ISSN:
1547-5816
eISSN:
1553-166X
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Journal of Industrial & Management Optimization
October 2006 , Volume 2 , Issue 4
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2006, 2(4): 351-371
doi: 10.3934/jimo.2006.2.351
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Abstract:
In this paper we propose a new arithmetic and a novel order relation for interval numbers. We shall show that this new interval arithmetic satisfies some operational properties and has the merit that it reduces uncertainties coming from classic ones. We shall also show that the order relation introduced in this paper is a linear or partial order (satisfying reflexivity, anti-symmetry and transitivity), and has the property of comparability. This is in contrast to the existing interval orders which do not have the property of comparability. Numerical examples on profit intervals and uncertain interval data will be presented to demonstrate the usefulness and applicability of these new arithmetic and order relations.
In this paper we propose a new arithmetic and a novel order relation for interval numbers. We shall show that this new interval arithmetic satisfies some operational properties and has the merit that it reduces uncertainties coming from classic ones. We shall also show that the order relation introduced in this paper is a linear or partial order (satisfying reflexivity, anti-symmetry and transitivity), and has the property of comparability. This is in contrast to the existing interval orders which do not have the property of comparability. Numerical examples on profit intervals and uncertain interval data will be presented to demonstrate the usefulness and applicability of these new arithmetic and order relations.
2006, 2(4): 373-385
doi: 10.3934/jimo.2006.2.373
+[Abstract](2084)
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Abstract:
In this paper we present a novel approach to a class of constrained nonlinear programming problems with interval objective functions. In this approach we first introduce a new concept of optimal solutions to the nonlinear programming problem, based on a new linear order relation for interval numbers proposed in Part 1 of this series. We then propose two efficient methods for the solution of the nonlinear optimization problem with respect to the new interval order relation. Comparisons between our approach and some existing methods will be given using illustrative examples. The numerical results show the superiority of our method to the existing ones.
In this paper we present a novel approach to a class of constrained nonlinear programming problems with interval objective functions. In this approach we first introduce a new concept of optimal solutions to the nonlinear programming problem, based on a new linear order relation for interval numbers proposed in Part 1 of this series. We then propose two efficient methods for the solution of the nonlinear optimization problem with respect to the new interval order relation. Comparisons between our approach and some existing methods will be given using illustrative examples. The numerical results show the superiority of our method to the existing ones.
2006, 2(4): 387-398
doi: 10.3934/jimo.2006.2.387
+[Abstract](1639)
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Abstract:
Second and third shift considerations are important for production planning and scheduling in most manufacturing facilities. A careful literature survey revealed that multiple workshift considerations have not been previously discussed for general aggregate production planning models. In this paper, we propose a mixed integer logic for permitting the opening or closing of second and third shifts of production within a generic model having linear and quadratic cost components. We illustrate the results of the model on a hypothetical data set. The model permits different costs for opening and closing the extra shifts and provides for a minimum required work force size for the extra shifts.
Second and third shift considerations are important for production planning and scheduling in most manufacturing facilities. A careful literature survey revealed that multiple workshift considerations have not been previously discussed for general aggregate production planning models. In this paper, we propose a mixed integer logic for permitting the opening or closing of second and third shifts of production within a generic model having linear and quadratic cost components. We illustrate the results of the model on a hypothetical data set. The model permits different costs for opening and closing the extra shifts and provides for a minimum required work force size for the extra shifts.
2006, 2(4): 399-423
doi: 10.3934/jimo.2006.2.399
+[Abstract](1768)
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Abstract:
In this paper, we obtain results on the good ''choice of techniques'' in the long-run in a model proposed by Robinson, Solow, and Srinivasan. We study this model without an assumption that a utility function which represents the preferences of the planner is concave and establish the existence of good programs.
In this paper, we obtain results on the good ''choice of techniques'' in the long-run in a model proposed by Robinson, Solow, and Srinivasan. We study this model without an assumption that a utility function which represents the preferences of the planner is concave and establish the existence of good programs.
2006, 2(4): 425-434
doi: 10.3934/jimo.2006.2.425
+[Abstract](1928)
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Abstract:
Recently, Frommer, Lang, and Schnurr [4] presented an existence test, which can be used to prove the existence of a zero of a continuous mapping from $\R^n $ to $\R^n$. The existence test relies on Miranda's theorem and was shown to be more powerful than the Moore test [10]. In this paper, we show that under additional assumptions, the Moore and Kioustelidis test [11] can be applied successfully in more cases than that of Frommer, Lang, and Schnurr [4]. For instance, these assumptions are fulfilled concerning the linear complementarity problem with interval data.
Recently, Frommer, Lang, and Schnurr [4] presented an existence test, which can be used to prove the existence of a zero of a continuous mapping from $\R^n $ to $\R^n$. The existence test relies on Miranda's theorem and was shown to be more powerful than the Moore test [10]. In this paper, we show that under additional assumptions, the Moore and Kioustelidis test [11] can be applied successfully in more cases than that of Frommer, Lang, and Schnurr [4]. For instance, these assumptions are fulfilled concerning the linear complementarity problem with interval data.
2006, 2(4): 435-450
doi: 10.3934/jimo.2006.2.435
+[Abstract](2688)
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Abstract:
In this paper, we develop a global computational approach to a class of optimal control problems governed by impulsive dynamical systems and subject to continuous state inequality constraint. We show that this problem is equivalent to an optimal control problem governed by ordinary differential equations with periodic boundary conditions and subject to a set of the continuous state inequality constraints. For this equivalent optimal control problem, a constraint transcription method is used in conjunction with a penalty function to construct an appended new cost functional. This leads to a sequence of approximate optimal control problems only subject to periodic boundary conditions. Each of these approximate problems can be solved as an optimization problem using gradient-based optimization techniques. However, these techniques are designed only to find local optimal solutions. Thus, a filled function method is introduced to supplement the gradient-based optimization method. This leads to a combined method for finding a global optimal solution. A numerical example is solved using the proposed approach.
In this paper, we develop a global computational approach to a class of optimal control problems governed by impulsive dynamical systems and subject to continuous state inequality constraint. We show that this problem is equivalent to an optimal control problem governed by ordinary differential equations with periodic boundary conditions and subject to a set of the continuous state inequality constraints. For this equivalent optimal control problem, a constraint transcription method is used in conjunction with a penalty function to construct an appended new cost functional. This leads to a sequence of approximate optimal control problems only subject to periodic boundary conditions. Each of these approximate problems can be solved as an optimization problem using gradient-based optimization techniques. However, these techniques are designed only to find local optimal solutions. Thus, a filled function method is introduced to supplement the gradient-based optimization method. This leads to a combined method for finding a global optimal solution. A numerical example is solved using the proposed approach.
2006, 2(4): 451-466
doi: 10.3934/jimo.2006.2.451
+[Abstract](1912)
+[PDF](259.1KB)
Abstract:
The successive projection algorithms were introduced by von Neuman [11] and Brègman [3]. These algorithms have a broad applicability in medical imaging, computerized tomography or signal processing. In this paper we focus on a particular projection sequence in which the points are successively projected onto the convex set from which they are the most distant. Among other things, we analyze the convergence of the algorithm and propose a finite termination criterion allowing for the analysis of the complexity of the algorithm. In line with the seminal work by Brègman, an application to linear optimization problems is proposed. This issue illustrates that successive projection algorithms may have some interest for very large finite dimensional problems.
The successive projection algorithms were introduced by von Neuman [11] and Brègman [3]. These algorithms have a broad applicability in medical imaging, computerized tomography or signal processing. In this paper we focus on a particular projection sequence in which the points are successively projected onto the convex set from which they are the most distant. Among other things, we analyze the convergence of the algorithm and propose a finite termination criterion allowing for the analysis of the complexity of the algorithm. In line with the seminal work by Brègman, an application to linear optimization problems is proposed. This issue illustrates that successive projection algorithms may have some interest for very large finite dimensional problems.
2006, 2(4): 467-479
doi: 10.3934/jimo.2006.2.467
+[Abstract](1864)
+[PDF](279.8KB)
Abstract:
This paper uses an ant system (AS) meta-heuristic optimization method to solve the problem of structure optimization of series-parallel production systems. In the considered problem, redundant machines (elements) and buffers in process are included in order to attain a desirable level of reliability. A procedure which determines the minimal cost system configuration is proposed. In this procedure, multiple choices of producing machines and buffers are allowed from a list of product available in the market. The elements of the system are characterized by their cost, estimated average up and down times, productivity rates and buffers capacities. The reliability is defined as the ability to satisfy the consumer demand which is represented as a piecewise cumulative load curve. The proposed meta-heuristic is used as an optimization technique to seek for the optimal design configuration. The advantage of the proposed AS approach is that allows machines and buffers with different parameters to be allocated.
This paper uses an ant system (AS) meta-heuristic optimization method to solve the problem of structure optimization of series-parallel production systems. In the considered problem, redundant machines (elements) and buffers in process are included in order to attain a desirable level of reliability. A procedure which determines the minimal cost system configuration is proposed. In this procedure, multiple choices of producing machines and buffers are allowed from a list of product available in the market. The elements of the system are characterized by their cost, estimated average up and down times, productivity rates and buffers capacities. The reliability is defined as the ability to satisfy the consumer demand which is represented as a piecewise cumulative load curve. The proposed meta-heuristic is used as an optimization technique to seek for the optimal design configuration. The advantage of the proposed AS approach is that allows machines and buffers with different parameters to be allocated.
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