bi level programming

In this paper, a mathematical negotiation mechanism is designed to minimize the negotiators’ costs in a distributed procurement problem at two echelons of an automotive supply chain. The buyer’s costs are procurement cost and shortage penalty in a one-period contract. On the other hand, the suppliers intend to solve a multi-period, multi-product production planning to minimize their costs. Such a mechanism provides an alignment among suppliers’ production planning and order allocation, also supports the partnership with the valued suppliers by taking suppliers’ capacities into account. Such a circumstance has been modeled via bi-level programming, in which the buyer acts as a leader, and the suppliers individually appear as followers in the lower level. To solve this nonlinear bi-level programming model, a hybrid algorithm by combining the particle swarm optimization (PSO) algorithm with a heuristic algorithm based on A* search is proposed. The heuristic A* algorithm is embedded to solve the mixed-integer nonlinear programming (MINLP) sub-problems for each supplier according to the received variable values determined by PSO system particles (buyer’s request for quotations (RFQs)). The computational analyses have shown that the proposed hybrid algorithm called PSO-A* outperforms PSO-SA and PSO-Greedy algorithms.

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.

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.

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.

The 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.

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.

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.

The integration of the distributed energy resources into a single entity can do with virtual power plants. VPP is a cluster of dispatchable and non- dispatchable resource with flexible loads which distributed in allover the grid that aggregated and acts as a unique power plant. Flexible load is able to change the consumption so demand response program is applied to use them to improvement of the power system performance. Virtual power plant generation has uncertainty and it make hard to schedule the VPP. To deal this matter Information gap decision theory hint us to optimal schedule of the VPP. To show the effects of VPP and DRP on power system operation cost a bi-level unit commitment with regard the VPPs and DRP is solved in modified IEEE 24 bus reliability test system. Results in presence of VPP and DRP in both IGDT strategies are compared with disregard VPP and DRP and effectiveness of the proposed model is reflected.

Fast 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.

In this paper, we focus on solving the integrated energy and flexiramp procurement problem in the day-ahead market. The problem of energy and ramp procurement could be perfectly analyzed through Stackelberg concept, because of its hierarchical nature of the decision-making process. Such a circumstance is modeled via a bi-level programming, in which suppliers act as leaders and the ISO appear as the follower. The ISO intends to minimize the energy and spinning reserve procurement cost, and the suppliers aim to maximize their profit. To solve the proposed model, a fuzzy max-min approach is applied to maximize the players’ utilities. The objectives and suppliers’ dynamic offers, determined regarding the market clearing prices, are reformulated through fuzzy utility functions. The proposed approach is an effective and simple alternative to the KKT method, especially for problems with non-convex lower-level.

This study aims in providing a new approach regarding design of a closed loop supply chain network through emphasizing on the impact of the environmental government policies based on a bi-level mixed integer linear programming model. Government is considered as a leader in the first level and tends to set a collection rate policy which leads to collect more used products in order to ensure a minimum distribution ratio to satisfy a minimum demands. In the second level, private sector is considered as a follower and tries to maximize its profit by designing its own closed loop supply chain network according to the government used products collection policy. A heuristic algorithm and an adaptive genetic algorithm based on enumeration method are proposed and their performances are evaluated through computational experiences. The comparison among numerical examples reveals that there is an obvious conflict between the government and CLSC goals. Moreover, it shows that this conflict should be considered and elaborated in uncertain environment by applying Min-Max regret scenario based robust optimization approach. The results show the necessity of using robust bi-level programming in closed loop supply chain network design under the governmental legislative decisions as a leader-follower configuration.

The present study aimed to design a bi-objective bi-level mathematical model for multi-dimensional cellular manufacturing system. Minimizing the total number of voids and balancing the assigned workloads to cells are regarded as two objectives of the upper level of the model. However, the lower level attempts to maximize the workers' interest to work together in a special cell. To this aim, two nested bi-level metaheuristics including particle swarm optimization (NBL-PSO) and a population-based simulated annealing algorithm (NBL-PBSA) were implemented to solve the model. In addition, the goal programming approach was utilized in the upper level procedure of these algorithms. Further, nine numerical examples were applied to verify the suggested framework and the TOPSIS method was used to find the better algorithm. Furthermore, the best weights for upper level objectives were tuned by using a weight sensitivity analysis. Based on computational results, all three objectives were different from their ideal goals when decisions about inter and intra-cell layouts, and cell formation to balance the assigned workloads by considering voids and workers' interest were simultaneously madeby considering a wide assumption-made problem closer to the real world. Finally, NBL-PBSA could perform better than NBL-PSO, which confirmed the efficiency of the proposed framework.

Nowadays, 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.

This 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.

Motorized 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.

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.