• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.15-25
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• 2022

In 1963, Graham introduced a problem to find integer partitions such that the reciprocal sum of their parts is 1. Inspired by Graham's work and classical partition identities, we show that there is an integer partition of a sufficiently large integer n such that the reciprocal sum of the parts is 1, while the parts satisfy certain congruence conditions.

• Journal of Integrative Natural Science
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• v.12 no.2
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• pp.44-46
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• 2019

In this paper, we study a ratio variation of Nicomachus' theorem by computing ${\sum\limits_{k=1}^{n}}[rk]$ and ${\sum\limits_{k=1}^{n}}[rk]^3$, where r is a rational number.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2003.05a
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• pp.369-374
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• 2003

Disassembly scheduling is the problem or determining the ordering and disassembly schedules or used or end-of-life products while satisfying the demand or their parts or components over a certain planning horizon. To represent and optimally solve the basic case. i e ., single product type without parts commonality, an Integer programming model is suggested for the objective or minimizing the sum or purchase, setup, inventory holding, and disassembly operation costs. Then, the basic model is offended to consider the cases or single and multiple product types with parts commonality. Computational experiments done on the examples obtained from the literature show that the integer programming approach suggested in this paper works well.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1017-1024
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• 2023

A celebrated result in the study of integer partitions is the identity due to Lehmer whereby the number of partitions of n with an even number of even parts minus the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct odd parts. Inspired by Lehmer's identity, we prove explicit formulas for evaluating generating functions for sequences that enumerate integer partitions of fixed width with an even/odd number of even parts. We introduce a technique for decomposing the even entries of a partition in such a way so as to evaluate, using a finite sum over q-binomial coefficients, the generating function for the sequence of partitions with an even number of even parts of fixed, odd width, and similarly for the other families of fixed-width partitions that we introduce.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1279-1287
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• 2022

A partition of n is complete if every positive integer from 1 to n can be represented by the sum of its parts. The concept of complete partitions has been extended in several ways. In this paper, we consider the number of k-relaxed r-complete partitions of n and the number of double-complete partitions of n.

• Management Science and Financial Engineering
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• v.11 no.1
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• pp.63-78
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• 2005

Disassembly scheduling is the problem of determining the ordering and disassembly schedules of used or end-of-life products while satisfying the demand of their parts or components over a certain planning horizon. This paper considers the case of the assembly product structure for the cost-based objective of minimizing the sum of purchase, setup, inventory holding, and disassembly operation costs. To represent and solve the problem optimally, this paper presents an integer programming model, which is a reversed form of the multi-level lot sizing formulation. Computational experiments on an example derived from the literature and a number of randomly generated test problems are done and the results are reported.

• Journal of Intelligence and Information Systems
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• v.4 no.1
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• pp.1-10
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• 1998

Process planning is a very complex task and requires the dynamic informatioon of shop foor and market situations. Process plan selection is one of the main problems in the process planning. In this paper, we propose a new process plan selection model considering operation flexibility for the computer aided process planing. The model is formulated as a 0-1 integer programming considering realistic shop factors such as production volume, machining time, machine capacity, transportation time and capacity of tractors such as production volume, machining time, machine capacity, transportation time capacity of transfer device. The objective of the model is to minimize the sum of the processing and transportation time for all parts. A genetic algorithm a, pp.oach is developed to solve the model. The efficiency of the proposed a, pp.oach is verified with numerical examples.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.23 no.54
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• pp.13-26
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• 2000

Machining route selection to produce parts should be based on shop flow information because of input data at scheduling tasks and is one of the main problem in process planning. This paper addresses the problem of machining route selection in multi-stage process with machine group included a similar function. The model proposed is formulated as 0-1 integer programing considering the relation of parts and machine table size, avaliable time of each machine for planning period, and delivery date. The objective of the model is to minimize the sum of processing, transportation, and setup time for all parts. Genetic algorithm approach is developed to solve this model. The efficiency of the approach is examined in comparison with the method of branch and bound technique for the same problem. Also, this paper is to solve large problem scale and provide it if the multiple machining routes are existed an optimal solution.

• Journal of Korean Institute of Industrial Engineers
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• v.21 no.3
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• pp.329-343
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• 1995

Traditionally, machining process sequence was influenced and constrained by the design information obtained from CAD data base, i.e., class of operations, geometric shape, tooling, geometric tolerance, etc. However, even though all the constraints from design information are considered, there may exist more than one way to feasibly machine parts. This research is focused on the integrated problem of operations sequencing and machine tools selection in the presence of the product mix and their production volumes. With the transitional costs among machining operations, the operation sequencing problem can be formulated as a well-known Traveling Salesman Problem (TSP). The transitional cost between two operations is expressed as the sum of total machining time of the parts on a machine for the first operation and transportation time of the parts from the first machine to a machine for the second operation. Therefore, the operation sequencing problem formulated as TSP cannot be solved without transitional costs for all operation pairs. When solved separately or serially, their mutual optima cannot be guaranteed. Machining operations sequencing and machine tool selection problems are two core problems in process planning for discretely machined parts. In this paper, the interrelated two problems are integrated and analyzed, zero-one integer programming model for the integrated problem is formulated, and the solution methods are developed using a Tabu Search technique.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 2005.05a
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• pp.715-722
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• 2005

Disassembly scheduling is the problem of determining the quantity and timing of disassembling used products while satisfying the demand of their parts or components over a planning horizon. The case of single product type with assembly structure is considered for the objective of minimizing the sum of disassembly operation and inventory holding costs. In particular, the resource capacity constraint is explicitly considered. The problem is formulated as an integer programming model, and a two-stage heuristic with construction and improvement algorithms is suggested in this paper. To show the performance of the heuristic, computational experiments are done on a number of randomly generated problems, and the test results show that the algorithm can give near optimal solutions within a very short amount of computation time.