# Bi-level programming

## Top PDF Bi-level programming:

### Integrating Goal Programming, Taylor Series, Kuhn-Tucker Conditions, and Penalty Function Approaches to Solve Linear Fractional Bi-level Programming Problems

Abstract. In this paper, we integrate goal programming (GP), Tay- lor Series, Kuhn-Tucker conditions and Penalty Function approaches to solve linear fractional bi-level programming (LFBLP)problems. As we know, the Taylor Series is having the property of transforming fractional functions to a polynomial. In the present article by Taylor Series we obtain polynomial objective functions which are equivalent to fractional objective functions. Then on using the Kuhn-Tucker optimality condition of the lower level problem, we transform the linear bilevel programming problem into a corresponding single level programming. The complemen- tary and slackness condition of the lower level problem is appended to the upper level objective with a penalty, that can be reduce to a single objective function. In the other words, suitable transformations can be applied to formulate FBLP problems. Finally a numerical example is given to illustrate the complexity of the procedure to the solution.

### A robust bi-level programming model for designing a closed-loop supply chain considering government's collection policy

work design problem under the governmental legislative decisions as a leader-follower conguration. Indeed, the government as the leader seeks to set the widely used product collection rate policy by ensuring the achievement of at least a predened satised demand rate. On the other hand, the private sector as a follower sets its CLSC design to determine the loca- tion of distribution and collection centers among a set of candidate sites and to obtain the highest net prot subject to the government regulation. Heuristic and genetic algorithms based on enumeration were proposed for the model. Numerical examples were randomly generated and used to test and evaluate eciency of the solution approaches. Computational results showed that the proposed genetic algorithm could obtain a near-optimal solution in large-scale instances in a reasonable amount of time, compared with the enumeration approach. Besides, a min-max regret and min-sum regret based bi-level programming approaches were proposed to incorporate uncertainty of demands. The numerical comparisons conrm their necessity as well as their eciency. Considering the uncertainty of other parameters in the proposed model is suggested as for a future study. Moreover, the application of other stochastic and robust approaches to bi-level programming can be considered as another future work direction.

### An Effective Branch-and-cut algorithm in Order to Solve the Mixed Integer Bi-level Programming

An interactive process in which a central unit (leader) coordinates a lower level unit (follower) is called hierarchical organizations. When the follower afforded some level of autonomy, this process becomes more complex to implement, coordinate and optimize. Moreover, in some instances, the objectives of the follower may conflict with those of the leader. In mathematical programming environment, these interactive processes are existed and known as bi-level or multi-level programming. Linear bi-level programming problems (BLPP) generally involve a hierarchy of two optimization problems, in the following form:

### A bi-level programming approach to coordinating pricing and ordering decisions in a multi-channel supply chain

This paper investigates the Stackelberg equilibrium for pricing and ordering decisions in a multi- channel supply chain. We study a situation where a manufacturer is going to open a direct online channel in addition to n existing traditional retail channels. It is assumed that the manufacturer is the leader and the retailers are the followers. The situation has a hierarchical nature and is formulated as a bi-level programming problem. The upper level problem is a mathematical model dealing with decisions of the manufacturer, while the lower level is a Nash equilibrium model determining the retail prices and order quantities by formulating the competition between the physical retailers. We consider a price-sensitive linear demand model with an additive uncertain part and analyze the optimal decisions for each sales channel. To enable supply chain coordination, we propose a particular revenue-sharing contract. This contract enables the retailers to set pricing and ordering policies that are equivalent to those in an integrated supply chain. Finally, we examine the impact of the model parameters on the equilibrium with a comprehensive numerical study.

### A Modified Simplex Method for Solving Linear-Quadratic and Linear Fractional Bi-Level Programming Problem

The bi-level programming problem (BLP) is a suitable method for solving the real and complex problems in applicable areas such as management, economics, policies and planning and so on. There are several forms of the BLP as an NP- hard problem. The linear-quadratic bi-level programming (LQBP) and the linear-fractional bi-level programming (LFBP) are important forms of the BLP. In this paper, we attempt to develop two effective approaches, one based on modified simplex method and the other based on the genetic algorithm for solving the LQBP and LFBP. To obtain efficient upper bound and lower bound we employ the Karush -Kuhn -Tucker (KKT) conditions for transforming the LQBP into a single level problem. By using the proposed penalty functions, the single problem is transformed to an unconstraint problem and then it is solved by modified simplex method and genetic algorithm. The proposed approach achieves efficient and feasible solution and it is evaluated by comparing with references.

### Pricing and advertising decisions in a dominant-retailer supply chain: A multi-follower bi-level programming approach

This study considered a two-stage supply chain consisting of one dominant retailer and multiple com- petitive manufactures which produce several perishable and substitutable products. The aim of this study was investigation of coordination of pricing and cooperative advertising in a decentralized supply chain. In this decentralized chain, the dominant retailer has more power to control other members' decisions. Hence, the former plays the leader role, and the manufacturers are his/her followers. Each member decides on the prices, advertising expenditures, as well as production or purchase amount. This problem was modeled as a multi-follower bi-level programming model.

### Pricing decisions in a two-echelon decentralized supply chain using bi-level programming approach

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.

### A bi-level programming model for decentralized manufacturer-distributer supply chain considering cooperative advertising

y i j = 0 or 1; i = 1; 2; ; I; j = 1; 2; ; J: The optimal solution to a bi-level programming prob- lem is referred to as Stackelberg Equilibrium. Consid- ering the presence of non-linear functions as objective and constraints in both the proposed models and the complexity of solving a bi-level programming problem that, even in its simplest form (linear objective function and constraint), has been proved to be NP-hard, two genetic algorithms with hierarchical structure are developed to provide near optimal solutions for the bi-level programming models in the next section. It is worth noting that although algorithms based on a local search structure [31-35] can be considered to deal with the research problem, algorithms accompanied by a population-based structure present better perfor- mance. The reason lies in the fact that a basic local search algorithm providing a single solution in each iteration might not be able to generate good quality solutions for such a complex problem in a reasonable time.

### Research of Flow Allocation Optimization in Hybrid Software Defined Networks Based on Bi-level Programming

As a transition from Traditional Networks to Software Defined Networks, Hybrid Software Defined Networks (SDN) shows significant research value. Hybrid SDN successfully faces the challenges that SDN comes with, such as robustness and scalability. In this paper, we describe the network architecture of flow-based hybrid software defined networks. We show how to distribute different flows in flow-based hybrid SDN to realize optimization of the entire network, using models of bi-level programming and stochastic user equilibrium.

### Coordination of pricing and cooperative advertising for perishable products in a two-echelon supply chain: A bi-level programming approach

Since the structure of a bi-level programming problem (BLPP) facilitates the formulation of the problems that involve a hierarchical decision making process, the considered problem will be modeled as a bi-level programming model in contrast to a single level model. Also, bi-level programming is practical for consecutive decision making. In the literature, it is mostly assumed that the manufacturer is the dominant player as the leader (e.g. Wang 2002,Corebtt et al. 2004). In some cases, it is recognized that the dominant manufacturer is not appropriate (see Chu and Messinger 1997). In this case, no member is dominant. In the case of powerful international retailer chain (such as Wal-Mart, Tesco, etc) the retailer is dominant (see Tsay 2002, Ertek and Griffin 2002). Because it is assumed that the manufacture plays a more significant role than the retailer, the manufacturer is considered as the first level (leader) and the retailer as the second one (follower).

### Chance Constrained Linear Plus Linear Fractional Bi level Programming Problem

In this paper, the concept of Pramanik and Banerjee is extended to chance constrained linear plus linear fractional bi- level programming problem (CCLPLFBLPP). In bi-level programming problem (BLPP), two types of decision makers (DMs) i.e. upper level decision maker (ULDM) and lower level decision maker (LLDM) execute their decision in hierarchical way. Each level DM independently controls a set of decision variables. Candler and Townsley [16] as well as Fortuny –Amart and McCarl [17] developed the formal BLPP. After that many researchers [18, 19] studied BLPP in various perspectives. Sakawa and Nishizaki [20, 21] introduced linear fractional BLPP using interactive fuzzy programming. Pramanik and Dey [22] presented bi-level linear fractional programming problem based on FGP using first order Taylor’s series approximation.

### A Metaheuristic Framework for Bi-level Programming Problems with Multi-disciplinary Applications

Abstract Bi-level programming problems arise in situations when the decision maker has to take into account the responses of the users to his decisions. Several problems arising in engineering and economics can be cast within the bi-level pro- gramming framework. The bi-level programming model is also known as a Stack- leberg or leader-follower game in which the leader chooses his variables so as to optimise his objective function, taking into account the response of the follower(s) who separately optimise their own objectives, treating the leader’s decisions as ex- ogenous. In this chapter, we present a unified framework fully consistent with the Stackleberg paradigm of bi-level programming that allows for the integration of meta-heuristic algorithms with traditional gradient based optimisation algorithms for the solution of bi-level programming problems. In particular we employ Differ- ential Evolution as the main meta-heuristic in our proposal. We subsequently apply the proposed method (DEBLP) to a range of problems from many fields such as transportation systems management, parameter estimation and game theory. It is demonstrated that DEBLP is a robust and powerful search heuristic for this class of problems characterised by non smoothness and non convexity.

### Designing the optimum plan for regenerating the pedestrian network of historic districts using bi-level programming (Case study: Historical-Cultural district of Tehran, Iran)

The main purpose of this paper is to tackle three drawbacks in the literature: 1- No model exists to anticipate the distribution of the pedestrians with different behavior patterns in a large scale network. 2- There is no systematic decision making platform to choose the optimize option of urban projects in a sidewalk network, considering the limitations and objectives. In fact, the urban planners lay the schemes based on their experience and knowledge. The effectiveness of the process is bounded by the constraints of the human mind. 3-The decisions are not made systematically, since their effects are not exactly and quantitatively measured. Considering these problems, we first aim to propose a macroscopic origin-destination pedestrian model to consider the heterogeneity of the travelers and applicable to the large-scale networks. Second, we present a model as a decision making structure for urban planners. In this paper, a bi-level programming model is presented in which the lower level is a multi-class user equilibrium pedestrian assignment algorithm and the upper level problem is a multi-objective optimization model based on the Non-dominated Sorting Genetic Algorithm (NSGA-II). The paper is structured as follows. In the next section, the pedestrian network design problem and the proposed model are presented. Section 3 describes the framework and algorithmic design of the proposed model. In order to evaluate the performance of the method, numerical examples are provided in section 4. Finally, some concluding remarks are given in the last section.

### An application of data mining classification and bi-level programming for optimal credit allocation

This paper investigates credit allocation policy making and its effect on economic development using bi-level programming. There are two challenging problems in bi-level credit allocation; at the first level government/public related institutes must allocate the credit strategically concerning sustainable development to regions and industrial sectors. At the second level, there are agent banks, which should allocate the credit tactically to individual applicants based on their own profitability and risk using their credit scoring models. There is a conflict of interest between these two stakeholders but the cooperation is inevitable. In this paper, a new bi-level programming formulation of the leader-follower game in association with sustainable development theory in the first level and data mining classifier at the second level is used to mathematically model the problem. The model is applied to a national development fund (NDF) as a government related organization and one of its agent banks. A new algorithm called Bi-level Genetic fuzzy apriori Algorithm (BGFAA) is introduced to solve the bilateral model. Experimental results are presented and compared with a unilateral policy making scenario by the leader. Findings show that although the objective functions of the leader are worse in the bilateral scenario but agent banks collaboration is attracted and guaranteed.

### A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows

The BP problem may be seen as a single programming problem with the upper-level variables being constrained by the lower-level solutions. In this sense, a BP problem is similar to a non- linear programming (NP) problem. However, in a BP problem, the evaluation of the upper- level objective function requires solving the lower-level optimisation problem whose functional form is generally unknown. A further complication is that a BP problem is in general non-convex. This implies the potential existence of local minimum solutions and so a global minimum may be difficult to find. The EP problem is similar to a multiple-objective NP problem in that there are two objectives. However, in a NP problem, there is only one decision maker who chooses all variables so as to optimise several objectives. In an EP problem, on the other hand, there are two decision makers, each having control of only one set of variables. (A more general EP problem can have more than two parties.) It is clear that most standard algorithms for NP problems may not be applicable to BP and EP problems.

### Bi-level programming for supplier selection under quantity discount policy

BLP is a proper model to deal with non- cooperative decision-makers in decision-making pro- cesses [3]. The aim of the BLP is to formulate problems with hierarchical structure and two levels of decision- making [4]. The leader, at a higher level of hierarchy, follows a specic strategy, and the follower, at a lower level of hierarchy, determines its strategy subsequently. In such a non-cooperative game, each decision-maker controls a set of decision variables and tries to opti- mize his/her own objective function [5]. Stackelberg reported the follower as the dominated member of the game which is controlled by the leader. Given the estimation of the followers' reaction, the leader takes the rst step [6]. According to its characteristics, the BLP approach has been applied in so many elds, i.e. environmental economic problem [7], optimal design engineering [8], mechanical engineering, decentralized resource planning, logistics and transportation [9], civil engineering, electric power markets [10], and so on.

### A Non-linear Integer Bi-level Programming Model for Competitive Facility Location of Distribution Centers

The facility location problem is a strategic decision-making for a supply chain which determines the profitability and sustainability of its components. This paper deals with a scenario where two supply chains, consisting of a producer, a number of distribution centers and several retailers provided with similar products, compete to maintain their market shares by opening new distribution centers because of increasing demand. The competition problem is formulated as a non-linear integer bi-level mathematical model, where the upper level represents the decisions of the leader producer and the lower level administrates the decisions of the follower producer. It has been shown that even in small- scale problems, bi-level mathematical programming problems are strongly NP-hard, so an adapted bi- level ant colony algorithm with inter-level information sharing is developed to solve the problem.To evaluate the performance of the developed ant colony algorithm, the upper bound of the competitive facility location problem is determined by solving the upper-level problem as an integer linear programming model without considering the follower’s decision. Comparing the computational results of the developed ant colony algorithm with those of the determined upper bounds shows the satisfactory capability of the proposed approach for solving even medium- and large-scale problems.

### On the Solution of Rough Goal Bi-Level Multi-Objective Linear Programming Problem

Bi-level programming is a powerful and robust technique for solving hierarchical decision making problem. It has been applied in many real life problems such as agriculture, bio-fuel production, economic systems, finance, engineering, banking, management sciences, and transportation problem ([1], [15], [17], [18], [21]). Goal programming is one powerful tool that has been proposed for the modeling, analysis and solution of multi-objective optimization problems [6]. Dauer and Krueger in [4] suggested an iterative goal programming approach for solving multi objective nonlinear programming problems [2].

### A bi-level decision model for customer churn analysis

Bi-level decision problems have been studied for decades. A large part of the research on bi-level decision problems has centered on linear bi-level decision problems, for which nearly two dozen algorithms have been developed. The well- known ones include the Kuhn-Tucker approach (Bard 1998), the Kth-best algorithm (Zhang, Lu and Dillon 2006), the Branch-and-bound algorithm (Lu, Shi and Zhang 2006b), and heuristic based approaches (Murata and Ishibuchi 1995). For bi-level decision problems with non-linear formats for either the objective functions or constraints functions from the leader or follower, particle swarm optimization is an emerging and competitive solution strategy that has the advantages of high computation efficiency and effectiveness (Gao, Zhang and Lu 2009) . Current research into bi-level programming techniques includes nonlinear (Gao, Zhang and Lu 2009), multi-leader (Zhang, et al 2010), multi-follower (Lu, Shi and Zhang 2006a), multi-objective (Feng and Wen 2005), and fuzzy bi-level decision problems (Zhang and Lu 2007).