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Lot sizing and cutting stock problems generally arise in manufacturing as a two stage connected process. Although these two problems have been addressed separately by many authors, recently, inspired by practical applications, some studies have emerged analyzing their integration. The mono-objective version of the integrated lot-sizing and cutting stock problem can be considered as an enhancement that minimizes the global production cost. However, it does not include the multiple criteria that can arise from the inclusion of multiple stakeholders in a modern, distributed manufacturing process. This paper aims to introduce a new extended goal programming model for the integrated lot-sizing and cutting stock problem which models the arising multiple criteria by a set of goals representing the interests of different stakeholders in the manufacturing process. This formulation allows for the consideration of the balance between the conflicting goals of multiple stakeholders and the cost efficiency of the overall process. In order to efficiently solve the proposed model, a column generation based heuristic procedure is applied. The trade-offs among the various criteria related to the problem are assessed, and a series of computational experiments are performed. The computational results compare individual criteria weighting schemes and evaluate the goals’ sensitivity using performance profiles and time-to-target plots methodologies.

The glass cutting problem proposed by Saint Gobain for the 2018 ROADEF challenge includes some specific constraints that prevent the direct application of procedures developed for the standard cutting problem. On the one hand, the sheets to be cut have defects that make them unique and they must be used in a given order. On the other hand, pieces are grouped in stacks and the pieces in each stack must be cut in order. There are also some additional characteristics due to the technology being used, especially the requirement for a three-stage guillotine cutting process. Taking into account the sequencing constraints on sheets and pieces, we have developed a beam search algorithm, using a tree structure in which at each level the partial solution is increased by adding some new elements until a complete solution is built. We have developed a randomized constructive algorithm for building these new elements and explored several alternatives for the local and the global evaluation. An improvement procedure, specifically designed for the problem, has also been added. The computational study, using the datasets provided by the company, shows the efficiency of the proposed algorithm for short and long running times.

In this paper, we address a new variant of the cutting stock problem, in which skiving is allowed and setup costs are considered. Specifically, the output sheets are generally longer but narrower than the input coils. To satisfy the demand of each sheet, combining two or more coils is allowed. Moreover, because changing from one pattern to another involves considerable time and cost, minimizing the number of different patterns or setups is vital. Thus, the objective of our study is to minimize the material cost and the number of setups. We propose an integer programming (IP) formulation for the problem that contains an exponential number of binary variables and column-dependent constraints. The linear programming (LP) relaxation is solved using a column-and-row generation framework that involves a knapsack subproblem and a nonlinear IP subproblem. For the latter, we propose a decomposition-based exact solution method with pseudo-polynomial time. To obtain IP solutions, we develop a diving heuristic based on matching level. The computational experiments show that these algorithms are efficient and of high quality.

The aim of this paper is to classify the literature related to the integration between the lot-sizing and cutting stock problem. A deterministic mathematical model, that considers multiple dimensions of integration and comprises several aspects found in practice, is proposed. This model is used as a framework to classify the current literature in this field. The main classification of the literature is organized around two types of integration. In a planning horizon which consists of multiple periods, the inventory provides a link between the periods. This integration across time periods constitutes the first type of integration. The proposed model also considers the production of different types of items at three different levels: objects are fabricated or purchased and next cut into pieces which are then assembled into final products. The integration between these three production levels constitutes the second type of integration. The classification indicates that, even though many papers in the current literature analyze an integrated lot-sizing and cutting stock problem, they vary widely with respect to the level of integration on the time and production level dimensions. Furthermore, our analysis indicates that the current models also consider varying assumptions with respect to the inventory, the production capacities and the setups. Finally, we point out some possible interesting areas for future research.

Motivated by a firm in the technical textile industry, we study a two-dimensional cutting stock problem with sequence dependent setup times and permissible tolerances. We provide a sequential heuristic with feedback loop based on the approach of Gilmore and Gomory and formulate the sequencing problem as a mixed integer program. We derive a lower bound algorithm and demonstrate the near-optimal performance of our heuristic. Finally, we use real data to test our heuristic and illustrate its applicability to a problem of realistic size.

Among various complexities affecting the performance of an optimization algorithm, the search space dimension is a major factor. Another key complexity is the discreteness of the search space. When these two complexities are present in a single problem, optimization algorithms have been demonstrated to be inefficient, even in linear programming (LP) problems. In this paper, we consider a specific resource allocation problem constituting to an integer linear programming (ILP) problem which, although comes from a specific industry, is similar to other practical resource allocation and assignment problems. Based on a population based optimization approach, we present a computationally fast method to arrive at a near-optimal solution. Compared to two popular softwares ( glpk, Makhorin, 2012 and CPLEX, Gay, 2015 ), which are not able to handle around 300 and 2000 integer variables while continuing to run for hours, respectively, our proposed method is able to find a near-optimal solution in less than second on the same computer. Moreover, the main highlight of this study is to propose a customized population based optimization algorithm that scales in almost a linear computational complexity in handling 50,000 to one billion (109) variables. We believe that this is the first time such a large-sized real-world constrained problem is ever handled using any optimization algorithm. We perform sensitivity studies of our method for different problem parameters and these studies clearly demonstrate the reasons for such a fast and scale-up application of the proposed method.

We consider the 0/1 Collapsing Knapsack Problem (CKP) and a generalization involving more than a capacity constraint (M-CKP). We propose a novel ILP formulation and a problem reduction procedure together with an exact approach. The proposed approach compares favorably to the methods available in the literature and manages to solve to optimality very large size instances particularly for CKP and 2-CKP.

The two-dimensional vector packing problem with piecewise linear cost function (2DVPP-PLC) models a practical packing problem faced by many companies that use courier services. We propose a branch-and-price algorithm to solve the 2DVPP-PLC exactly. The column generation procedure is a key component that affects the performance of a branch-and-price algorithm. The pricing problem exhibits an interesting structure that allows us to decompose it into subproblems that form a lattice. We explore dominance relations on the lattice and design an efficient algorithm for the pricing problem. Experimental results show that our branch-and-price algorithm is capable of solving 2DVPP-PLC test instances effectively.

The Multiple Knapsack Assignment Problem (MKAP) has been solved successfully using heuristics, but important special cases have not yet been dealt with. Motivated by an emergency relocation problem, an optimal polynomial algorithm and a high performing heuristic that take advantage of the special structure are provided for two special cases of the MKAP.

In an algorithm for a problem whose candidate solutions are selections of objects, an ejection chain is a sequence of moves from one solution to another that begins by removing an object from the current solution. The quadratic multiple knapsack problem extends the familiar 0–1 knapsack problem both with several knapsacks and with values associated with pairs of objects. A hybrid algorithm for this problem extends a local search algorithm through an ejection chain mechanism to create more powerful moves. In addition, adaptive perturbations enhance the diversity of the search process. The resulting algorithm produces results that are competitive with the best heuristics currently published for this problem. In particular, it improves the best known results on 34 out of 60 test problem instances and matches the best known results on all but 6 of the remaining instances.

Given a collection of n items (elements) and an associated symmetric distance dij between each pair of items i and j, we seek a subset P of these items (with a given cardinality p) so that the minimum pairwise distance among the selected items is maximized. This problem is known as the max–min diversity problem or the p-dispersion problem, and it is shown to be np-hard. We define a collection of node packing problems associated with each instance of this problem and employ a binary search among these node packing problems to devise an effective procedure for solving the original problem. We employ existing integer programming techniques, i.e., branch-and-bound and strong valid inequalities, to solve these node packing problems. Through a computational experiment we show that this approach can be used to solve relatively large instances of the p-dispersion problem, i.e., instances with more than 1000 items. We also discuss an application of this problem in the context of locating traffic sensors in a highway network.

This paper presents an exact algorithm for solving the knapsack sharing problem with common items. In literature, this problem is also denominated the Generalized Knapsack Sharing Problem (GKSP). The GKSP is NP-hard because it lays on the 0–1 knapsack problem and the knapsack sharing problem. The proposed exact method is based on a rigorous decomposition technique which leads to an intense simplification of the solution procedure for the GKSP. Furthermore, in order to accelerate the procedure for finding the optimum solution, an upper bound and several reduction strategies are considered. Computational results on two sets of benchmark instances from literature show that the proposed method outperforms the other approaches in most instances.

We consider the one-dimensional skiving stock problem which is strongly related to the dual bin packing problem: find the maximum number of items with minimum length L that can be constructed by connecting a given supply of m ∈ N smaller item lengths l 1 , … , l m with availabilities b 1 , … , b m . For this optimization problem, we present three new models (the arcflow model, the onestick model, and a model of Kantorovich-type) and investigate their relationships, especially regarding their respective continuous relaxations. To this end, numerical computations are provided. As a main result, we prove the equivalence between the arcflow model, the onestick approach and the existing pattern-oriented standard model. In particular, this equivalence is shown to hold for the corresponding continuous relaxations, too.

We study the operations scheduling problem encountered in the process of making and distributing emergency supplies. The lead times of a multi-echelon process, including shipping time, assembly time, and waiting time for raw materials must be explicitly modeled. The optimization problem is to find an inventory allocation and a production/assembly plan together with a shipping schedule for inbound supplies and outbound deliveries so that the total tardiness in customer order fulfillment is minimized. We define the problem as a mixed integer programming model, perform a structure analysis of the problem, and then propose a new search heuristic for the problem. This proposed heuristic finds a feasible solution to the problem by solving a series of linear programming relaxation problems, and is able to terminate quickly. Observations from an extensive empirical study are reported.

In this study, we investigate a two-dimensional cutting stock problem in the thin film transistor liquid crystal display industry. Given the lack of an efficient and effective mixed production method that can produce various sizes of liquid crystal display panels from a glass substrate sheet, thin film transistor liquid crystal display manufacturers have relied on the batch production method, which only produces one size of liquid crystal display panel from a single substrate. However, batch production is not an effective or flexible strategy because it increases production costs by using an excessive number of glass substrate sheets and causes wastage costs from unused liquid crystal display panels. A number of mixed production approaches or algorithms have been proposed. However, these approaches cannot solve industrial-scale two-dimensional cutting stock problem efficiently because of its computational complexity. We propose an efficient and effective genetic algorithm that incorporates a novel placement procedure, called a corner space algorithm, and a mixed integer programming model to resolve the problem. The key objectives are to reduce the total production costs and to satisfy the requirements of customers. Our computational results show that, in terms of solution quality and computation time, the proposed method significantly outperforms the existing approaches.

The redundancy allocation problem is the problem of finding an optimal allocation of redundant components subject to a set of resource constraints. The problem studied in this paper refers to a series-parallel system configuration and allows for component mixing. We propose a new modeling/solution approach, in which the problem is transformed into a multiple choice knapsack problem and solved to optimality via a branch and cut algorithm. The algorithm is tested on well-known sets of benchmark instances. All instances have been solved to optimality in milliseconds or very few seconds on a normal workstation.

The primary objective in the one-dimensional cutting stock problem is to minimize material cost. In real applications it is often necessary to consider auxiliary objectives, one of which is to reduce the number of different cutting patterns (setups). This paper first presents an integer linear programming model to minimize the sum of material and setup costs over a given pattern set, and then describes a sequential grouping procedure to generate the patterns in the set. Two sets of benchmark instances are used in the computational test. The results indicate that the approach is efficient in improving the solution quality.

This paper investigates the 2D guillotine knapsack packing problem, in which the objective is to select and cut a set of rectangles from a sheet with fixed size and maximize the total profit of the selected rectangles. The orientation of the rectangles is fixed. And the guillotine cut, in which the cut must be parallel to the sides of the sheet to divide it into two completely separated sheets, is required. Two well-known methods, namely the top-down and bottom-up approaches, are combined into a coherent algorithm to address this problem. Computational experiments on benchmark test sets show that the approach finds the optimal solution for almost all moderately sized instances and outperforms all existing approaches for larger instances.

We consider a stochastic knapsack problem in which the event of overflow results in the problem ending with zero return. We assume that there are n types of items available where each type has infinite supply. An item has an exponentially distributed random weight with a known mean depending on its type and the item’s value is proportional to its weight with a given factor depending on the item’s type. We have to make a decision on each stage whether to stop, or continue to put an item of a selected type in the knapsack. An item’s weight is learned when placed to the knapsack. The objective of this problem is to find a policy that maximizes the expected total values. Using the framework of dynamic programming, the optimal policy is found when n = 2 and a heuristic policy is suggested for n > 2 .

We formulate the multiple knapsack assignment problem (MKAP) as an extension of the multiple knapsack problem (MKP), as well as of the assignment problem. Except for small instances, MKAP is hard to solve to optimality. We present a heuristic algorithm to solve this problem approximately but very quickly. We first discuss three approaches to evaluate its upper bound, and prove that these methods compute an identical upper bound. In this process, reference capacities are derived, which enables us to decompose the problem into mutually independent MKPs. These MKPs are solved euristically, and in total give an approximate solution to MKAP. Through numerical experiments, we evaluate the performance of our algorithm. Although the algorithm is weak for small instances, we find it prospective for large instances. Indeed, for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0.1% within one CPU second.

This paper presents two new dynamic programming (DP) algorithms to find the exact Pareto frontier for the bi-objective integer knapsack problem. First, a property of the traditional DP algorithm for the multi-objective integer knapsack problem is identified. The first algorithm is developed by directly using the property. The second algorithm is a hybrid DP approach using the concept of the bound sets. The property is used in conjunction with the bound sets. Next, the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set. It means that the upper bound set is underestimated. Then, an extended upper bound set is proposed on the basis of the set of linear relaxation solutions. The efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set. The numerical results obtained from different types of bi-objective instances show the effectiveness of the proposed approach.

In this article, we review published studies that consider the solution of the one-dimensional cutting stock problem (1DCSP) with the possibility of using leftovers to meet future demands, if long enough. The one-dimensional cutting stock problem with usable leftovers (1DCSPUL) is a problem frequently encountered in practical settings but often, it is not dealt with in an explicit manner. For each work reviewed, we present the application, the mathematical model if one is proposed and comments on the computational results obtained. The approaches are organized into three classes: heuristics, item-oriented, or cutting pattern-oriented.

We consider a variant of the multidimensional assignment problem (MAP) with decomposable costs in which the resulting optimal assignment is described as a set of disjoint stars. This problem arises in the context of multi-sensor multi-target tracking problems, where a set of measurements, obtained from a collection of sensors, must be associated to a set of different targets. To solve this problem we study two different formulations. First, we introduce a continuous nonlinear program and its linearization, along with additional valid inequalities that improve the lower bounds. Second, we state the standard MAP formulation as a set partitioning problem, and solve it via branch and price. These approaches were put to test by solving instances ranging from tripartite to 20-partite graphs of 4 to 30 nodes per partition. Computational results show that our approaches are a viable option to solve this problem. A comparative study is presented.

Two-staged patterns are often used in manufacturing industries to divide stock plates into rectangular items. A heuristic algorithm is presented to solve the rectangular two-dimensional single stock size cutting stock problem with two-staged patterns. It uses the column-generation method to solve the residual problems repeatedly, until the demands of all items are satisfied. Each pattern is generated using a procedure for the constrained single large object placement problem to guarantee the convergence of the algorithm. The computational results of benchmark and practical instances indicate the following: (1) the algorithm can solve most instances to optimality, with the gap to optimality being at most one plate for those solutions whose optimality is not proven and (2) for the instances tested, the algorithm is more efficient (on average) in reducing the number of plates used than a published algorithm and a commercial stock cutting software package.

This paper presents a backward state reduction dynamic programming algorithm for generating the exact Pareto frontier for the bi-objective integer knapsack problem. The algorithm is developed addressing a reduced problem built after applying variable fixing techniques based on the core concept. First, an approximate core is obtained by eliminating dominated items. Second, the items included in the approximate core are subject to the reduction of the upper bounds by applying a set of weighted-sum functions associated with the efficient extreme solutions of the linear relaxation of the multi-objective integer knapsack problem. Third, the items are classified according to the values of their upper bounds; items with zero upper bounds can be eliminated. Finally, the remaining items are used to form a mixed network with different upper bounds. The numerical results obtained from different types of bi-objective instances show the effectiveness of the mixed network and associated dynamic programming algorithm.

We present a novel generic programming implementation of a column-generation algorithm for the generalized staff rostering problem. The problem is represented as a generalized set partitioning model, which is able to capture commonly occurring problem characteristics given in the literature. Columns of the set partitioning problem are generated dynamically by solving a pricing subproblem, and constraint branching in a branch-and-bound framework is used to enforce integrality. The pricing problem is formulated as a novel three-stage nested shortest path problem with resource constraints that exploits the inherent problem structure. A very efficient implementation of this pricing problem is achieved by using generic programming principles in which careful use of the C++ pre-processor allows the generator to be customized for the target problem at compile-time. As well as decreasing run times, this new approach creates a more flexible modeling framework that is well suited to handling the variety of problems found in staff rostering. Comparison with a more-standard run-time customization approach shows that speedups of around a factor of 20 are achieved using our new approach. The adaption to a new problem is simple and the implementation is automatically adjusted internally according to the new definition. We present results for three practical rostering problems. The approach captures all features of each problem and is able to provide high-quality solutions in less than 15 minutes. In two of the three instances, the optimal solution is found within this time frame.

The optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and, consequently, originates an unnecessarily large number of iterations of the method. To overcome this drawback, variations of the standard approach use interior points of the dual feasible set instead of optimal solutions. In this paper, we focus on a variation known as the primal-dual column generation technique which uses a primal-dual interior point method to obtain well-centered non-optimal solutions of the restricted master problems. We show that the method converges to an optimal solution of the master problem even though non-optimal solutions are used in the course of the procedure. Also, computational experiments are presented using linear-relaxed reformulations of three classical integer programming problems: the cutting stock problem, the vehicle routing problem with time windows, and the capacitated lot sizing problem with setup times. The numerical results indicate that the appropriate use of a primal-dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times, on average. Furthermore, the results show that the larger the instance, the better the relative performance of the primal-dual column generation technique. (C) 2012 Elsevier B.V. All rights reserved.

The accessibility arc upgrading problem (AAUP) is a network upgrading problem that arises in real-life decision processes such as rural network planning. In this paper, we propose a linear integer programming formulation and two solution approaches for this problem. The first approach is based on the knapsack problem and uses the knowledge gathered from an analytical study of some special cases of the AAUP. The second approach is a variable neighbourhood search with strategic oscillation. The excellent performance of both approaches is demonstrated using a large set of randomly generated instances. Finally, we stress the importance of a proper allocation of scarce resources in accessibility improvement. (c) 2012 Elsevier B.V. All rights reserved.

Three-staged guillotine patterns are widely used in the manufacturing industry to cut stock plates into rectangular items. The cutting cost often increases with the number of cuts required. This paper focuses on the rectangular two-dimensional cutting stock problem, where three-staged guillotine patterns are used, and the objective is to minimize the sum of plate and cutting costs. The column generation framework is used to solve the problem. It uses a pattern-generation procedure to obtain the patterns. The cutting cost is considered in both the pattern-generation procedure and the objective of the linear programming formulation. The computational results indicate that the approach can reduce the number of cuts, without increasing the plate cost.

This paper considers a multi-product newsvendor problem with multiple constraints. Multiple constraints in the problem make it more challenging to solve. Previous research has attempted to solve the problem by considering two-constraint case or/and using approximation techniques or active sets methods. The solution methods in literature for solving multi-constraint problem are limited or cumbersome. In this paper, by analyzing structural properties of the multi-constraint multi-product newsvendor problem, we develop a multi-tier binary solution method for yielding the optimal solution to the problem. The proposed method is applicable to the problem with any continuous demand distribution and more than two constraints, and its computational complexity is polynomial in the number of products. Numerical results are presented for showing the effectiveness of our method.

Three new bounds for periodicity theorems on the unbounded Knapsack problem are developed. Periodicity theorems specify when it is optimal to pack one unit of the best item (the one with the highest profit-to-weight ratio). The successive applications of periodicity theorems can drastically reduce the size of the Knapsack problem under analysis, theoretical or empirical. We prove that each new bound is tight in the sense that no smaller bound exists under the given condition.

A heuristic algorithm for the one-dimensional cutting stock problem with usable leftover (residual length) is presented. The algorithm consists of two procedures. The first is a linear programming procedure that fulfills the major portion of the item demand. The second is a sequential heuristic procedure that fulfills the remaining portion of the item demand. The algorithm can balance the cost of the consumed bars, the profit from leftovers and the profit from shorter stocks reduction. The computational results show that the algorithm performs better than a recently published algorithm.

We consider two types of orthogonal, oriented, rectangular, two-dimensional packing problems. The first is the strip packing problem, for which four new and improved level-packing algorithms are presented. Two of these algorithms guarantee a packing that may be disentangled by guillotine cuts. These are combined with a two-stage heuristic designed to find a solution to the variable-sized bin packing problem, where the aim is to pack all items into bins so as to minimise the packing area. This heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space. The results of the algorithms are compared to those of seven level-packing heuristics from the literature by means of a large number of strip-packing benchmark instances. It is found that the new algorithms are an improvement over known level-packing heuristics for the strip packing problem. The advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable-sized bin packing problem. However, they do provide results faster than many existing algorithms.

Pre-processing operations that reduce the size of a problem may be decisive for solving or not solving practical instances of a NP-hard problem. In this article we review some properties suggested in the literature for the minimization of open stacks problem that can be used in pre-processing operations to reduce the instances sizes. We also present a new pre-processing technique that may be very effective in reducing the size of an instance. We present computational tests with the suggested pre-processing operations applied on sets of MOSP instances of the literature and we show that the reductions obtained can be significant.

The one-dimensional cutting stock problem is the problem of cutting stock material into shorter lengths, in order to meet demand for these shorter lengths while minimizing waste. In industrial cutting operations, it may also be necessary to fill the orders for these shorter lengths before a given due date. We propose new optimization models and solution procedures which solve the cutting stock problem when orders have due dates. We evaluate our approach using data from a large manufacturer of reinforcement steel and show that we are able to solve industrial-size problems, while also addressing common cutting considerations such as aggregation of orders, multiple stock lengths and cutting different types of material on the same machine. In addition, we evaluate operational performance in terms of resulting waste and tardiness of orders using our model in a rolling horizon framework.

An iterative scheme which is based on a dynamic fixation of the variables is developed to solve the 0-1 multidimensional knapsack problem. Such a scheme has the advantage of generating memory information, which is used on the one hand to choose the variables to fix either permanently or temporarily and on the other hand to construct feasible solutions of the problem. Adaptations of this mechanism are proposed to explore different parts of the search space and to enhance the behaviour of the algorithm. Encouraging results are presented when tested on the correlated instances of the 0-1 multidimensional knapsack problem.

In this work we consider a one-dimensional cutting stock problem in which the non-used material in the cutting patterns may be used in the future, if large enough. This feature introduces difficulties in comparing solutions of the cutting problem, for example, up to what extent a minimum leftover solution is the most interesting one when the leftover may be used. Some desirable characteristics of good solutions are defined and classical heuristic methods are modified, so that cutting patterns with undesirable leftover (not large enough to be used, nor too small to be acceptable waste) are redesigned. The performance of the modified heuristics is observed by solving instances from the literature, practical instances and randomly generated instances.

The cutting stock problem (CSP) is one of the most fascinating problems in operations research. The problem aims at determining the optimal plan to cut a number of parts of various length from an inventory of standard-size material so to satisfy the customers demands. The deterministic CSP ignores the uncertain nature of the demands thus typically providing recommendations that may result in overproduction or in profit loss. This paper proposes a stochastic version of the CSP which explicitly takes into account uncertainty. Using a scenario-based approach, we develop a two-stage stochastic programming formulation. The highly non-convex nature of the model together with its huge size prevent the application of standard software. We use a solution approach designed to exploit the specific problem structure. Encouraging preliminary computational results are provided.

The classical economic lot-sizing problem assumes that a single supplier and a single transportation mode are used to replenish the inventory. This paper studies an extension of this problem where several suppliers and transportation modes are available. The decision-making process in this case involves identifying (i) the timing for an order; (ii) the choice of shipment modes; and (iii) the order size for each mode. The problem is defined as a network flow problem with multiple setups cost function and additional side constraints. This study provides an MIP formulation for the problem. We also provide an additional formulation of the problem by redefining its decision variables and show that the dual of the corresponding LP-relaxation has a special structure. We take advantage of the structure of the dual problem to develop a primal–dual algorithm that generates tight lower and upper bounds. Computational results demonstrate the effectiveness of the algorithm.

This paper presents a two-stage approach for pattern generation and cutting plan determination of the one-dimensional cutting stock problem. Calculation of the total number of patterns that will be cut and generation of the cutting patterns are performed in the first stage. On the other hand, the second stage determines the cutting plan. The proposed approach makes use of two separate integer linear programming models. One of these models is employed by the first stage to generate the cutting patterns through a heuristic procedure with the objective of minimizing trim loss. The cutting patterns obtained from Stage 1 are then fed into the second stage. In this stage, another integer linear programming model is solved to form a cutting plan. The objective of this model is to minimize a generalized total cost function consisting of material inputs, number of setups, labor hours and overdue time; subject to demand requirements, material availability, regular and overtime availability, and due date constraints. The study also demonstrates an implementation of the proposed approach in a coronary stent manufacturer. The case study focuses on the cutting phase of the manufacturing process followed by manual cleaning and quality control activities. The experiments show that the proposed approach is suitable to the conditions and requirements of the company.

This paper describes a specific local search approach to solve a problem arising in logistics which we prove to be NP-hard. The problem is a complex scheduling or vehicle routing problem where we have to schedule the tours of concrete mixer vehicles over a working day from concrete-producing depots to concrete-demanding customers and vice versa. We give a general mixed integer programming model which is too hard to solve for state of the art mixed integer programming optimizers in the case of the usually huge problem instances coming from practice. Therefore we present a certain local search approach to be able to handle huge practical problem instances.

We consider the allocation of a limited budget to a set of activities or investments in order to maximize return from investment. In a number of practical contexts (e.g., advertising), the return from investment in an activity is effectively modeled using an S-curve, where increasing returns to scale exist at small investment levels, and decreasing returns to scale occur at high investment levels. We demonstrate that the resulting knapsack problem with S-curve return functions is NP-hard, provide a pseudo-polynomial time algorithm for the integer variable version of the problem, and develop efficient solution methods for special cases of the problem. We also discuss a fully-polynomial-time approximation algorithm for the integer variable version of the problem.

This paper deals with a procurement problem of missiles involving the efficient assignment of the missiles to some targets. Within a fixed amount of budget, a leader purchases several types of missiles, by which he aims to damage as much value as possible a follower hides into some facilities later. The effectiveness of the missile depends on the type of missile and facility. A payoff of the game is the expected amount of destroyed value. The problem is generalized as a two-person zero-sum game of distributing discrete resources with a leader and a follower. Our problem is to derive a Stackelberg equilibrium for the game. This type of game has an abundance of applications. The problem is first formulated into an integer programming problem with a non-separable objective function of variables and it is further equivalently transformed into a maximin integer knapsack problem. We propose three exacts methods and an approximation method for an optimal solution.

The two-dimensional guillotine bin packing problem consists of packing, without overlap, small rectangular items into the smallest number of large rectangular bins where items are obtained via guillotine cuts. This problem is solved using a new guillotine bottom left (GBL) constructive heuristic and its agent-based (A–B) implementation. GBL, which is sequential, successively packs items into a bin and creates a new bin every time it can no longer fit any unpacked item into the current one. A–B, which is pseudo-parallel, uses the simplest system of artificial life. This system consists of active agents dynamically interacting in real time to jointly fill the bins while each agent is driven by its own parameters, decision process, and fitness assessment. A–B is particularly fast and yields near-optimal solutions. Its modularity makes it easily adaptable to knapsack related problems.

This paper considers the minimization version of a class of nonconvex knapsack problems with piecewise linear cost structure. The items to be included in the knapsack have a divisible quantity and a cost function. An item can be included partially ill the given quantity range and the cost is a nonconvex piecewise linear function of quantity. Given a demand, the optimization problem is to choose an optimal quantity for each item such that the demand is satisfied and the total cost is minimized. This problem and its close variants are encountered in manufacturing planning, supply chain design, volume discount procurement auctions, and many other contemporary applications. Two separate mixed integer linear programming formulations of this problem are proposed and are compared with existing formulations. Motivated by different scenarios in which the problem is useful, the following algorithms are developed: (1) a fast polynomial time, near-optimal heuristic using convex envelopes; (2) exact pseudo-polynomial time dynamic programming algorithms; (3) a 2-approximation algorithm; and (4) a fully polynomial time approximation scheme. A comprehensive test suite is developed to generate representative problem instances with different haracteristics. Extensive computational experiments show that the proposed formulations and algorithms are faster than the existing techniques.

We consider a class of knapsack problems that include setup costs for families of items. An individual item can be loaded into the knapsack only if a setup cost is incurred for the family to which it belongs. A mixed integer programming formulation for the problem is provided along with exact and heuristic solution methods. The exact algorithm uses cross decomposition. The proposed heuristic gives fast and tight bounds. In addition, a Benders decomposition algorithm is presented to solve the continuous relaxation of the problem. This method for solving the continuous relaxation can be used to improve the performance of a branch and bound algorithm for solving the integer problem. Computational performance of the algorithms are reported and compared to CPLEX.

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm – for staged patterns – also uses a recurrence formula. We show that if the items are not so small compared to the dimensions of the bin, then these algorithms require polynomial time. Using these algorithms we solved all instances of the RK problem found at the OR-LIBRARY, including one for which no optimal solution was known. We also consider the Two-dimensional Cutting Stock problem. We present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns. We propose two strategies to tackle the residual instances. We also investigate a variant of this problem where the bins have different sizes. At last, we study the Two-dimensional Strip Packing problem. We also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed. In this case we solve instances for two-, three- and four-staged patterns. We report on some computational experiments with the various algorithms we propose in this paper. The results indicate that these algorithms seem to be suitable for solving real-world instances. We give a detailed description (a pseudo-code) of all the algorithms presented here, so that the reader may easily implement these algorithms.

In this paper, we study the problem of synchronized scheduling of assembly and air transportation to achieve accurate delivery with minimized cost in consumer electronics supply chain. This problem was motivated by a major PC manufacturer in consumer electronics industry. The overall problem is decomposed into two sub-problems, which consist of an air transportation allocation problem and an assembly scheduling problem. The air transportation allocation problem is formulated as an integer linear programming problem with the objective of minimizing transportation cost and delivery earliness tardiness penalties. The assembly scheduling problem seeks to determine a schedule ensuring that the orders are completed on time and catch the flights such that the waiting penalties between assembly and transportation is minimized. The problem is formulated as a parallel machine scheduling problem with earliness penalties. The computational complexities of the two sub-problems are investigated. The air transportation allocation problem with split delivery is shown to be solvable. The parallel machine assembly scheduling problem is shown to be NP-complete. Simulated annealing based heuristic algorithms are presented to solve the parallel machine problem.

This paper describes a new placement method based on pattern matching for 2D tiling problems. Tiling problem can be considered as a special case of bin packing. In the proposed method, the representation of the figures and the board is based on directional chain codes. Contrary to other works that the area has been used for the board and the figures, the proposed method is based on usage of their boundaries instead. With this representation, consideration of the area has been replaced with that of the exact string matching. With the proposed knowledge representation, rotation and reflection of the figures can be considered easily. The results of a hybrid approach of genetic algorithm and simulated annealing have been shown. This new method, introduces a novel approach for handling and solving a variety of 2D-packing problems.

In a steel tube mill where an endless stream of steel tube is supplied from a manufacturing facility, trim waste is never made regardless of cutting patterns used and the standard cutting stock problem seems meaningless. Therefore, the continuous stock cutting problem with setup is introduced to minimize the sum of cutting time and pattern changing time to meet the given demand. We propose a new configuration of cutting machines to achieve higher production efficiency, namely the open-ended configuration as opposed to the traditional closed-ended configuration, thereby two variants of the problem are defined. We propose linear formulations for both problems using binary expansion of the number of pieces of different types in a pattern. Furthermore, we define the time for pattern change as a linear function of the number of knives used in the pattern to be more realistic. Computational studies suggest that the open-ended cutting machine may improve the production time by up to 44% and that our linear formulations are more efficient than the existing ones.