP instances, our method proves optimality for 746, being the first exact algorithm to solve all instances with up to two vehicles. 3FP-B&C finds 949 best known solutions (BKS) with 153 new BKS (NBKS). For the large-size set, our method provides two new optimal solutions (OPT), and finds 82% of BKS, being 70% of NBKS for instances with up to five vehicles. This result is more than twice the number of BKS considering all heuristic methods from the literature combined. Finally, our 3FP-B&C finds the best lower bounds (BLB) for 1,169/1,316 instances, outperforming all previous exact algorithms. On the PRP, our method obtained 278 OPT out of the 336 instances of benchmark set of small- and medium-size instances being 19 new ones in addition to 335 BKS (74 NBKS) and 313 BLB (52 new ones). On another set of PRP with medium- and large-size instances, our algorithm finds 1,105 BKS out of 1,440 instances with 584 NBKS. Besides that, our 3FP-B&C is the first exact algorithm to solve the instances with an unlimited fleet, providing the first lower bounds for this subset with an average optimality gap of 0.61%. We also address a very large-size instance set, the second exact algorithm to address this set, outperforming the previous approach by far. Finally, a comparative analysis of each front shows the gains of the integrated approach. History: This paper has been accepted for the Transportation Science Special Issue: DIMACS Implementation Challenge: Vehicle Routing. Funding: C. M. Schenekemberg was supported by the São Paulo Research Foundation (FAPESP) [Grant 2020/07145-8]. A. A. Chaves was supported by FAPESP [Grants 2018/15417-8 and 2016/01860-1] and Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 312747/2021-7 and 405702/2021-3]. L. C. Coelho was supported by the Canadian Natural Sciences and Engineering Research Council [Grant 2019-00094]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0261 .

An oracle-based algorithm for robust planning of production routing problems in closed-loop supply chains of beverage glass bottles

The demand for glass bottles is exhibiting an upward trend over time. The manufacturing of glass bottles is costlier in terms of time and resources and is associated with a higher level of heat generation and environmental pollution compared to recycling processes. In response to the aforementioned challenges, companies that use glass bottles need to implement strategies to manage their reverse supply chains in conjunction with their traditional supply chains, as the economic and environmental benefits of returned products are unquestionable. Closed-loop supply chains (CLSCs) integrate forward and reverse flows of products and information. This integration helps companies to have a broader view of the whole chain. Despite these advantages, managing CLSCs can be challenging as they are exposed to many uncertainties regarding supply and demand processes, travel times, and quantity/quality of returned products.In this study, we consider the production planning, inventory management, and vehicle routing decisions of a CLSC of beverage glass bottles. We propose an MILP model and rely on a multi-stage adjustable robust optimization (ARO) formulation to deal with the randomness in both the demand for filled bottles and the requests for pickups of empty bottles. We develop an exact oracle-based algorithm to solve the ARO problem and propose a heuristic search algorithm to reduce the solution time. Our numerical experiments not only show the incompetency of the customary method, namely the affine decision rule approach, but also illustrate how our algorithms can solve the small-size problems and significantly improve the quality of the obtained solution for large problems. Furthermore, our numerical results show that robust plans tend to be sparse, meaning the routes are chosen so that empty bottles are transported to production sites in such a way that fewer new bottles need to be ordered. Thus, robust planning makes the CLSCs more environmentally friendly.

A robust optimization approach for the production-routing problem with lateral transshipment and outsourcing

Despite the fact that there is a large body of literature on the Production Routing Problem (PRP), we were struck by the dearth of research on outsource planning and lateral transshipment. This paper presents a mixed-integer linear programming model for incorporating outsourcing, lateral transshipment, back ordering, lost sales, and time windows into production routing problems. Then a robust optimization model will be introduced to overcome the detrimental effects of demand uncertainty. Considering the scale and complexity of the suggested problem, addressing it in a reasonable time was a challenge. Therefore, three matheuristic algorithms, including Genetic Algorithm (GA), Simulated Annealing (SA), and Modified Simulated Annealing (MSA), are developed for solving large-scale problems. Eventually, computational experiments on disparate instances are performed, and the results show the effectiveness and efficiency of the proposed algorithms. In other words, our recommended algorithms outperform the CPLEX solver in terms of the quality and time of obtaining the solutions.

A Memetic Algorithm for the multi-product Production Routing Problem

This article addresses the Production Routing Problem (PRP), which consists of determining, in an integrated way, production and inventory planning, and vehicle routing to minimize the costs involved. In the problem, a plant is responsible for producing several types of products to meet the known demand of a set of customers using a homogeneous fleet of vehicles over the planning horizon. In the literature, evolutionary approaches have not been explored in depth for the PRP, specifically for the problem with multiple products. Thus, this work mitigates this gap, presenting a novel Memetic Algorithm and testing its effectiveness on randomly generated sets of instances, comparing the results obtained with a commercial optimization solver. In our solution approach, several classic operators from the literature were implemented. Furthermore, we propose four novel genetic operators. In addition, we evaluated the proposed method’s performance in classical instances of literature considering a single item. The computational experiments were carried out to assess the impact of the numerous parameter combinations involving the metaheuristic, and, from statistical analyses, we evidence the proposed technique’s robustness. Computational experiments showed that our proposed method outperforms the commercial solver Gurobi in determining feasibly high-quality solutions, mainly on large instances for the PRP with multiple items.

A green production routing problem for medical nitrous oxide: Model and solution approach

The impact of climate change and the global efforts to address it have made environmental considerations increasingly important. Industries and transportation, being major energy consumers, play a significant role in global warming and must reduce their pollutants. This paper presents a bi-objective production routing problem aimed at minimizing the total Greenhouse Gas (GHG) emissions and cost, taking into account real-world constraints and data from a single-plant, multi-distribution center, gaseous-pharmaceutical supply chain with a heterogeneous delivery fleet. The modeling takes into account the GHG emissions from the production process of the product being delivered. A branch and cut method is used in combination with multi-objective fuzzy goal programming to solve the problem. The results show that the proposed model reduces total costs while also reducing GHG emissions compared to the current method used in the case study.

The effect of different mathematical formulations on a matheuristic algorithm for the production routing problem

We perform an experimental study to evaluate the performance of a matheuristic for the production routing problem (PRP). First, we develop a basic matheuristic that prescribes starting from a partial initial solution, completing it using a sequence of constructive heuristics, and improving it using a general-purpose mixed-integer programming heuristic. Next, we investigate the effect of three state-of-the-art mathematical formulations on the proposed matheuristic convergence. The formulations are implemented and tested with and without the use of valid inequalities. In addition, by suggesting different techniques to generate a feasible starting solution for our matheuristic, we assess the contribution of an initial solution to the matheuristic’s overall performance. We conduct extensive computational experiments on benchmark data instances for the PRP. The results show that a proper choice of an embedded mathematical formulation depends on the data instances’ features, such as the number of customers and the length of the planning horizon. The comparisons undertaken in this study indicate that having a better initial solution does not necessarily lead to finding a better final solution.

The cyclic production routing problem

This paper introduces the Cyclic Production Routing Problem (CPRP). The CPRP is an extension of the well-known NP-hard Production Routing Problem (PRP), which is a hard-to-solve combinatorial optimisation problem with numerous practical applications in the field of freight transportation, logistics and supply chain management. Under the PRP setting, a manufacturer is responsible for determining production decisions, as well as the timing and quantity of replenishment services offered to a set of geographically dispersed customers over a multi-period time horizon. The problem calls for jointly optimising the production, inventory, distribution and routing decisions. In this paper, the basic PRP model is modified to generate repeatable cyclic production and delivery schedules. A two-commodity flow formulation is proposed along with valid inequalities. Extensive comparisons between the basic PRP and the proposed cyclic variant on well-known benchmark instances are provided. The new variant is significantly harder to solve, especially when the vehicle fleet is limited. From a managerial perspective, the generation of cyclic production-routing schedules significantly increases all costs, whereas the number of vehicle routes required to implement a cyclic schedule is higher.

The inventory replenishment policy in an uncertain production-inventory-routing system

<p style='text-indent:20px;'>This study introduces an uncertain programming model for the integrated production routing problem (PRP) in an uncertain production-inventory-routing system. Based on uncertainty theory, an uncertain programming model is proposed firstly and then transformed into a deterministic and equivalent model. The study further probes into different types of replenishment policies under the condition of uncertain demands, mainly the uncertain maximum level (UML) policy and the uncertain order-up to level (UOU) policy. Some inequalities are put forward to define the UML policy and the UOU policy under the uncertain environments, and the influences brought by uncertain demands are highlighted. The overall costs with optimal solution of the uncertain decision model grow with the increase of the confidence levels. And they are simultaneously affected by the variances of uncertain variables but rely on the value of confidence levels. Results show that when the confidence levels are not less than 0.5, the cost difference between the two policies begins to narrow along with the increase of the confidence levels and the variances of uncertain variables, eventually being trending to zero. When there are higher confidence levels and relatively large uncertainty in realistic applications, in which the solution scale is escalated, being conducive to its efficiency advantage, the comprehensive advantages of the UOU policy is obvious.</p>

Rolling horizon-based heuristics for solving a production-routing problem with price-dependent demand

This paper studies a production-routing problem with price-dependent demand (PRP-PD). The problem is formulated as a mixed-integer nonlinear program (MINLP) and solved within an outer approximation (OA) framework where the primal and master problems are solved iteratively. In order to solve the computationally expensive master problem, which is a mixed-integer program (MIP), four different rolling horizon-based heuristics are developed. The efficiency of the proposed heuristic algorithms is benchmarked against the existing methods in the literature to solve the PRP-PD for both linear and nonlinear demand functions. Our results show that by adopting proper freezing and simplification strategies, the rolling horizon-based heuristic can solve the large problem instances significantly faster than the existing methods in the literature, with little to no reduction in solution quality. For large problem instances with linear demand functions, the proposed heuristic provides solutions within 1.5–3 min. The quality of these solutions deviates at most 3.5% from the solutions provided by the existing exact methods, which take more than 2 h. Our solutions are also higher in quality compared to the near-optimal solutions provided in the literature, and they are achieved with a 90% reduction in computational time. With nonlinear demand functions, one-third of the instances require more than 2 h to be solved by the exact methods in the literature, while they are solved within 0.5–3.5 min using our heuristic, with less than 2% deviation in solution quality.

Integrated production and inventory routing planning of oxygen supply chains

In this work, we address a production and inventory routing problem for a liquid oxygen supply chain comprising production facilities, distribution network, and distribution resources. The key decisions of the problem involve production levels of production plants, delivery schedule and routing through heterogeneous vehicles, and inventory strategies for national stock-out prevention. Due to the problem complexity, we propose a two-level hybrid solution approach that solves the problem using both exact and metaheuristic methods. At the upper level, we develop a mixed-integer linear programming (MILP) model that determines production and inventory decisions and customer allocation. In the lower level, the original problem is reduced to several multi-trip heterogeneous vehicle routing problems by fixing the optimal production, inventory, and allocation decisions and clustering customers. A well-recognised metaheuristic, guided local search method, is adapted to solve the low-level routing problems. A real-world case study in the UK illustrates the applicability and effectiveness of the proposed optimisation framework.

Efficient matheuristics to solve a rich production-routing problem

We present a rich production-routing problem having limited production and storage capacities at the plant, limited storage capacity at the clients, a heterogeneous fleet subjected to a maximum riding time, and allowing for back-orders to meet unfulfilled demands at penalty cost. As the problem scales quickly with the number of customers, periods, products, and vehicles, three hybrid two-level decomposition approaches using a top-down strategy were devised. The top tier determines the production and inventory levels, and the distribution of goods via CPLEX, that is, it makes tactical decisions, while the bottom tier heuristically routes a heterogeneous fleet in each period, that is, it makes operational decisions. The proposed methods rely on an iterated local search framework that combines tailored perturbation schemes prioritizing either tactical or operational decisions, or both. The main new feature of the algorithms is the adoption of an implicit cost that estimates the delivery routing costs when making production, holding, and transportation decisions. This implicit cost serves as an important guide to obtain improved solutions. The algorithms were tested over an extensive set of instances, and the results demonstrated that all methods overcome CPLEX by obtaining more, better, and faster solutions with much less computational effort. The devised heuristic, which prioritizes operational-level decisions during the perturbation phase, attained the best overall results.