• Journal of the Korean Operations Research and Management Science Society
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• v.16 no.2
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• pp.13-35
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• 1991

The purpose of this study is to develop an efficient exact algorithm for the problem of scheduling n in dependent jobs on m unequal parallel processors to minimize makespan. Efficient solutions are already known for the preemptive case. But for the non-preemptive case, this problem belongs to a set of strong NP-complete problems. Hence, it is unlikely that the polynomial time algorithm can be found. This is the reason why most investigations have bben directed toward the fast approximate algorithms and the worst-case analysis of algorithms. Recently, great advances have been made in mathematical theories regarding Lagrangean relaxation and the subgradient optimization procedure which updates the Lagrangean multipliers. By combining and the subgradient optimization procedure which updates the Lagrangean multipliers. By combining these mathematical tools with branch-and-bound procedures, these have been some successes in constructing pseudo-polynomial time algorithms for solving previously unsolved NP-complete problems. This study applied similar methodologies to the unequal parallel processor problem to find the efficient exact algorithm.

• Management Science and Financial Engineering
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• v.17 no.2
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• pp.1-21
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• 2011

This paper considers an integrated inventory-distribution system with a fleet of heterogeneous vehicles employed where a single warehouse distributes a single type of products to many spatially distributed retailers to satisfy their dynamic demands. The problem is to determine order planning at the warehouse, and also vehicle schedules and delivery quantities for the retailers with the objective of minimizing the sum of ordering cost at the warehouse, inventory holding cost at both the warehouse and retailers, and transportation cost. For the problem, we give a Mixed Integer Programming formulation and develop a Lagrangean heuristic procedure for computing lower and upper bounds on the optimal solution value. The Lagrangean dual problem of finding the best Lagrangrean lower bound is solved by subgradient optimization. Computational experiments on randomly generated test problems showed that the suggested algorithm gives relatively good solutions in a reasonable amount of computation time.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 1993.10a
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• pp.133-133
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• 1993

• Journal of the Korean Operations Research and Management Science Society
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• v.17 no.1
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• pp.107-112
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• 1992

A slight extension of the classical saddle point and the global optimality condition has been discussed relative to some algorithmic implications. It also involves an economic interpretation which shows satisfying, rather than optimizing, decision making behavior under bounded rationality.

• Journal of Korean Institute of Industrial Engineers
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• v.12 no.2
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• pp.21-31
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• 1986

Consider a capacitated facility location problem in which the demands of customers are all equal and so are the capacities of facilities. Under the restriction that each customer's uniform demand be met by exactly one facility, the objective is to select a set of facilities to open, and to assign customer's demand to them so as to minimize the total cost which includes fixed costs of opening facilities as well as variable assignment costs. The problem is modelled as a pure zero-one program which may be viewed as a variant of well-known capacitated facility location problems. The purpose of this study is to develop efficient computational procedures for solving the pure zero-one facility location problems. Due to the special structure of our zero-one location problem with uniform demand, it can be converted to a location problem with the unimodular property. A Lagrangean relaxation algorithm is developed to solve the location problem. The algorithm is made efficient by employing a device which exploits the special structure of a surrogate constraint. The efficiency of the algorithm is analyzed through computational experiments with some test problems.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2004.10a
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• pp.375-388
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• 2004

As today's business environment has become more and more competitive, forward as well as backward flows of products among members belonging to a supply chain have been increased. The backward flows of products, which are common in most industries, result from increasing amount of products that are returned, recalled, or need to be repaired. Effective management for these backward flows of products has become an important issue for businesses because of opportunities for simultaneously enhancing profitability and customer satisfaction from returned products. Since third party logistics service providers (3PLs) are playing an important role in reverse logistics operations, the 3PLs should perform two simultaneous logistics operations for a number of different clients who want to improve their logistics operations for both forward and reverse flows. In this case, distribution networks have been independently designed with respect to either forward or backward flows so far. This paper proposes a mixed integer programming model for the design of network integrating both forward and reverse logistics. Since this network design problem belongs to a class of NP-hard problems, we present an efficient heuristic based on Lagrangean relaxation and apply it to numerical examples to test the validity of proposed heuristic.

• Journal of the Korean Operations Research and Management Science Society
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• v.15 no.1
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• pp.99-115
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• 1990

This paper is concerned with the unit commitment problem in an electric power system with both thermal and pumped-storage hydroelectric units. This is a mixed integer programming problem and the Lagrangean relaxation method is used. We show that the relaxed problem decomposes into two kinds of subproblems : a shortest-path problem for each thermal unit and a minimum cost flow problem for each pumped-storage hydroelectric unit. A method of obtaining an incumbenet solution from the solution of a relaxed problem is presented. The Lagrangean multipliers are updated using both subgradient and incremental cost. The algorithm is applied to a real Korean power generation system and its computational results are reported and compaired with other works.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2002.05a
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• pp.426-433
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• 2002

This paper deals with an offline routing problem, which can be used as an explicit routing procedure in MPLS(Multiprotocol Label Switching) network, for traffic engineering. This problem is formulated as an MIP(Mixed Integer Programming) with the objective function which is to minimize the sum of the maximum link utilization for load balancing (link utilization) and the routing cost. Constraints arc composed of link capacity restriction and demand requirement that has origin-destination pair, bandwidth requirement and hop restriction. The problem is proved to be NP-hard so that the Lagrangean relaxation method is applied to propose a Lagrangean heuristic. To test the effectiveness & efficiency of the proposed algorithm, computational experiments are performed with numerical instances. The experiment results show that the proposed algorithm solves the problem within a reasonable time.

• Journal of the Korean Operations Research and Management Science Society
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• v.15 no.1
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• pp.23-35
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• 1990

A mathematical model for a dynamic capacitate facility location problem is formulated by a mixed integer problem. The objective of the model is to minimize total discounted costs that include fixed charges and distributed costs. The Cross Decomposition method of Van Roy is extended and applied to solve the dynamic capacitated facility location problem. The method unifies Benders Decomposition and Lagrangean relaxation into a single framework. It successively solves a transportation problem and a dynamic uncapacitated facility location problem as two subproblems. Computational results are compared with those of general mixed integer programming.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 1993.10a
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• pp.43-43
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• 1993

항공기 일정계획문제는 항공기, 승무원, 항공기 정비, 공항사용 등의 현재흐름을 시간과 공간의 개념을 도입한 Network상에 표현하여 분석하는 문제로서 여러가지 의사결정이 복합적으로 관련되어 있는 문제이다. 기존 연구에서는 운영상에 발생하는 많은 제약조건을 주어진 가정으로 처리하여 비교적 단순한 수리모형으로 최적해를 구하고자 했으므로 실제 적용에 한계를 가지고 있다. 또한, 최적해법으로 사용했던 최대흐름 알고리즘(Maximum Flow Algorithm)이나 단계적 접근법(Sequential Approach)으로는 변수의 수가 증가함에 따른 해법의 한계점을 가지고 있다. 본 연구에서는 1978년 미국의 항공규제완화 조치 후 많이 사용되고 있는 Hub and Spoke 노선상에서 항공기를 할당하는 문제로서, 보다 현실적인 제약요인을 포함하는 수리모형을 수립하고, Lagrangean Relaxation 기법을 사용하여 해를 구하고자 하였다.