Multi-item distribution network design problems under volume discount on transportation cost

This study considers the distribution network design problems of multi-echelon, multi-item supply chains under volume (weight) discounts on transportation costs. Minimising the total network cost requires determining the following: (1) the service area of distribution centres (DCs); (2) the assignment of retail stores to DCs; and (3) the inventory volume at DCs. This study compares various replenishment policies and discusses the effects of such policies on distribution network design. In general, the results show that single-cluster replenishment is superior to joint cluster replenishment. However, joint cluster replenishment may be superior to single-cluster replenishment under volume (weight) discounts on transportation costs. The results additionally show that single-item replenishment is inferior to multi-item replenishment under volume (weight) discounts on transportation costs. All the problems are formulated as piecewise non-linear programming models with multiple variables. Algorithms are proposed for solving these piecewise non-linear programming problems. Several numerical studies demonstrate the solution procedures and the effects of changing parameters on decision-making.     

A study on crate sizing problems

This paper studies the crate sizing problem to predetermine optimal standard packaging crate sizes for a demand of assorted sizes of products to minimise space and costs. The crate sizing problem is a real world logistics scenario faced by an existing multinational corporation in the applied chemistry industry. Customer demands come in a mix of rolls to be packaged in individual crates and then shipped in shipping containers from plants to customers. Firstly, a linear crate length optimisation model is introduced with the objective to minimise the total loss of extra space inside the crates for a given distribution of customer demands. In order to take inventory costs of crates into consideration, a second problem is introduced to both find the optimal number of crate types and the corresponding lengths such that the total costs of packaging and inventory are minimised. The latter problem is first formulated as a non-linear optimisation model and then solved using dynamic programming approach to balance the trade-off between the number of crate types and the penalty cost for space wastage inside the crates.     

Optimization of evaporative fluid coolersOptimisation des refroidisseurs de liquide évaporatifs

In the paper the optimization of geometrical and operating parameters for evaporative fluid coolers has been presented. The algorithm of optimizational calculations contains:

• the mathematical model of heat and mass transfer in evaporative fluid coolers;

• model of operating costs based, among others, on the Zalewski–Gryglaszewski relations developed by the authors for evaluation of air pressure drops in the heat exchanger;

• model of evaporative heat exchangers production costs.

Optimization problem formulated for non-linear objective functions with inequality and equality constraints has been solved using Schittkowski's method based on quadratic programming. Two optimization problems have been considered. A solution to the first problem consists in the design of a heat exchanger with such geometrical parameters which ensure its maximum heat capacity at minimal total costs. A result to the second optimization problem are such thermal and flow parameters which will ensure minimum operating costs.The results of optimization calculations presented in the paper are accompanied by their experimental verification.     

凝聚函数法求解粘弹性本构参数及温度场联合辨识问题

提出利用准静态位移信息对粘弹性本构参数及温度场进行联合识别的求解策略。建立了适于敏度分析的粘弹性与温度场耦合问题的正演数值模型,并将其反演归结为一个带有多个不等式约束的非线性规划问题,采用凝聚函数法将此问题转化为一个可微的单约束优化问题。在此基础上,采用乘子法进行求解,给出了数值验证,探讨了信息误差对反演结果的影响。     

Frictionless geometrically non-linear contact using quadratic programming

The successive quadratic programming (SQP) method is used with the finite element method (FEM) to solve frictionless geometrically non-linear contact problems involving large deformations of the elastica in the presence of flat rigid walls. To formulate the SQP problems, the potential energy (PE) is expanded in a Taylor series of second order in displacement increments about a configuration near a contact solution. The SQP problems consist of minimizing the Taylor expansion of the PE subject to the inequality constraints which represent contact. The quadratic programming (QP) method is made part of a Newton–Raphson (NR) search in which the QP corrections are made when a NR step does not satisfy the constraints. A revised simplex method developed by Rusin is used to solve the QP problems. The elastica is modelled with a total Lagrangian FEM developed by Fried. Solutions are obtained for the end loaded buckled elastica in point contact with a rigid wall and for a uniformly loaded elastica in regional contact with a rigid wall. The problems are also solved using a penalty method. The results obtained for the point contact problem are compared to an analytical solution. Calculations were made to obtain numerical information on maximum load step size and the number of inverse operations required for each load step. Cases in which the elastica stiffened substantially as a result of the initiation of contact are also discussed.     

A NUMERICAL PROCEDURE FOR SENSITIVITY ANALYSIS IN GEOMETRIC PROGRAMMING

This paper presents a development of sensitivity analysis (post optimal) for non-linear optimization problems. The basis for this development is the optimization technique known as geometric programming. Efficient procedures are developed which relate changes in the coefficients to the new design variables. The procedure is used to analyze the design of a condenser.     

Supply chain network design with unreliable suppliers: a Lagrangian relaxation-based approach

This paper deals with the integrated facility location and supplier selection decisions for the design of supply chain network with reliable and unreliable suppliers. Two problems are addressed: (1) facility location/supplier selection; and (2) facility location/supplier reliability. We first consider the facility location and supplier selections problem where all the suppliers are reliable. The decisions concern the selection of suppliers, the location of distribution centres (DCs), the allocation of suppliers to DCs and the allocation of retailers to DCs. The objective is to minimise fixed DCs location costs, inventory and safety stock costs at the DCs and ordering costs and transportation costs across the network. The introduction of inventory costs and safety stock costs leads to a non-linear NP-hard optimisation problem. To solve this problem, a Lagrangian relaxation-based approach is developed. For the second problem, a two-period decision model is proposed in which selected suppliers are reliable in the first period and can fail in the second period. The corresponding facility location/supplier reliability problem is formulated as a non-linear stochastic programming problem. A Monte Carlo optimisation approach combining the sample average approximation scheme and the Lagrangian relaxation-based approach is proposed. Computational results are presented to evaluate the efficiency of the proposed approaches.     

Optimal leadtimes planning in serial production systems with earliness and tardiness costs

We consider a production system consisting of N processing stages. The actual leadtimes at the stages are stochastic. The objective is to determine the planned leadtimes at each stage so as to minimize the expected total inventory costs, tardiness penalties, and a backlog penalty for not meeting demand due date at the last stage. Recursive relationships are used for automatic generation and efficient computation of the objective function. The efficiency of the proposed algorithms allows us to obtain new insights regarding operating policies, leadtime delivery reliability, and production line design. The problem is formulated as a convex non-linear programming problem. The latter is then solved using classical convex optimization algorithms. For the special case of exponentially distributed leadtimes, the objective function is derived in a closed form.     

Synthesis of near-optimal topologically constrained property-based water network using swarm intelligence

The sustainability of water resources is one of the major concerns of the world. Industries are finding ways to minimize water costs and to reduce water pollution through efficient use of supplies. One of effective solutions is the use of water pinch technology or process integration techniques. Property integration emerged as the third trend of process integration technique (aside from energy and mass integration) to solve the problem of water network synthesis by considering physical stream properties such as pH or resistivity. Various techniques have been done to solve this property-based problem. In this paper, mixed integer non-linear programming (MINLP) problems with topological constraints is studied. Nowadays, various evolutionary algorithms are also used to solve such MINLP problems. In this study, particle swarm optimization is used to solve the property integration problem. A comparison is made between this algorithm and commercial genetic algorithm (GA) applications. Particle swarm optimization is observed to perform favorably compared to GA.     

AN OPTIMIZATION METHOD FOR THE DESIGN OF LARGE,HIGHLY CONSTRAINED COMPLEX SYSTEMS

A practical and efficient optimization method for the rational design of large, highly constrained complex systems is presented. The design of such systems is iterative and requires the repeated formulation and solution of an analysis model, followed by the formulation and solution of a redesign model. The latter constitutes an optimization problem. The versatility and efficiency of the method for solving the optimization problem is of fundamental importance for a successful implementation of any rational design procedure.

In this paper, a method is presented for solving optimization problems formulated in terms of continuous design variables. The objective function may be linear or non-linear, single or multiple. The constraints may be any mix of linear or non-linear functions, and these may be any mix of inequalities and equalities. These features permit the solution of a wide spectrum of optimization problems, ranging from the standard linear and non-linear problems to a non-linear problem with multiple objective functions (goal programming). The algorithm for implementing the method is presented in sufficient detail so that a computer program, in any computing language, can be written.     

An alternating optimization approach for mixed discrete non-linear programming

This article contributes to the development of the field of alternating optimization (AO) and general mixed discrete non-linear programming (MDNLP) by introducing a new decomposition algorithm (AO-MDNLP) based on the augmented Lagrangian multipliers method. In the proposed algorithm, an iterative solution strategy is proposed by transforming the constrained MDNLP problem into two unconstrained components or units; one solving for the discrete variables, and another for the continuous ones. Each unit focuses on minimizing a different set of variables while the other type is frozen. During optimizing each unit, the penalty parameters and multipliers are consecutively updated until the solution moves towards the feasible region. The two units take turns in evolving independently for a small number of cycles. The validity, robustness and effectiveness of the proposed algorithm are exemplified through some well known benchmark mixed discrete optimization problems.     

A natural gas cash-out problem: A bilevel programming framework and a penalty function method

One of the many complex problems that arise from the transmission and marketing of natural gas is when a shipper draws a contract with a pipeline company to deliver a certain amount of gas among several points. What is actually delivered is often different from the amount that had been originally agreed upon. This phenomenon is called an imbalance. When an imbalance occurs, the pipeline penalizes the shipper by imposing a cash-out penalty policy. Since this penalty is a function of the operating daily imbalances, an important decision-making problem for the shippers is how to carry out their daily imbalances so as to minimize their incurred penalty. In this paper, we introduce the problem of minimizing the cash-out penalty costs from the point of view of a natural gas shipping party. We present a mixed integer bilevel linear programming model and discuss its underlying assumptions. To solve it efficiently, we reformulate it as a standard mathematical program and describe a penalty-function algorithm functions for its solution. The algorithm is well-founded and its convergence is proved. Results of numerical experiments support the algorithm’s robustness providing a valuable solution technique for this very important and complex problem in the natural gas market.     

Algorithms for non-linear contact constraints with application to stability problems of rods and shells

In this paper a class of non-linear problems is discussed where stability as well as post-buckling behaviour is coupled with contact constraints. The contact conditions are introduced via a perturbed Lagrangian formulation. From this formulation the penalty and Lagrangian multiplier method are derived. Both algorithms are investigated together with an algorithm based on an augmented Lagrangian method. The resulting finite element formulation is applied to structural problems of beams and shells undergoing finite elastic deflections and rotations. For the examination of the post-buckling behaviour the arc-length method is used. The performance of the element formulation and a comparison of the different contact algorithms are demonstrated by numerical examples.     

OPTIMUM SHAPE DESIGN AND POSITIONING OF FEATURES USING THE BOUNDARY INTEGRAL EQUATION METHOD

A design optimization procedure is developed using the boundary integral equation (BIE) method for linear elastostatic two-dimensional domains. Optimal shape design problems are treated where design variables are geometric parameters such as the positions and sizing dimensions of entire features on a component or structure. A fully analytical approach is adopted for the design sensitivity analysis where the BIE is implicitly differentiated. The ability to evaluate response sensitivity derivatives with respect to design variables such as feature positions is achieved through the definition of appropriate design velocity fields for these variables. How the advantages of the BIE method are amplified when extended to sensitivity analysis for this category of shape design problems is also highlighted. A mathematical programming approach with the penalty function method is used for solving the overall optimization problem. The procedure is applied to three example problems to demonstrate the optimum positioning of holes and optimization of radial dimensions of circular arcs on structures.     

Design-point excitation for non-linear random vibrations

It has been shown in recent years that certain non-linear random vibration problems can be solved by well established methods of time-invariant structural reliability, such as FORM and importance sampling. A key step in this approach is finding the design-point excitation, which is that realization of the input process that is most likely to give rise to the event of interest. It is shown in this paper that for a non-linear, elastic single-degree-of-freedom oscillator subjected to white noise, the design-point excitation is identical to the excitation that generates the mirror image of the free-vibration response when the oscillator is released from a target threshold. This allows determining the design-point excitation with a single non-linear dynamic analysis. With a slight modification, this idea is extended to non-white and non-stationary excitations and to hysteretic oscillators. In these cases, an approximate solution of the design-point excitation is obtained, which, if necessary, can be used as a ‘warm’ starting point to find the exact design point using an iterative optimization algorithm. The paper also offers a simple method for computing the mean out-crossing rate of a response process. Several examples are provided to demonstrate the application and accuracy of the proposed methods. The methods proposed in this paper enhance the feasibility of approximately solving non-linear random vibration problems by use of time-invariant structural reliability techniques.     

A direct application of higher-order parametric programming techniques to structural optimization

Computational methods based on a sequence of parametric programming problems are presented for solving constrained optimization problems (COP) without any parameter. An auxiliary parametric programming problem (APPP) is formulated in order to solve COP. The procedure is started with an arbitrary initial solution which is the trivial solution of APPP corresponding to the initial value of the parameter. Then the optimal solution for the final value of the parameter, which is the optimal solution of COP, is estimated by Taylor's expansion with respect to the parameter where higher-order terms are incorporated. It is shown that the incorporation of the higher-order terms indeed leads to a faster convergence of the solution. As an extension of the method, a general algorithm is presented for optimum design problems with state variable constraints which are implicit functions of the design variables. Logarithmic penalty functions are incorporated and the weight coefficients for the penalty terms are updated continuously. The derivatives of the state variables with respect to the parameter and their sensitivity coefficients are expressed explicitly in terms of those of the design variables. Finally, a method of simultaneous analysis and optimization is developed for trusses with geometrical non-linearity.     

A consistent tangent stiffness matrix for three-dimensional non-linear contact analysis

A consistent tangent stiffness matrix for the analysis of non-linear contact problems is presented. The associated element has three or four nodes and establishes contact between three-dimensional structures like solids and shells. It accounts for the non-linear kinematics of large deformation analysis and guarantees a quadratic convergence rate. Two formulations, the penalty method and the Lagrange multiplier method, are investigated.     

Lagrange constraints for transient finite element surface contact

A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.     

JOINT CONFIGURATION OF BACKBONE AND LOGICAL NETWORKS ON A RECONFIGURABLE PACKET-SWITCHED NETWORK WITH UNRELIABLE LINKS

Abstract

This paper considers a problem of configuring both backbone and logical networks in a reconfigurable packet-switched network where links are subject to failures. The objective is to design feasible backbone and logical networks at least cost where the network cost includes backbone link capacity expansion cost and average packet delay penalty cost due to link failures. The problem is formulated as a zero-one non-linear mixed integer programming problem, for which an effective solution procedure is developed by using a Lagrangean relaxation technique for finding a lower bound and a heuristic method is exploited for improving the upper bound of an intermediate solution. The solution procedure is tested for its effectiveness with various numerical examples.     

Numerical solution of constrained non-linear algebraic equations

The numerical solution of n non-linear algebraic equations with n unknowns, when any or all of the variables are subjected to constraints, is discussed. Distinction is made between physical and absolute constraints. It is shown, using a set of test problems, that solution of constrained equations does require a special approach. It is found that the step length restricted Newton's method performs best for problems with absolute constraints. Physical constraints can be best handled using a continuation method or penalty functions.