• Journal of the Korean Operations Research and Management Science Society
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• v.15 no.2
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• pp.33-44
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• 1990

An efficient algorithm is developed for the linear programming relaxation of generalized multiple choice knaspack problem. The generalized multiple choice knaspack problem is an extension of the multiple choice knaspack problem whose relaxed LP problem has been studied extensively. In the worst case, the computational coimplexity of the proposed algorithm is of order 0(n. $n_{max}$)$^{2}$), where n is the total number of variables and $n_{max}$ denotes the cardinality of the largest multiple choice set. The algorithm can be easily embedded in a branch-and-bound procedure for the generalized multiple choice knapsack problem. A numerical example is presented and computational aspects are discussed.sed.

• Journal of Korean Institute of Industrial Engineers
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• v.23 no.4
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• pp.661-667
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• 1997

We consider a generalized problem of the continuous multiple choice knapsack problem and study on the LP relaxation of the candidate problems which are generated in the branch and bound algorithm for solving the generalized problem. The LP relaxed candidate problem is called the generalized continuous multiple choice linear knapsack problem and characterized by some variables which are partitioned into continuous multiple choice constraints and the others which only belong to simple upper bound constraints. An efficient algorithm of order O($n^2logn$) is developed by exploiting some structural properties and applying binary search to ordered solution sets, where n is the total number of variables. A numerical example is presented.

• Journal of Korean Institute of Industrial Engineers
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• v.22 no.3
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• pp.365-375
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• 1996

We consider an extension of the multiple choice linear knapsack problem and develop a fast algorithm of order $O(r_{max}n^2)$ by exploiting some new properties, where $r_{max}$ is the largest multiple choice number and n is the total number of variables. The proposed algorithm has convenient structures for the post-optimization in changes of the right-hand-side and multiple choice numbers. A numerical example is presented.

• Journal of Korean Institute of Industrial Engineers
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• v.21 no.4
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• pp.519-527
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• 1995

By finding some new properties, we develop an O($r_{max}n^2$) algorithm for the generalized multiple choice linear knapsack problem where $r_{max}$ is the largest multiple choice number and n is the total number of variables. The proposed algorithm can easily be embedded in a branch-and-bound procedure due to its convenient structure for the post-optimization in changes of the right-hand-side and multiple choice numbers. A numerical example is presented.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.21 no.45
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• pp.291-299
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• 1998

We present a variant for the generalized continuous multiple-choice knapsack problem[1], which additionally has the well-known generalized lower bound constraints. The presented problem is characterized by some variables which only belong to the simple upper bound constraints and the others which are partitioned into both the continuous multiple-choice constraints and the generalized lower bound constraints. By exploiting some extended structural properties, an efficient algorithm of order Ο($n^2$1og n) is developed, where n is the total number of variables. A numerical example is presented.

• Journal of Korean Elementary Science Education
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• v.23 no.4
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• pp.332-343
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• 2004

Multiple choice items have been widely used. However the difficulties in understanding and solving the items have not been known well. The purpose of this study is to analyze the difficulties and errors in the process of solving multiple choice items. Twelve multiple choice items were developed based on the Unit 5 Separation of Mixtures in the 4th grade. Four items which students had hardly given the correct answer were selected and six students were chosen for interview. Interview results were analyzed with regard to the errors in the process of solving the multiple choice items. The findings of this study are as follows: I) The students who misread and misunderstand the questions choose the incorrect answers. 2) Most of the students activate daily knowledge in the process of problem solving. 3) The students who have misconception with the daily knowledge or have no experiences choose incorrect answers, while students who activate both daily knowledge and school knowledge choose correct answer. 4) The students of high level commit errors mainly in the latter part of problem solving process, but the students of low level do from early.

• Journal of Korean Institute of Industrial Engineers
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• v.40 no.4
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• pp.396-403
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• 2014

The multi-divisional knapsack problem is defined as a binary knapsack problem where each mutually exclusive division has its own capacity. In this paper, we present an extension of the multi-divisional knapsack problem that has generalized multiple-choice constraints. We explore the linear programming relaxation (P) of this extended problem and identify some properties of problem (P). Then, we develop a transformation which converts the problem (P) into an LP knapsack problem and derive the optimal solutions of problem (P) from those of the converted LP knapsack problem. The solution procedures have a worst case computational complexity of order $O(n^2{\log}\;n)$, where n is the total number of variables. We illustrate a numerical example and discuss some variations of problem (P).

• Journal of Korean Institute of Industrial Engineers
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• v.9 no.2
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• pp.3-8
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• 1983

The multiple choice knapsack problem is modified. To solve this modified multiple choice knapsack problem, Lagrangian relaxation is used, and to take advantage of the special structure of subproblems obtained by decomposing this relaxed Lagrangian problem, a modified ranking algorithm is used. The K best rank order solutions obtained from each subproblem as a result of applying modified ranking algorithm are used to formulate restricted problems of the original problem. The optimality for the original problem of solutions obtained from the restricted problems is judged from the upper bound and lower bounds calculated iteratively from the relaxed problem and restricted problems, respectively.

• Journal of the Korean Operations Research and Management Science Society
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• v.17 no.1
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• pp.31-41
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• 1992

An extension of generalized linear multiple choice knapsack problem [1] is presented and an algorithm of order 0([n .n$_{max}$]$_{2}$) is developed by exploiting its extended properties, where n and n$_{max}$ denote the total number of variables and the cardinality of the largest multiple choice set, respectively. A numerical example is presented and computational aspects are discussed.sed.

• Communications of Mathematical Education
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• v.22 no.4
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• pp.489-503
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• 2008

It is not an easy matter to develop problems which help students understand mathematical concepts correctly and precisely. The aim of this paper is to review the merits and demerits of three problem types (i.e. one answer problems, multiple choice problems and proof problems) and to suggest some points that should be taken into consideration in problem making. First, we presented the merits and demerits of three types of problems by examining actual examples. Second, we discussed some examples of misleading problems and the ways to make desirable ones. Finally, on the basis of our examination and discussion, we suggested some points that should be kept in mind in problem making. The major suggestions are as follows; i) In making one answer problems, we should consider the possibility of sitting a solution by wrong precesses, ii) In formulating multiple choice tests which are layered for their easiness of grading, we should take into account the importance of checking whether the students are fully understanding the concepts, iii) We may depend on the previous research result that multiple choice tests for proof problems can be helpful for the students who have insufficient math background. Besides those suggestions, we made an overall proposal that we should endeavor to find ways to implement the demerits of each problem type and to develop instructive problems that can help students understanding of math.