bi-level programming
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A significant part of each system is determining system efficiency to conduct future planning-associated operations. Data Envelopment Analysis (DEA) is often employed to measure system efficiency. The paper considers a bi-level structure and proposes a new non-radial method by generalizing Russell's model to measure system efficiency. Furthermore, data from 33 branches of an Iranian state bank in 2021 are investigated to present an example of the application. The results indicate that among 33 branches, only two branches are regarded as efficient at both leader and follower levels.Keywords: Data Envelopment Analysis, Bi-Level Programming, Undesirable Data, Non-Radial Model, Bank
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The present study developed a bi-level mathematical model to determine optimal routes for repair teams in charge of inspecting urban traffic lights. In this model, the municipality as the leader locates the construction sites of urban spare parts warehouses and repair centers to minimize the costs of constructing the facilities. At the follower level, the contractor determines the optimal routes for the repair teams. Bi-level models are strongly NP-hard in type. A heuristic algorithm is therefore developed to solve numerical examples, in which the leader first determines different decision-making strategies for the follower through generating a set of justified solutions. In response to the leader’s set of strategies, the follower presents a set of corresponding solutions. The individual solutions of the follower are then entered into the leader model and the corresponding values of the objective function are calculated. A solution with the optimal numerical value for the leader is ultimately selected as the Stackelberg equilibrium. The efficiency of the proposed model and algorithm was evaluated by presenting the computational results obtained from solving several random numerical examples of small, medium and large dimensions through generating the Stackelberg equilibrium and establishing a relationship between the leader and follower levels. The present findings are recommended to be used as a management tool by policymakers in the urban management sector.
Keywords: Bi-Level Programming, Heuristic, Support Center Location, Location Arc Routing, Urban Services -
Empirical studies have indicated that linking advertising and pricing will bring significant advantages to the supply chain components. With the exponential extension of online social networks and society’s greater interest in receiving information from this space, many firms have been encouraged to use online social networks and maximize the effects of advertising campaigns; however, literature on designing this type of advertising and linking it with pricing in the supply chain is still rare. To fill this gap, this paper uses a data-driven support vector optimization framework to link influencer-based advertising and pricing in a two-echelon SC. Also, the impact of the passage of time and uncertainty on advertising message diffusion has been examined. The results show that advertising in social media is a complex task and is affected by various factors, such as the time of serving the primary and supporting ads. Based on our results, only after six weeks of releasing the primary ads did the effect of the advertisement decrease significantly. It seems that disseminating supporting advertising messages in advertising campaigns is vital. Also, results obtained from the data-driven robust optimization models show that the slightest change in the degree of conservatism significantly changes the profitability of the company (an increase of only 5% of the degree of conservatism increases profitability by about 1.4 on average), therefore, determining this coefficient has a significant effect on the performance advertising campaigns.
Keywords: Dynamic competitive pricing, time-sensitive advertising, polynomial Kernel, Support Vector Machine, Bi-level programming, data-driven programming -
International Journal of Supply and Operations Management, Volume:10 Issue: 2, Spring 2023, PP 151 -173The aim of this paper was to develop a binary bi-level optimization model for the emergency warehouse location-allocation problem in terms of national and regional levels. This type of modeling is suitable for countries where the design of the disaster emergency network is decentralized. The upper-level decision-maker makes a decision regarding the location and allocation of national warehouses through considering the location of regional warehouses and allocating them to the demand cities. Each regional warehouse can provide a service for the demand cities within a specified distance threshold, ultimately affecting the efficiency of the solution algorithms. The optimization model parameters were calculated in terms of the real data in Iran. To solve the small size problem, an exact method was proposed from the explicit complete enumeration. Due to the complexity of the model with the large size, two innovative hybrid genetic algorithms, namely HG-ES-1 and HG-ES-2, were suggested. The results obtained from solving the problems showed that the HG-ES-1 algorithm outperformed HG-ES-2. The findings further indicated the proper functioning of the solution approaches.Keywords: Bi pre-positioning, Disaster management, Bi-level programming, Hybrid genetic algorithms, Location-Allocation Problem
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مساله مکان یابی هاب از اساسی ترین و مهم ترین مسایل در حوزه ی تصمیم گیری و برنامه ریزی سیستم های حمل ونقل به شمار می رود. هدف اصلی مساله مکان یابی هاب، انتقال جریان بین نقاط تقاضا از طریق هاب یا هاب هایی است که نقش اساسی را در این میان ایفا می کنند. موضوع از کار افتادگی هاب ها در این مسایل از مواردی است که سال های اخیر مورد توجه برخی از محققان قرار گرفته است. در این تحقیق، مساله مکان یابی هاب سلسله مراتبی با وجود اختلال در هاب های شبکه بررسی شده و در قالب یک مدل برنامه ریزی دوسطحی، مد ل سازی می شود. مساله مکان یابی هاب ارایه شده از نوع مسایل حداکثر پوشش هاب است. از کار افتادگی هاب در این مدل به صورت عمدی رخ داده و باعث اختلال فعالیت ها در هاب های غیرمرکزی می شود. در سطح دوم تلاش می شود تا با از کار انداختن یک هاب، پوشش دهی مساله به کمترین میزان خود برسد، درحالی که سطح اول مساله می خواهد خسارت به وجود آمده را کاهش داده و پوشش دهی مساله را بالا ببرد. مساله مورد بررسی با استفاده از الگوریتم های شمارش کامل و شبیه سازی تبرید با داده های متفاوت حل شده است. نتایج محاسباتی حل مدل پیشنهادی برای مسایل نمونه از جمله تقاضا و تعداد هاب های متفاوت، فاکتور تخفیف بین هاب ها و شعاع پوشش مختلف بررسی شد. نتایج عددی نشان داد با افزایش شعاع پوشش، تعداد گره های مکان یابی شده و همچنین مقدار پوشش دهی مساله افزایش می یابد. همچنین نشان داده شد که روش فراابتکاری پیاده شده کارایی دارد و توانایی حل داده های بزرگ را نیز داراست.
کلید واژگان: مکان یابی هاب حداکثر پوشش، برنامه ریزی دوسطحی، هاب سلسله مراتبی، از کار افتادگی هابJournal of Industrial Engineering Research in Production Systems, Volume:10 Issue: 20, 2023, PP 33 -47The hub location problem is one of the most fundamental and crucial issues in transportation systems and decision-making. The primary purpose of a transportation network is to transfer traffic between demand points via a hub, and hubs are essential to this process. The failure of hubs has garnered considerable attention of researchers in recent years. This research examines the problem of hierarchical hub location using bi-level programming. This study presents a model for optimal hub coverage. In non-central hubs, disruptions are intentional. The objective function of the second level is to minimize the problem's coverage by disabling the hubs, whereas the objective function of the first level is to minimize the problem's damage while expanding its coverage. The studied problem was solved using the simulated annealing and the full enumeration method. The proposed model has been solved for a variety of different scenarios, including fluctuating demand and hub count, fluctuating discount factors between hubs, and fluctuating coverage radii. According to the numerical results, as the covering radius increases, the number of located nodes and the problem's coverage also increase. In conclusion, an analysis of the employed solution methods concludes that the proposed meta-heuristic method is both effective and applicable to larger data sets.
Keywords: Maximal Covering, Bi-level Programming, Hierarchical Hub Location, Disruption -
The linear fractional bi-level problems are strongly NP-hard and non-convex, which results in high computational complexity to find the optimal solution. In this paper, we propose an efficient algorithm for solving a class of non-linear bi-level optimization problems, where the upper and lower objectives are linear fractional. The main idea behind the proposed algorithm is to obtain a single objective optimization problem via Taylor approximation. The proposed algorithm is composed of four steps. In the first, the lower level of the problem is converted into the convex optimization problem by using auxiliary variables and approximation techniques. Next, a single objective optimization problem is obtained by adopting the dual Lagrange method and Karush-Kuhn-Tucker (KKT) conditions. The obtained problem is non-convex with high computational complexity challenging to solve. Hence, the Fischer-Burmeister function is applied to smooth the problem. Finally, the first-order Taylor approximation is adopted to transform the non-linear problem into the linear one. Numerical results confirm the effectiveness of the proposed algorithm in comparison with Estimation of Distribution Algorithm (EDA) in terms of convergence performance.
Keywords: Bi-level programming, linear fractional bi-level problem, Taylor approximation, dual Lagrange method, Fischer-Burmeister -
In this paper, a bi-level mathematical formulation for a pricing-inventory-routing problem in the context of sustainable closed-loop supply chains is developed. The two levels are entitled as the upper level model and the lower level model. The upper level model (the leader model) tries to minimize greenhouse gas (GHG) emissions while the lower level model (the follower model) focuses on profit maximization. To solve the problem, an enumeration heuristic method based on knapsack problem and genetic algorithm (GA) is devised. The results show that the heuristic method is capable of obtaining high-quality solutions in reasonable CPU-times.Keywords: Bi-level programming, Heuristic Method, pricing-routing-inventory, Closed loop supply chain, incentive loans
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Journal of Industrial Engineering and Management Studies, Volume:7 Issue: 2, Summer-Autumn 2020, PP 119 -138Fast growth of motorized transportation infrastructures in the cities is a consequence of the urbanization process. Despite the undeniable benefits of the developments, some unwelcome social-environmental damages have been occurred. On top of the list, the movements of the pedestrians and their participation in social activities have dramatically reduced as a result of the vehicles dominancy. Pedestrianization and walking-friendly schemes are the key answer to preserve the valuable element of the urban lifestyle. This need motivated the researchers to study and propose mathematical methods to model the dynamics and behavior of the pedestrians in response to their surroundings. However, most of the models in the literature are suitable for limited small-size area and cannot be applied for a large scale urban zone. In this paper, a fuzzy macroscopic pedestrian assignment model is proposed which is applicable for a large scale network and useful for urban master plans as a decision making framework. In addition, a bi-level mixed integer programming model is presented to optimize the pedestrian walking network via selecting some projects on the network, considering the behavior of the pedestrians. Finally, the problem is solved for a large scale pedestrian network in the city of Tehran. The results show the efficiency of the algorithm where spending half of the maximum possible cost has led to a welfare gain of 82.6 percent. The problem was efficiently solved within 12.5 days which is fairly acceptable for the strategic planning of such a large scale network. The numerical results verify the necessity of the model for urban master plan horizon.Keywords: pedestrian modeling, Bi-level programming, Decision Making, Fuzzy logic, NSGA-II
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Journal of Quality Engineering and Production Optimization, Volume:3 Issue: 2, Summer - Autumn 2018, PP 11 -26Nowadays, coordination between members in a supply chain has become very important and beneficial to channel members. Through cooperative advertising, manufacturers and retailers can jointly participate in promotional programs. This action not only reduces the cost of advertising, but also is important to create a link with local retailers in order to increase immediate sales at the retail level. In this article, the problem of cooperative advertising and pricing decisions in a multi-product manufacturer-retailer (oligopoly market) supply chain is investigated. Stackelberg game with leadership of the manufacturer is proposed to model the problem. In order to find optimal prices and advertising expenditure, the bi-level programming approach is implemented. Solutions for the first level are determined by a genetic algorithm and best responses of retailers to the generated solutions of the manufacturer are calculated by CPLEX. Finally, numerical experiments and sensitivity analysis are conducted in order to assess the efficiency of models and solution procedures. Results show that competition will lead to a lower retail price, which is preferable from the consumers’ point of view. Also, profit of the manufacturer and retailers will decrease if competition effect increases.Keywords: Cooperative advertising, Pricing, Stackelberg game, genetic algorithm, Bi-level programming
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Journal of Optimization in Industrial Engineering, Volume:11 Issue: 24, Summer and Autumn 2018, PP 35 -54This paper presents a competitive supply chain network design problem in which one, two, or three supply chains are planning to enter the price-dependent markets simultaneously in uncertain environments and decide to set the prices and shape their networks. The chains produce competitive products either identical or highly substitutable. Fuzzy multi-level mixed integer programming is used to model the competition modes, and then the models are converted into an integrated bi-level one to be solved, in which the inner part sets the prices in dynamic competition and the outer part shapes the network cooperatively.Finally, a real-world problem is investigatedto illustrate how the bi-level model works and discuss how price, market share, total income, and supply chain network behave with respect to key marketing activities such as advertising, promotions, and brand loyalty.Keywords: Competitive supply chain network design, Fuzzy multi-level mixed integer programming, Bi-level programming, Nash equilibrium
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Journal of Industrial Engineering and Management Studies, Volume:4 Issue: 1, Winter-Spring 2017, PP 34 -54Motorized transportation systems in the urban areas witnessed huge developments in the infrastructures thanks to the advances in various aspects of technology. This urbanization revolution has its own pros and cons. The resulting dominance of vehicles has limited the presence of people in public places and their participation in social activities, threatening the human based lifestyle of the cities. Historic districts are of most affected areas which withstand the unwanted consequences of such an experience. These areas play a substantial role in urban activities by providing great social activity and walking zones for pedestrians. Hence, in recent years, urban management has paid attention to this endanger regions in order to sustain and enhance their properties by introducing some pedestrianization plan as urban regeneration policies. To design an effective plan, it is necessary to figure out how people behave in response to their environment. Pedestrian modeling is the key to the problem and is studied in the past few decades, mostly in microscopic scale. In addition, a logical decision-making process is required to choose the option with the best outcome in this complex system, considering financial limits of strategic urban planning. In this paper, a macroscopic multi-class user equilibrium pedestrian assignment algorithm is proposed to anticipate the route choice behavior of the pedestrians in a network, and a decision making platform for the pedestrian network design is presented using bi-level mathematical mixed-integer programming and genetic algorithm. The presented model determines the best possible projects to be implemented on the network, considering the constraints of the historic districts. The model brings forward an intelligent framework to help the urban planners in spending the minimum cost, while maximizing some predefined objectives. The proposed method is applied to solve the problem in a test network and in a real case scenario for the historic district of the city of Tehran. The results prove the validity and the efficiency of the algorithm.Keywords: Complex System, Pedestrian Flow Modeling, mathematical modeling, Bi-level programming, Multi-objective optimization, Genetic Algorithm
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Nowadays, the necessity of manufacturers response to their customers needs and their fields of activities have extended widely. The cellular manufacturing systems have adopted reduced costs from mass-production systems and high flexibility from job-shop manufacturing systems, and therefore, they are very popular in modern manufacturing environments. Manufacturing systems, in addition to proper machinery and equipment, workforces and their performance play a critical role.
Staff creativity is an important factor in product development, and their interest in cooperating with each other in the work environment can help the growth and maturity of this factor. In this research, two important aspects of cellular manufacturing take into consideration: Cell formation and workforce planning. Cell formation is a strategic decision, and workforce planning is a tactical decision. Practically, these two sectors cannot be planned simultaneously, and decision making in this regard is decentralized. For this reason, a bi-level mathematical model is proposed. The first level aims to reduce the number of voids and exceptional elements, and the second level tends to promote the sense of interest between the workforces for working together, which will result in synergy and growth of the organization.Keywords: Cellular Manufacturing, Bi-Level Programming, KKT, Worker's Interest -
Pricing is one of the major aspects of decision making in supply chain. In the previous works mostly a centralized environment is considered indicating the retailers cannot independently apply their decisions on the pricing strategy. Although in a two-echelon decentralized environment it may be possible that supply chain contributors have encountered with different market power situations which provide that some of them try to impose their interests in pricing and/or volume of the products. In such situations the leader-follower Stackelberg game or more specifically bi-level programming seems to be the best approach to overcome the problem. Furthermore, in this study we consider the impacts of disruption risk caused by foreign exchange uncertainty on pricing decisions in a multi-product two-echelon supply chain. Also it is assumed that the market is partitioned to domestic and international retailers with segmented market for each retailer. The purpose of this paper is to introduce decisions policy on the pricing such that the utility of both manufacturer and retailers is met. Since the proposed bi-level model is NP-hard, a simulated annealing method combining with Tabu search is proposed to solve the model. A numerical example is presented to investigate the effect of foreign exchange variation on the decision variables through different scenarios. The results from numerical example indicate that the international retailers are indifferent to the manufacture undergoes changes where the domestic retailers react to changes, dramatically.Keywords: Bi-level programming, Decentralized supply chain, pricing, Disruption risk, Simulated annealing, Tabu search
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International Journal of Research in Industrial Engineering, Volume:3 Issue: 1, Spring 2014, PP 1 -10In this paper a partial interdiction problem on a capacitated hierarchical system is studied. We consider an attacker who can interdict facilities at different levels and each interdiction level causes a specified reduction in the capacity of a facility depending upon its service level in the hierarchy. First, the interdictor identifies her interdiction strategy whose aim is to cause the most demand satisfaction cost subject to her budgetary limitation. Subsequently, the defender tries to optimize the objective function which is similar to the attacker’s one but in the opposite direction. The defender is responsible for choosing the least cost strategy in order to satisfy all customers’ demand. She can achieve this goal by two ways: (1) allocating their demand to the hierarchical facilities subject to their residual capacity, (2) benefiting from outsourcing option. This problem can be regarded as a static Stackelberg game between a malicious interdictor as the leader and a system defender as the follower. In this paper we propose a bi-level mathematical formulation in order to model the problem. To solve this problem with exhaustive enumeration, CPLEX has been used.Keywords: hierarchical capacitated facilities, Interdiction, Outsourcing, Bi-level programming
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