Global optimization for generalized geometric programming problems with discrete variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management, but most existing methods for solving GGP actually only consider continuous variables. This article presents a new branch-and-bound algorithm for globally solving GGP problems with discrete variables. For minimizing the problem, an equivalent monotonic optimization problem (P) with discrete variables is presented by exploiting the special structure of GGP. In the algorithm, the lower bounds are computed by solving ordinary linear programming problems that are derived via a linearization technique. In contrast to pure branch-and-bound methods, the algorithm can perform a domain reduction cut per iteration by using the monotonicity of problem (P), which can suppress the rapid growth of branching tree in the branch-and-bound search so that the performance of the algorithm is further improved. Computational results for several sample examples and small randomly generated problems are reported to vindicate our conclusions.     

Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results

A classical problem within the field of structural optimization is to find the stiffest truss design subject to a given external static load and a bound on the total volume. The design variables describe the cross sectional areas of the bars. This class of problems is well-studied for continuous bar areas. We consider here the difficult situation that the truss must be built from pre-produced bars with given areas. This paper together with Part I proposes an algorithmic framework for the calculation of a global optimizer of the underlying non-convex mixed integer design problem. In this paper we use the theory developed in Part I to design a convergent nonlinear branch-and-bound method tailored to solve large-scale instances of the original discrete problem. The problem formulation and the needed theoretical results from Part I are repeated such that this paper is self-contained. We focus on the implementation details but also establish finite convergence of the branch-and-bound method. The algorithm is based on solving a sequence of continuous non-convex relaxations which can be formulated as quadratic programs according to the theory in Part I. The quadratic programs to be treated within the branch-and-bound search all have the same feasible set and differ from each other only in the objective function. This is one reason for making the resulting branch-and-bound method very efficient. The paper closes with several large-scale numerical examples. These examples are, to the knowledge of the authors, by far the largest discrete topology design problems solved by means of global optimization.     

Minimizing the Range of Lateness on a Single Machine

This paper gives an optimizing algorithm to minimize the range of lateness, that is the difference between the maximum and minimum lateness, of jobs in a single-machine sequencing problem. The procedure is based on a branch-and-bound technique. One example has been solved to illustrate the method.     

Topology and Geometry Optimization of Branched Sheet Metal Products

In order to optimize branched sheet metal profiles consisting of several chambers, decisions concerning topology and geometry have to be made. This leads to a problem entailing discrete and nonlinear features. We describe an integrated approach combining both aspects. The underlying idea is to use a branch-and-bound algorithm. In each node of the branch-and-bound tree, a nonlinear optimization problem has to be solved. We describe how the branch-and-bound tree is constructed, i.e., how the topology decision can be classified in a meaningful way. Moreover, we explain how to approach the nonlinear optimization problem arising in the nodes of the tree. We conclude by presenting a numerical example. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decision CPM: Network Reduction and Solution

The Decision Critical Path Method (DCPM) applies to problems having a discrete time-cost tradeoff and including the possibility of interdependency constraints between the discrete job alternatives. Previously, it has been formulated as an integer-programming problem and solved by integer programming or by heuristic techniques. This paper shows that a variant of the usual critical path method may be used to reduce DCPM networks to equivalent networks containing only decision nodes and maximal distances between them. This operation, along with feasibility tests on the interdependency constraints, allows substantial simplification. A branch-and-bound method is then developed to solve the reduced problem.     

Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems

This two part paper considers the classical problem of finding a truss design with minimal compliance subject to a given external force and a volume bound. The design variables describe the cross-section areas of the bars. While this problem is well-studied for continuous bar areas, we treat here the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single bar area, i.e., a 0/1-problem. In contrast to heuristic methods considered in other approaches, Part I of the paper together with Part II present an algorithmic framework for the calculation of a global optimizer of the underlying large-scaled mixed integer design problem. This framework is given by a convergent branch-and-bound algorithm which is based on solving a sequence of nonconvex continuous relaxations. The main issue of the paper and of the approach lies in the fact that the relaxed nonlinear optimization problem can be formulated as a quadratic program (QP). Here the paper generalizes and extends the available theory from the literature. Although the Hessian of this QP is indefinite, it is possible to circumvent the non-convexity and to calculate global optimizers. Moreover, the QPs to be treated in the branch-and-bound search tree differ from each other just in the objective function. In Part I we give an introduction to the problem and collect all theory and related proofs for the treatment of the original problem formulation and the continuous relaxed problems. The implementation details and convergence proof of the branch-and-bound methodology and the large-scale numerical examples are presented in Part II.     

Adapting Branch-and-Bound for Real-World Scheduling Problems

Many sequencing and scheduling problems can be formulated as 0-1 integer programs and, in theory, solved using a branch-and-bound approach. In practice, real-world instances of these problems are usually solved using heuristics. In this paper we demonstrate the benefits of incorporating an intuitive, user-controlled variable-tolerance into a depth-first branch-and-bound algorithm. The tolerance comprises two components: a variable depth tolerance and a variable breadth tolerance. A sample scheduling problem is thoroughly analysed to illustrate empirically parameter impact on solution quality and execution time. Then, results based on several real-world problems are discussed.     

Minimizing total weighted late work in the resource-constrained project scheduling problem

This paper deals with resource-constrained project scheduling problem under the weighted late work criterion. Late work objective functions estimate the quality of a schedule based on durations of late parts of activities, not taking into account the amount of delay for fully late activities. It is assume that a project contains activities interrelated by finish-to-start type precedence relations with time lag of zero, which require one or more constrained renewable resources. The objective is to schedule each activity such that the total weighted late work is minimized. The problem has been formulated using a linear integer programming model and solved by the CPLEX. Also, a set of priority rules have been designed to quickly generate a set of initial solutions. In order to solve the problem optimally, a depth-first branch-and-bound algorithm is applied based on idea of minimal delaying alternatives. The branching order of nodes that belong to the same level of the search tree is determined on the basis of the developed priority rules. This results in generation six different versions of the branch-and-bound algorithm. Computational results on randomly generated problem sets are provided to analyze the efficiency of the priority rules and the branch-and-bound algorithm.     

Single Machine Scheduling and Selection to Minimize Total Flow Time with Minimum Number Tardy

Emmons considered the problem of sequencing N jobs on a single machine to minimize total flow time with the minimum number of tardy jobs. He proposed an effective branch-and-bound algorithm for this problem. In this paper, we show that Emmons' algorithm can be extended to a more difficult scheduling problem which includes an optimal selection of jobs as well.     

Siting and Sizing of Facilities under Probabilistic Demands

In this paper a discrete location model for non-essential service facilities planning is described, which seeks the number, location, and size of facilities, that maximizes the total expected demand attracted by the facilities. It is assumed that the demand for service is sensitive to the distance from facilities and to their size. It is also assumed that facilities must satisfy a threshold level of demand (facilities are not economically viable below that level). A Mixed-Integer Nonlinear Programming (MINLP) model is proposed for this problem. A branch-and-bound algorithm is designed for solving this MINLP and its convergence to a global minimum is established. A finite procedure is also introduced to find a feasible solution for the MINLP that reduces the overall search in the binary tree generated by the branch-and-bound algorithm. Some numerical results using a GAMS/MINOS implementation of the algorithm are reported to illustrate its efficacy and efficiency in practice.     

A branch-and-bound algorithm with fuzzy inference for a permutation flowshop scheduling problem

This paper considers an m-machine permutation flowshop scheduling problem of minimizing the makespan. This classical scheduling problem is still important in modern manufacturing systems, and is well known to be intractable (i.e., NP-hard). In fact branch-and-bound algorithms developed so far for this problem have not come to solve large scale problem instances with over a hundred jobs. In order to improve the performance of branch-and-bound algorithms this paper proposes a new dominance relation by which the search load could be reduced, and notices that it is based on a sufficient precondition. This suggests that the dominance relation holds with high possibility even if the precondition approximately holds, thus being more realistic. The branch-and-bound algorithm proposed here takes advantage of this possibility for obtaining an optimal solution as early as possible in the branch-and-bound search. For this purpose this paper utilizes membership functions in the context of the fuzzy inference. Extensive numerical experiments that were executed through Monte Carlo simulations and benchmark tests show that the developed branch-and-bound algorithm can solve 3-machine problem instances with up to 1000 jobs with probability of over 99%, and 4-machine ones with up to 900 jobs with over 97%.     

Job-shop scheduling with resource-time models of operations

The job-shop problems with allocation of continuously-divisible nonrenewable resource is considered. The mathematical models of operations are linear, decreasing functions with respect to an amount of resource. The objective is sequencing operations and allocation of constrained resource such that the project duration is minimized. Thus, the problem considered is a generalization of the classical job-shop problem. Some properties of the optimal solution are presented. The algorithm of solving this problem is based on the disjunctive graphs theory and branch-and-bound technique. The theory of the algorithm is based on the critical path concept using the segment system approach. The special feature of the algorithm is that it gives a complete solution which is associated with each node of the enumeration tree. Possible generalizations of the results presented are indicated.     

A discrete model for optimal operation of fossil-fuel generators of electricity

This paper presents a new discrete approach to the price-based dynamic economic dispatch (PBDED) problem of fossil-fuel generators of electricity. The objective is to find a sequence of generator temperatures that maximizes profit over a fixed-length time horizon. The generic optimization model presented in this paper can be applied to automatic operation of fossil-fuel generators or to prepare market bids, and it works with various price forecasts. The model’s practical applications are demonstrated by the results of simulation experiments involving 2009 NYISO electricity market data, branch-and-bound, and tabu-search optimization techniques.     

Operational scheduling of refined product pipeline with dual purpose depots

Pipelines are one of the most economic and safe ways to transport oil derivatives to depots near the local markets. This is so, especially when there is a need for carrying a huge amount of petroleum products to long distance points. A pipeline may connect several refineries and depots where the connected points are able to inject products or receive them, or both (dual purpose depots). The main point of pipeline scheduling is to identify an optimal sequence of batches that satisfy demand with minimum costs. This paper presents an MILP continuous formulation for pipeline scheduling. The proposed model results in the reduction of problem size with respect to available models, and increases the accuracy. Computational results and data are reported.     

Siting and Sizing of Facilities under Probabilistic Demands

In this paper a discrete location model for non-essential service facilities planning is described, which seeks the number, location, and size of facilities, that maximizes the total expected demand attracted by the facilities. It is assumed that the demand for service is sensitive to the distance from facilities and to their size. It is also assumed that facilities must satisfy a threshold level of demand (facilities are not economically viable below that level). A Mixed-Integer Nonlinear Programming (MINLP) model is proposed for this problem. A branch-and-bound algorithm is designed for solving this MINLP and its convergence to a global minimum is established. A finite procedure is also introduced to find a feasible solution for the MINLP that reduces the overall search in the binary tree generated by the branch-and-bound algorithm. Some numerical results using a GAMS/MINOS implementation of the algorithm are reported to illustrate its efficacy and efficiency in practice.     

Optimal scheduling in film production to minimize talent hold cost

The problem of minimizing the cost due to talent hold days in the production of a feature film is considered. A combinatorial model is developed for the sequencing of shooting days in a film shoot. The problem is shown to be strongly NP-hard. A branch-and-bound solution algorithm and a heuristic solution method for large instances of the problem (15 shooting days or more) are developed and implemented on a computer. A number of randomly generated problem instances are solved to study the power and speed of the algorithms. The computational results reveal that the heuristic solution method is effective and efficient in obtaining near-optimal solutions.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OPG-0036424. The authors are thankful to two anonymous referees for their helpful comments on an earlier version of this paper.     

A branch-reduce-cut algorithm for the global optimization of probabilistically constrained linear programs

We consider probabilistically constrained linear programs with general distributions for the uncertain parameters. These problems involve non-convex feasible sets. We develop a branch-and-bound algorithm that searches for a global optimal solution to this problem by successively partitioning the non-convex feasible region and by using bounds on the objective function to fathom inferior partition elements. This basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partition elements and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires solving linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.     

New computational results on the discrete time/cost trade-off problem in project networks

We describe a new exact procedure for the discrete time/cost trade-off problem in deterministic activity-on-the-arc networks of the CPM type, where the duration of each activity is a discrete, nonincreasing function of the amount of a single resource (money) committed to it. The objective is to construct the complete and efficient time/cost profile over the set of feasible project durations. The procedure uses a horizon-varying approach based on the iterative optimal solution of the problem of minimising the sum of the resource use over all activities subject to the activity precedence constraints and a project deadline. This optimal solution is derived using a branch-and-bound procedure which computes lower bounds by making convex piecewise linear underestimations of the discrete time/cost trade-off curves of the activities to be used as input for an adapted version of the Fulkerson labelling algorithm for the linear time/cost trade-off problem. Branching involves the selection of an activity in order to partition its set of execution modes into two subsets which are used to derive improved convex piecewise linear underestimations. The procedure has been programmed in Visual C ++ under Windows NT and has been validated using a factorial experiment on a large set of randomly generated problem instances.     

Solving the uncapacitated multiple allocation hub location problem by means of a dual-ascent technique

This paper deals with the uncapacitated multiple allocation hub location problem. The dual problem of a four-indexed formulation is considered and a heuristic method, based on a dual-ascent technique, is designed. This heuristic, which is reinforced with several specifical subroutines and does not require any external linear problem solver, is the core tool embedded in an exact branch-and-bound framework. Besides, the heuristic provides the branch-and-bound algorithm with good lower bounds for the nodes of the branching tree. The results of the computational experience (with the classical CAB and AP data sets) are included, showing the great effectiveness of this approach: instances with up to 120 nodes are solved.