Discrete Sizing Research Articles

Simultaneous lot sizing and scheduling for multi-product multi-level production

The paper deals with the multi-level lot sizing and scheduling problem for job shop production in capacitated, dynamic and deterministic cases. Demand is to be fulfilled without backlogging. The multi-level lot sizing and scheduling models of the literature are small bucket problems such as the discrete lot sizing and scheduling problem (DLSP) and the proportional lot sizing and scheduling problem (PLSP). In contrast we develop a big bucket model for multi-product multi-level production. The multi-level general lot sizing and scheduling problem with multiple machines (MLGLSP_MM) is based on the general lot sizing and scheduling problem (GLSP) for single-level production. Our focus is on minimising the sum of sequence dependent setup costs, inventory costs, production costs and the costs of maintaining the setup conditions of the machines. Inventory costs will be calculated exactly. In addition sequence dependent setup times are considered. Finally, our approach minimises the lead-time of semi-finished goods and the throughput time of finished goods.

Multi-database exploration of large design spaces in the framework of cascade evolutionary structural sizing optimization

In discrete sizing optimization of truss and frame structures the design variables take values from databases, which are usually populated with a relatively small number of cross-section types and sizes. The aim of this work is to allow the use of large-size databases in discrete structural sizing optimization problems, in order to enrich the set of design variable options and increase the potential of achieving high-quality optimal designs. For this purpose, the concept of coarse database is introduced, according to which smaller-size versions of an appropriately ordered large database can be constructed. This concept is combined with the idea of cascading, which allows a single optimization problem to be tackled with a number of autonomous optimization stages. Under this context, several coarse versions of the same full-size database are formed, in order to utilize a different database in each cascade stage executed with an evolutionary optimization algorithm. The first optimization stages of the resulting multi-database cascade procedure make use of the coarsest database versions available and serve the purpose of basic design space exploration. The last stages exploit finer databases (including the original full-size database) and aim in fine tuning the achieved optimal solution. Based on the reported numerical results, multi-database cascading proves to be an effective tool for the handling of large databases and corresponding extensive design spaces in the framework of discrete structural sizing optimization applications.

Strong turnpike policies in the single-item capacitated lot-sizing problem with periodical dynamic parameter

This article considers optimization problems in a discrete capacitated lot sizing model for a single product with limited backlogging. The demand as well as the holding and backlogging costs are assumed to be periodical in time. Nothing is assumed about types of the cost functions. It is shown that there exists an optimal infinite inverse policy and a strong turnpike policy. A forward algorithm for computing optimal policies relative to the class of batch ordering type policies is derived. Some backward procedure is adopted to determine a strong turnpike policy. The algorithm is simple, and it terminates after the a number of steps equal to the turnpike horizon. Some remarks on the existence of rolling horizontal plans and forecast horizons are also given. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 775–790, 1997

Optimal discrete sizing of truss structures subject to buckling constraints

The selective dynamic rounding (SDR) algorithm previously developed by the authors, and based on a dual step rounding approach, is used for the optimal sizing design of truss structures subject to linear buckling constraints. The algorithm begins with a continuous optimum followed by a progressive freezing of individual variables while solving the remaining continuous problems. The allowable member stresses are predicted by the linear regression of the tabular section properties, while the exact allowable compressive stresses are back-substituted for those variables fixed on discrete values in each intermediate mixed-discrete nonlinear problem. It is shown that a continuous design based on the regression analysis of section effectiveness vs. area is effective as a starting point for the dual step discrete optimization phase. A range of examples is used to illustrate that with “conservative” regression, discrete designs can be achieved which are significantly lighter than those in which the variables have been rounded up.

Optimal wire sizing and buffer insertion for low power and a generalized delay model

We present efficient, optimal algorithms for timing optimization by discrete wire sizing and buffer insertion. Our algorithms are able to minimize a cost function subject to given timing constraints; we focus on minimization of dynamic power dissipation, but the algorithm is also easily adaptable to, for example, area minimization. In addition, the algorithm efficiently computes the complete, optimal power-delay trade-off curve for added design flexibility. An extension of our basic algorithm accommodates a generalized delay model which takes into account the effect of signal slew on buffer delay which can contribute substantially to overall delay. To the best of our knowledge, our approach represents the first work on buffer insertion to incorporate signal slew into the delay model while guaranteeing optimality. The effectiveness of these methods is demonstrated experimentally.

Improved two-point function approximations for design optimization

Two-point approximations are developed by utilizing both the function and gradient information of two data points. The objective of this work is to build a high-quality approximation to realize computational savings in solving complex optimization and reliability analysis problems. Two developments are proposed in the new twopoint approximations: 1) calculation of a correction term by matching with the previous known function value and supplementing it to the first-order approximation for including the effects of higher order terms and 2) development of a second-order approximation without the actual calculation of second-order derivatives by using an approximate Hessian matrix. Several highly nonlinear functions and structural examples are used for demonstrating the new two-point approximations that improved the accuracy of existing first-order methods. EVELOPMENTof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades.13 More accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run. Recent developments in constraint approximations addressed methods for eigenvalues, dynamic response, static forces, structural reliability, etc. Some of the typical methods are summarized in the following. Kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements, stresses, and forces with respect to sizing and topology variables. Vanderplaats and Salajegheh5 used two levels of approximations for discrete sizing and shape design of structures, first for solving the continuous problem and next the discrete one using a dual theory. Canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently. Thomas et al.7 presented an approximation for the frequency response of damped structures by using the concept of Ref. 6. A multivariate spline approximation was developed by Wang and Grandhi8 making use of the previously generated exact finite element analyses. Similarly, a Hermite approximation for n-dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points.9 Canfield10 used multipoint data in building an approximate Hessian matrix and constructed a second-order approximation for improving the accuracy. Toropov et al.11 approximated functions as a linear combination or products of variables, and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points. Fadel et al. 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point's exact values. This approach did not compare the function values of the previous point. Wang and Grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value, and the gradient information of the previous point was not utilized. The proposed paper develops an improved two-point approximation utilizing both function and gradient information of two data points. In the new two-point approximation, two developments are proposed: 1) calculation of a correction term based on two function values and

Specially Structured Uncapacitated Facility Location Problems

This paper considers a specially structured uncapacitated facility location problem. We show that several problems, including certain tool selection problems, substitutable inventory problems, supplier sourcing problems, discrete lot sizing problems, and capacity expansion problems, can be formulated as instances of the problem. We also show that the problem with m facilities and n customers can be solved in O(mn), as a shortest path problem on a directed graph.

Integrative versus separate cycle scheduling heuristics for capacitated discrete lot sizing and sequencing problems

This study investigates the effect of integrating lot sizing and lot sequencing decisions for a capacitated production system with a discrete shipping schedule. We compare integrative cycle scheduling heuristics, which explicitly consider both decisions, with traditional separate approaches. Experiments are conducted for both zero and positive setup cost situations with different daily demand variations, weekly demand patterns and number of items. The results of the total inventory cost and CPU time performance indicate the superiority of the integrative heuristics, the importance of an intelligent sequencing scheme, and the influential role of demand variations and the number of items. Insights from this study can be applied to a number of manufacturing settings, from corrugated paper making to component subassembly operations.

Integrated discrete and configuration optimization of trusses using genetic algorithms

The application of genetic algorithms to integrated discrete and configuration optimization of trusses is presented. It is mathematically formulated as a constrained nonlinear optimization problem with a mix of discrete sizing and continuous configuration variables. The components of genetic algorithms are described. The discrete sizing variables are treated by constructing mapping relationships between binary digit strings and discrete values by the medium of unsigned decimal integers. Careful attention has been paid to modify the genetic algorithms to promote computational efficiency. The solution procedures are stated. Several examples are solved in two ways to verify that considerable weight saving can be achieved by adding in configuration design variables, and the results also show the efficiency and robustness of genetic algorithms.

Total Optimal Synthesis Method for Truss Structures Subject to Static and Frequency Constraints

Abstract: In a previous study, we presented an efficient optimal structural synthesis method for truss structures in which the design variables are coordinates of the panel points, cross‐sectional areas, and discrete material kinds subjected to stress and displacement constraints. In this paper, the synthesis method is extended to solve design problems subject to stress, displacement, and fundamental natural‐frequency constraints. The design problem is formulated in terms of discrete material kinds and continuous shape and sizing variables and which are approximated by a convex and separable subproblem. The approximate subproblem is expressed in direct and/or reciprocal design variables and shape, material, and sizing sensitivities. Each subproblem is solved by a two‐stage minimization process. In the first stage, the continuous shape and sizing variables are optimized by a dual method. Second, the discrete material and continuous sizing variables are improved by a discrete sensitivity analysis. Using the proposed two‐stage minimization procedure, both the discrete material kinds and the continuous shape and sizing variables can be systematically improved to obtain an optimal solution. The rigorousness, reliability, and efficiency of the method are illustrated by applying it to the minimum cost design of truss structures subject to stress, displacement, and fundamental natural‐frequency constraints.

Comparative Design of Orthogonally Stiffened Plates for Production and Structural Integrity—Part 1: Size Optimization

Five alternative configurations of orthogonally stiffened plates are compared in order to identify the total cost optimum design including material and fabrication cost. Size optimization is performed within the limitations of structural component standardization for each of the five alternatives. The five optimal structures are then compared in terms of weight, fabrication, and total cost. Discrete sizing optimization is performed in this paper with two design variables, i.e., plate thickness and standardized beam cross section. Constraints are imposed on secondary and tertiary stresses computed by finite-element analysis (FEA); and on primary stresses to prevent plate buckling, stiffener tripping, and overall buckling. Confidence is established in the FEA results by making comparisons with FEA results using the effective breadth method and orthotropic plate theory. Producibility constraints dictated by standardization in shipyard practice are imposed as well.

Optimal sizing, geometrical and topological design using a genetic algorithm

A genetic algorithm is applied for the optimal design of skeletal building structures accounting for discrete sizing, geometrical and topological variables. An approximate designfitness evaluation technique is investigated with the aim to improve the numerical efficiency of the genetic search. Two design examples are presented to illustrate the principles involved.

Optimum design of trusses with discrete sizing and shape variables

The objective here is to present a method for optimizing truss structures with discrete design variables. The design variables are considered to be sizing variables as well as coordinates of joints. Both types of variables can be discrete simultaneously. Mixed continuous-discrete variables can also be considered. To increase the efficiency of the method, the structural responses, such as forces and displacements are approximated in each design cycle. The approximation of responses is carried out with respect to the design variables or their reciprocals. By substituting these approximate functions of the responses into the original design problem, an explicit high quality approximation is achieved, the solution of which does not require the detailed finite element analysis of the structure in each sub-optimization iteration. First it is assumed that all the design variables are continuous and a continuous variable optimization is performed. With the results of this step, the branch and bound method is employed on the same approximate problem to achieve a discrete solution. The numerical results indicate that the method is efficient and robust in terms of the required number of structural analyses. Several examples are presented to show the efficiency of the method.

A hybrid optimal synthesis method for truss structures considering shape, material and sizing variables

AbstractA generalized hybrid optimal synthesis method is developed for truss structures in which the design variables are the co‐ordinates of the panel points, cross‐sectional areas and discrete material kinds of member elements. The primary design problem is formulated in terms of the shape, material and sizing variables and approximated to a sequence of convex and separable subproblems by using mixed direct/reciprocal design variables. Each subproblem is solved by a dual method in which the continuous shape and sizing variables are optimized by a Newton‐type algorithm and the discrete material and continuous sizing variables are improved by a discrete sensitivity analysis. Using the proposed method, discrete material kinds, shape and sizing variables can be systematically improved to obtain an optimum solution. The rigourness, reliability and efficiency of the method are demonstrated by applying it to the minimum cost design of trusses subjected to stress and displacement constraints.

Lot sizing for material requirements planning systems- an incremental cost approach

SUMMARY An algorithm based on incremental cost approach for determining the optimal production policy in case of single-product discrete lot sizing is developed. This algorithm is extended to multi-product discrete lot sizing with capacity and inventory constraints when the capacity is specified in time units. Both the algorithms developed are illustrated with examples.

Discrete-continuous variable structural synthesis using dual methods

Approximation concepts and dual methods are extended to solve structural synthesis problems involving a mix of discrete and continuous sizing type of design variables. Pure discrete and pure continuous variable problems can be handled as special cases. The basic mathematical programming statement of the structural synthesis problem is converted into a sequence of explicit approximate primal problems of separable form. These problems are solved by constructing continuous explicit dual functions, which are maximized subject to simple nonnegativity constraints on the dual variables. A newly devised gradient projection type of algorithm called DUAL 1, which includes special features for handling dual function gradient discontinuities that arise from the discrete primal variables, is used to find the solution of each dual problem. Computational implementation is accomplished by incorporating the DUAL 1 algorithm into the ACCESS 3 program as a new optimizer option. The power of the method set forth is demonstrated by presenting numerical results for several example problems, including a pure discrete variable treatment of a metallic swept wing and a mixed discrete-continuous variable solution for a thin delta wing with fiber composite skins.

Discrete Sizing of Trusses for Optimal Geometry

The optimal configuration and discrete member sizes are automatically determined to minimize the cost of three-dimensional indeterminate truss structures under multiple loading conditions. Member areas and joint coordinates are used as design variables. Design constraints include Euler buckling and specified limits on member stresses, member sizes, and joint displacements. Design variables may be linked and members can be deleted. The design process is separated into geometry modifications using the complex method and suboptimization using the stress ratio method and an analogous scaling procedure for displacement constraints. For each geometry change, discrete member sizes are selected from a table of allowable values. The method is applied to the cost minimization of a 25-member space truss using discrete steel angle sections that are chosen from a table satisfying AISC code requirements, and comparisons are made with minimum weight designs and with continuous variable methods.

A rate-zonal centrifugation procedure for screening particle populations by sequential product recovery utilizing edge-unloading zonal rotors

A new rate-zonal density gradient centrifugation procedure for sequential product recovery was used to analyze the sedimentation characteristics of subcellular particles of rat liver and tumor cells utilizing a B-XXIX edge-unloading zonal rotor. Particle sedimentation was measured over a range of forces necessary for sedimenting mitochondria, lysosomes, and peroxisomes. Even though the populations overlapped, a discrete sizing of the various particles was achieved. Ninety percent of the mitochondria had a s o 20 ω of 2.8–7.3×10 4 S. Lysosomes were distributed throughout the whole range of particle sizes separated. However, 80% of the lysosomes shown to bind the element gallium were found in a small-granule zone, the population of the particles having a s o 20 ω of 0.8–2.2×10 4 S.

Analytical differential centrifugation: An analysis of the sedimentation properties of synaptosomes, mitochondria and lysosomes from rat brain homogenates

A newly developed analytical differential centrifugation procedure for use in angle rotors was used to analyze the sedimentation of certain brain subcellular particles. Particle sedimentation was measured over a wide range of applied forces, and a curve descriptive of this sedimentation was computed. Precise conditions for complete and fractional sedimentation of mitochondria, choline acetyltransferase-containing nerve-end particles, and lysosomal activities from rat brain cerebral cortex homogenates were established. Mitochondria, nerve-end particles, and lysosomal activities began to sediment together, but all mitochondrial cytochrome oxidase sedimented before nerve-end particles and lysosomes. Even though populations overlapped, a discrete sizing of mitochondria and nerve-end particles was accomplished. Mitochondria sedimentation was similar to an ideal particle even though brain mitochondria are composed of at least two populations. These data are compared with those used by other investigators to provide a basis for making a decision on significant differences among diversified separation conditions.