Integer Programming Model Research Articles

Deep Learning-Based Prediction and Revenue Optimization for Online Platform User Journeys

<abstract> <p>In today's digital landscape, businesses must allocate online resources efficiently. Data-driven AI methods are increasingly adopted for customer journey management. This study enhances existing frameworks with three key propositions, integrating deep learning and optimization to create a three-step revenue optimization model using online customer data. First, we apply K-means clustering to analyze online user data, constructing a behavior model. Then, convolutional neural networks (CNN) and long short-term memory (LSTM) networks predict user behavior and conversion values from sequential data. Finally, the heuristic algorithm optimizes revenue within budget constraints based on conversions. From an academic perspective, our study provides an empirical, theory-grounded model for service and marketing management. Technologically, we identify three key findings: stacking LSTM with CNN effectively processes sequential online user data, outperforming traditional machine learning methods; optimization methods and decision trees improve model interpretability and address marketing attribution challenges by understanding user behavior and channel impacts; and traditional integer programming models fall short in solving high-dimensional online channel planning problems, necessitating heuristic algorithms. Our model aids companies in setting online channel standards and budgets, offering valuable insights and practical guidance to decision-makers.</p> </abstract>

VENDOR MANAGED INVENTORY POLICY FOR PERISHABLE PRODUCTS UNDER A COMBINATION OF FIFO-LIFO STOCK CONTROL

For retailing systems, arranging the perishable products has always been a challenge. Although more profit is made when products are sold with a FIFO policy, now customers taste in a competitive market which requires new products to be offered to them has changed the game. This paper considers a hybrid of FIFO-LIFO strategies for controlling a retailer's stock and its impact on the profits of the vendor managed inventory contracts. The problem is formulated as a multi-period, multi-store integer programming model which aims to maximize the total profit. The proposed model considers different selling prices and different holding costs for perishable products according to their remaining shelf life. A dynamic programming with a backward strategy is proposed to find the optimal order quantity for each store. The numerical experiments confirm the applicability and efficiency of the model illustrating impact of the crucial parameters of the model on the stocking policy.

Development of a Linear Integer Programming Model for Solving Cutting Stock Problem in the Manufacturing Industry

Cutting stock problem is a problem generally encountered in many manufacturing industries such as the furniture, clothing, glass production, leather, paper, textile, metals industries amongst others most especially during mass production. The problem usually arises during roll slitting whereby a large role is to be cut into smaller pieces. The challenge may be further compounded due to constraints and different variants that arises from product customization, process and machinery as well as customer requirements and quality issues. To address this problem, this study develops a linear integer programming technique and provides a practical guided approach. First mathematical formulation (having objective function and constraints) comprising of a list of q orders, that requires p j pieces was formulated and a list of all possible cuts combination of configuration and patterns were derived. The linear integer program was solved in the MATLAB 2022 environment. The results indicated that the large rolls can be cut into the desired length, number, patterns and configuration in a time effective manner using the developed linear integer programming technique. The results further show that for the total wastes generated was 24 for the optimal solutions. This number of wastes can be considered to be minimal considering the volume of demand and patterns generated from the stock material. Furthermore, 6 out of the 7 pattern optimal solutions generated zero waste. This study provides theoretical and empirical findings that can assist the manufacturing industries in minimizing time and cost variables while addressing stock cutting problems.

A Novel Mixed Integer Programming Model With Precedence Relation-Based Decision Variables for Non-Cyclic Scheduling of Cluster Tools

A Novel Mixed Integer Programming Model With Precedence Relation-Based Decision Variables for Non-Cyclic Scheduling of Cluster Tools

Integer programming model for distance-edge-monitoring problem

The paper considers the recently introduced distance-edge-monitoring problem. For a given graph G = (V,E), the set M is called distance-edge-monitoring if it is a subset of V and for every edge e of E there is a vertex x of M and a vertex y of V such that e belongs to all the shortest paths between x and y. The goal is to find the smallest distance-edge-monitoring set of the graph. We propose the first-ever integer programming model for this problem and present empirical results for five instance classes with graphs up to 4096 vertices and 1599797 edges. The proposed model was able to solve all instances of four instance classes optimally, while the remaining fifth class of graphs proves to be more difficult for the proposed model.

Truck-drone joint path planning for post-disaster emergency material deployment considering fairness

As the expert gear of the emergency rescue system, drones are frequently utilized to distribute supplies following a calamity. The cost and effectiveness of rescue efforts as well as equitable distribution should be taken into account when allocating emergency supplies to disaster-affected areas. This work explores the emergency material allocation problem for truck-drone joint transportation with dynamic energy restrictions based on taking the fairness of emergency material allocation into consideration. In order to guarantee the equitable distribution of materials, the psychological stress experienced by the victims at each catastrophe site is measured using the relative deprivation cost. An adaptive large-scale neighborhood search method serves as the foundation for the creation of a two-stage heuristic algorithm, which reduces the overall cost of the system. The integer programming model MIP is built for this purpose. The research findings can serve as a useful guide for developing a just and effective emergency drone rescue system, and the testing results demonstrate the viability and effectiveness of the two-stage heuristic algorithm.

Research on optimization of flight crew scheduling considering pilot fatigue

Safety is a top concern for the civil aviation industry, and the risk of safety accidents will increase due to pilot fatigue. To ensure the safety of civil aviation, this paper proposes a method to solve the crew scheduling problem considering pilot fatigue. In order to reflect individual differences and fatigue levels of pilots, an improved three-stage alertness calculation model is first proposed based on subjective and objective perspectives to represent pilots’ alertness levels and fatigue working duration quantitatively. Then, for the crew scheduling problem considering pilot fatigue, a mixed integer programming model is constructed to simultaneously achieve the optimization objectives of reducing the overall scheduling cost and crew fatigue working duration. Next, since the actual crew scheduling problem is large-scale, a solution algorithm based on a column generation framework is developed to improve the quality and efficiency of solving the large-scale crew scheduling problem. Furthermore, in the case study, we collected actual data from an airline company to validate the effectiveness of our proposed method. Finally, through multiple experimental comparisons and analyses, to balance the two optimization objectives mentioned above, it is more reasonable to handle pilot fatigue working duration with soft constraints. Sensitivity analysis reveals the variation rules of the crew cost and fatigue, providing some valuable managerial insights for the crew scheduling problem considering pilot fatigue.

Bin Packing Problem with Item Time Window Conflict Dependent Profits

Abstract The problem of packing with items with conflicting time-window dependence on profit considering departure time is studied in the context of consolidation of goods in transport, which is a practical problem in the logistics industry. The objective is to select the loading plan that is most profitable based on the volume and time windows of the items. The time windows of items in the same box must intersect, and the intersection of the items’ time windows affects the box’s departure time and hence the profit-dependent time. The problem is identified as an integer programming model for decision making, and two modeling strategies are proposed to address it by improving the GRASP heuristic. Multi-class examples are used in experiments to confirm that the improved algorithm can solve the problem efficiently, find the near-optimal solution in a shorter time, and enhance customer satisfaction.

Multi-mode multi-skill resource-constrained project scheduling problem with differentiated professional capabilities

Motivated by a practical situation in a digital transformation project, this paper considers a resource-constrained project scheduling problem with multiple modes, multiple skill types, and differentiated professional capabilities. In the proposed problem, each project activity has one or more alternative execution modes associated with a trade-off between processing time and resource consumption. In an execution mode, an activity requires a certain number of employees with specific skill types and required professional capabilities. A mixed integer programming model is developed to minimize the total project duration. Since this problem is NP-hard, an efficient immunoglobulin-based artificial immune system (EIAIS) algorithm with a new encoding and decoding scheme and novel components is proposed. The effectiveness of the proposed EIAIS algorithm is tested on randomly generated instances. Computational results show that the proposed EIAIS algorithm has better performance than the existing algorithms.

Mixed Integer Programming Model Based on Data Algorithms in Sustainable Supply Chain Management

Mixed Integer Programming Model Based on Data Algorithms in Sustainable Supply Chain Management

Mixed integer programming model with constraints for integrated container terminal operations

The objective of a container terminal is to complete operations with each docked vessel as quickly as possible to minimize the waiting time of the vessel at the port and thus maximize the turnover, i.e., the number of containers handled. Optimization of complex operations at a container terminal is computationally challenging. This paper considers the complex problem of assigning mooring cranes, scheduling mooring cranes, assigning warehouse location and dispatching vehicles at a container terminal. The main objective is to minimize the turnaround time of ships and maximize the throughput of the terminal. In this paper, constraints are considered and a mixed integer programming model is proposed for the complex problem of integrated operations in a container terminal.

A cooperative evolutionary algorithm with simulated annealing for integrated scheduling of distributed flexible job shops and distribution

Production and distribution are two essential parts in supply chains. An integration of production and distribution has received amount of attention from both academia and industry. This article investigates an integrated scheduling problem of distributed flexible job shops and distribution. First, a mixed integer programming model is developed to minimize maximum completion time. Second, a cooperative evolutionary algorithm with simulated annealing methods is proposed in consideration of the problem's characteristics. A three-link integer string is designed to represent three parts, i.e., factory assignment, machine assignment and operation sequence, in a solution. Three populations are constructed to search the above three parts via genetic operations and simulated annealing methods, respectively. A cooperation strategy is designed to generate a schedule. Then, a heuristic rule is devised to determine the delivery scheme for distribution. Finally, experiments are carried out via extending a set of benchmark instances. The proposed algorithm is compared to four state-of-the-art algorithms and an exact solver CPLEX. The results and discussions verify that the developed model and method have strong competitiveness for solving the considered problems.

A simultaneous time and fuel minimization robust possibilistic multiobjective programming approach for truck-sharing scheduling in container terminals under uncertainty

The issue of integrated scheduling and sequencing operation of unloading and loading equipment in container ports has been one of the most important issues concerning time efficiency. In addition, with the emergence of green harbor concepts, the inclusion of criteria for minimizing energy consumption, fuel and emission reduction are among the other issues that have been noticed by planners in the field of energy efficiency. Furthermore, due to the complexity and scope of activities of a container terminal, uncertainty in operational parameters such as transportation time, time of readiness and entry of work into the system and the velocity of the transportation fleet are inevitable in this operational environment. Therefore, this research with the aim of sharing trucks among loading and unloading equipment, proposes a robust multi-objective integer programming model for the synchronized scheduling of truck operations with other handling equipment to decrease the fuel consumption of trucks and the flow time of containers, considering the uncertainty in operational parameters as fuzzy numbers. To find the Pareto solutions for this model, the ε-Constraint technique is employed. Finally, the performance of the model in deterministic and uncertain modes is evaluated, compared and analyzed employing the inputs gathered from Shahid Rajaei port. The findings demonstrate that using this model will result in a substantial decrease in both fuel consumption and flow time of containers in comparison to the current procedure. Additionally, results will demonstrate the extent to which the terminal's fuel and time consumption will increase under conditions of uncertainty in operational parameters when the optimal plans derived from the robust model are implemented.

Research on optimal cutting scheme based on three-stage guillotine cut

Smart manufacturing has become a major part of the core competitiveness of enterprises, and better scheduling optimization methods are obtained in order to reduce the material loss in the manufacturing process. In this paper, we construct a mixed integer programming model, iterate from local optimum to global optimum by genetic algorithm, and finally solve the question B of the 2022 China Graduate Student Mathematical Modeling Competition. We get the results that the utilization rate of each scheduling solution is higher than 90%, which shows the effectiveness of the model and algorithm, and can greatly improve the utilization rate of plates.

Integrating Statistical Simulation and Optimization for Redundancy Allocation in Smart Grid Infrastructure

It is a critical issue to allocate redundancy to critical smart grid infrastructure for disaster recovery planning. In this study, a framework to combine statistical prediction methods and optimization models for the optimal redundancy allocation problem is presented. First, statistical simulation methods to identify critical nodes of very large-scale smart grid infrastructure based on the topological features of embedding networks are developed, and then a linear integer programming model based on generalized assignment problem (GAP) for the redundancy allocation of critical nodes in smart grid infrastructure is presented. This paper aims to contribute to the field by employing a general redundancy allocation problem (GRAP) model from high-order nonlinear to linear model transformation. The model is specifically implemented in the context of smart grid infrastructure. The innovative linear integer programming model proposed in this paper capitalizes on the logarithmic multiplication property to reframe the inherently nonlinear resource allocation problem (RAP) into a linearly separable function. This reformulation markedly streamlines the problem, enhancing its suitability for efficient and effective solutions. The findings demonstrate that the combined approach of statistical simulation and optimization effectively addresses the size limitations inherent in a sole optimization approach. Notably, the optimal solutions for redundancy allocation in large grid systems highlight that the cost of redundancy is only a fraction of the economic losses incurred due to weather-related outages.

An extended ε$\varepsilon $‐constraint method for a bi‐objective assortment optimization problem

AbstractThe paper deals with assortment optimization when customers purchase products using a nonparametric choice model. Each customer is presented with a preference list and will buy an available highest‐ranked offered product on the preference list. The paper presents a novel bi‐objective integer programming model for assortment optimization following a ranking‐based consumer choice model. The objectives are maximizing the expected revenue and maximizing customer satisfaction. This problem is recognized as an NP‐hard problem. After developing the bi‐objective model, an improved version of the augmented ‐constraint method is proposed to solve the problem and retrieve the Pareto‐optimal solutions. In each iteration, this algorithm solves a single‐objective linear integer programming sub‐problem. Our approach aims to obtain a complete set of Pareto‐optimal solutions. Finally, we present computational experiments on synthetic data. Several numerical illustrations are provided at the end to demonstrate the application of the proposed methodology. The computational results provide the decision‐makers with valuable information regarding the factors that mainly affect the solutions.

VRPSolverEasy: A Python Library for the Exact Solution of a Rich Vehicle Routing Problem

The optimization community has made significant progress in solving vehicle routing problems (VRPs) to optimality using sophisticated branch-cut-and-price (BCP) algorithms. VRPSolver is a BCP algorithm with excellent performance in many VRP variants. However, its complex underlying mathematical model makes it hardly accessible to routing practitioners. To address this, VRPSolverEasy provides a Python interface to VRPSolver that does not require any knowledge of mixed integer programming modeling. Instead, routing problems are defined in terms of familiar elements, such as depots, customers, links, and vehicle types. VRPSolverEasy can handle several popular VRP variants and arbitrary combinations of them. History: Accepted by Ted Ralphs, Area Editor for Software Tools. This paper has been accepted for the INFORMS Journal on Computing Special Issue on Software Tools for Vehicle Routing. Funding: This work was supported by Faperj [Grant E-26/202.887/2017] and Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant 305684/2022-1]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0103 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0103 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Optimal en-route charging station locations for electric vehicles with heterogeneous range anxiety

This paper addresses a new optimal charging station location problem for intercity highway networks where electric vehicles are of heterogeneous driving ranges and usually need to be charged multiple times in their long-haul trips. Driving range heterogeneity can be measured in practice by sampling the driving population under a variety of physical, environmental and psychological conditions and is characterized in this study by a continuous distribution of some appropriate forms. The behavioral assumption underlying this problem is that all individual drivers choose their optimal route-and-charge choices if they can make a trip by driving an electric vehicle or cancel their trips or switch to other transportation modes if they cannot. By explicitly taking into account the impact of limited driving ranges on both route and trip/mode choices, we constructed a mixed integer linear programming model for formulating the charging station location problem, the goal of which is to maximize the networkwide travel efficiency and preference subject to a limited infrastructure investment budget. To tackle this integer programming model, we developed a branch-and-bound algorithm and a neighborhood search heuristic, in both of which a multi-criterion label-correcting algorithm is embedded for deriving the underlying route-and-charge flow pattern. For justifying the effectiveness and efficiency of the proposed algorithms, a synthetic network and two real-world networks are employed as numerical examples. The computational results obtained from the numerical analysis show that the heuristic is capable of obtaining optimal solutions in most test scenarios and is much more computationally efficient than the branch-and-bound algorithm, with a computing time that is only about one tenth of the latter.

Integrated scheduling optimization of AGV and double yard cranes in automated container terminals

Double yard cranes in one yard block refer to two yard cranes that run in different rails and can pass each other. They can both reach to the seaside and landside of the yard blocks. But interference between the double yard cranes occurs during picking up or putting down containers, which has to be solved when assigning tasks to the double yard cranes. AGVs assignment have also to be considered when scheduling yard cranes. This paper focuses on the integrated optimization of AGV and double yard cranes in automated container terminals, aiming to increase terminal efficiency through the coordination of multiple sub-operations. A mixed integer programming model is developed, which considers the interference of the double yard cranes. The objective is to minimize the completion time of all the tasks. To solve the problem, a branch and bound (B&B) based method is designed. Specifically, a heuristic algorithm is used to assign container tasks to AGVs, and B&B method is utilized to assign container tasks to double yard cranes. This paper derives pruning strategies and acceleration strategies based on model analysis to improve the efficiency of the algorithm. The effectiveness and validity of the proposed model and algorithm is verified through small-scale and large-scale experiments. Results are compared with that of commercial solver Gurobi.

Modeling a supply chain for carbon capture and offshore storage—A German–Norwegian case study

Carbon capture and storage (CCS) for industrial emission point sources is one of the potential instruments to achieve net-zero carbon dioxide (CO2) goals. However, emission point sources and storage formations are often far from each other, which requires capable CO2 transportation infrastructure. While pipeline transportation promises low cost for high and stable flows of CO2, ship transportation may be more expensive but also more flexible with regards to transport quantities and storage locations. Here, we present a mixed integer programming (MIP) model to provide decision support for a CCS Supply Chain Design Problem (CCS-SCDP) with the goal of minimizing total supply chain costs. We apply the model to four future CO2 supply scenarios, capturing CO2 from German industrial sources and bringing them to the Northern Lights unloading port in Kollsnes, Norway, for storage in a submarine geological formation. Our analysis reveals that the fraction of transportation costs of total supply chain costs drop considerably from 22 to 10 percent by economies of scale if annual capture volume increases. For low capture volumes, a ship-based solution is cheaper, while an offshore pipeline solution is favored for larger capture volumes. Accordingly, the potential gains from economies of scale in a pipeline-based solution must be balanced against potential lock-in effects in the investment decision for a CCS supply chain.