Solving the CLSP by a Tabu Search Heuristic

The multi-item, single-level, capacitated, dynamic lot-sizing problem, commonly abbreviated as CLSP, is considered. The problem is cast in a tight mixed-integer programming model (MIP); tight in the sense that the gap between the optimal value of MIP and that of its linear programming relaxation (LP) is small. The LP relaxation of MIP is then solved by column generation. The resulting feasible solution is further improved by adopting the corresponding set-up schedule and re-optimizing variable costs by solving a minimum-cost network flow (trans-shipment) problem. Subsequently, the improved solution is used as a starting solution for a tabu search procedure, with the worth of moves assessed using the same trans-shipment problem. Results of computational testing of benchmark problem instances are presented. They show that the heuristic solutions obtained are effective, in that they are extremely close to the best known solutions. The computational efficiency makes it possible to solve realistically large problem instances routinely on a personal computer; in particular, the solution procedure is most effective, in terms of solution quality, for larger problem instances.     

Production planning via scenario modelling

Several Linear Programming (LP) and Mixed Integer Programming (MIP) models for the production and capacity planning problems with uncertainty in demand are proposed. In contrast to traditional mathematical programming approaches, we use scenarios to characterize the uncertainty in demand. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield a nonanticipative or implementable policy. Such an approach makes it possible to model nonstationarity in demand as well as a variety of recourse decision types. Two scenario-based models for formalizing implementable policies are presented. The first model is a LP model for multi-product, multi-period, single-level production planning to determine the production volume and product inventory for each period, such that the expected cost of holding inventory and lost demand is minimized. The second model is a MIP model for multi-product, multi-period, single-level production planning to help in sourcing decisions for raw materials supply. Although these formulations lead to very large scale mathematical programming problems, our computational experience with LP models for real-life instances is very encouraging.     

Mathematical programming based heuristics for the 0–1 MIP: a survey

The 0–1 mixed integer programming problem is used for modeling many combinatorial problems, ranging from logical design to scheduling and routing as well as encompassing graph theory models for resource allocation and financial planning. This paper provides a survey of heuristics based on mathematical programming for solving 0–1 mixed integer programs (MIP). More precisely, we focus on the stand-alone heuristics for 0–1 MIP as well as those heuristics that use linear programming techniques or solve a series of linear programming models or reduced problems, deduced from the initial one, in order to produce a high quality solution of a considered problem. Our emphasis will be on how mathematical programming techniques can be used for approximate problem solving, rather than on comparing performances of heuristics.     

An LP-based heuristic for two-stage capacitated facility location problems

In this paper, a linear programming based heuristic is considered for a two-stage capacitated facility location problem with single source constraints. The problem is to find the optimal locations of depots from a set of possible depot sites in order to serve customers with a given demand, the optimal assignments of customers to depots and the optimal product flow from plants to depots. Good lower and upper bounds can be obtained for this problem in short computation times by adopting a linear programming approach. To this end, the LP formulation is iteratively refined using valid inequalities and facets which have been described in the literature for various relaxations of the problem. After each reoptimisation step, that is the recalculation of the LP solution after the addition of valid inequalities, feasible solutions are obtained from the current LP solution by applying simple heuristics. The results of extensive computational experiments are given.     

Minimizing makespan in parallel flowshops

In this study, a new class of proportional parallel flow shop problems with the objective of minimizing the makespan has been addressed. A special case for this problem in which jobs are processed on only one machine as opposed to two or more machines in a flow shop, is the well-known multiple processor problem which is NP-complete. The parallel processor problem is a restricted version of the problems addressed in this paper and hence are NP-complete. We develop and test heuristic and simulation approaches to solve large scale problems, while using exact procedures for smaller problems. The performance of the heuristics relative to the LP lower bound as well as a comparison with the truncated integer programming solution are reported. The performance of the heuristics and the simulation results were encouraging.     

A hybrid adaptive large neighborhood search heuristic for lot-sizing with setup times

This paper presents a hybrid of a general heuristic framework and a general purpose mixed-integer programming (MIP) solver. The framework is based on local search and an adaptive procedure which chooses between a set of large neighborhoods to be searched. A mixed integer programming solver and its built-in feasibility heuristics is used to search a neighborhood for improving solutions. The general reoptimization approach used for repairing solutions is specifically suited for combinatorial problems where it may be hard to otherwise design suitable repair neighborhoods. The hybrid heuristic framework is applied to the multi-item capacitated lot sizing problem with setup times, where experiments have been conducted on a series of instances from the literature and a newly generated extension of these. On average the presented heuristic outperforms the best heuristics from the literature, and the upper bounds found by the commercial MIP solver ILOG CPLEX using state-of-the-art MIP formulations. Furthermore, we improve the best known solutions on 60 out of 100 and improve the lower bound on all 100 instances from the literature.     

Modelling the mobile target covering problem using flying drones

This paper addresses the mobile targets covering problem by using unmanned aerial vehicles (UAVs). It is assumed that each UAV has a limited initial energy and the energy consumption is related to the UAV’s altitude. Indeed, the higher the altitude, the larger the monitored area and the higher the energy consumption. When an UAV runs out of battery, it is replaced by a new one. The aim is to locate UAVs in order to cover the piece of plane in which the target moves by using a minimum number of UAVs. Each target has to be monitored for each instant time. The problem under consideration is mathematically represented by defining mixed integer non-linear optimization models. Heuristic procedures are defined and they are based on restricted mixed integer programming (MIP) formulation of the problem. A computational study is carried out to assess the behaviour of the proposed models and MIP-based heuristics. A comparison in terms of efficiency and effectiveness among models and heuristics is carried out.     

Feasibility pump 2.0

Finding a feasible solution of a given mixed-integer programming (MIP) model is a very important ${\mathcal{NP}}$ -complete problem that can be extremely hard in practice. Feasibility Pump (FP) is a heuristic scheme for finding a feasible solution to general MIPs that can be viewed as a clever way to round a sequence of fractional solutions of the LP relaxation, until a feasible one is eventually found. In this paper we study the effect of replacing the original rounding function (which is fast and simple, but somehow blind) with more clever rounding heuristics. In particular, we investigate the use of a diving-like procedure based on rounding and constraint propagation—a basic tool in Constraint Programming. Extensive computational results on binary and general integer MIPs from the literature show that the new approach produces a substantial improvement of the FP success rate, without slowing-down the method and with a significantly better quality of the feasible solutions found.     

A robust optimization model for multi-product two-stage capacitated production planning under uncertainty

Production planning (PP) is one of the most important issues carried out in manufacturing environments which seeks efficient planning, scheduling and coordination of all production activities that optimizes the company’s objectives. In this paper, we studied a two-stage real world capacitated production system with lead time and setup decisions in which some parameters such as production costs and customer demand are uncertain. A robust optimization model is developed to formulate the problem in which minimization of the total costs including the setup costs, production costs, labor costs, inventory costs, and workforce changing costs is considered as performance measure. The robust approach is used to reduce the effects of fluctuations of the uncertain parameters with regards to all the possible future scenarios. A mixed-integer programming (MIP) model is developed to formulate the related robust production planning problem. In fact the robust proposed model is presented to generate an initial robust schedule. The performance of this schedule could be improved against of any possible occurrences of uncertain parameters. A case from an Iran refrigerator factory is studied and the characteristics of factory and its products are discussed. The computational results display the robustness and effectiveness of the model and highlight the importance of using robust optimization approach in generating more robust production plans in the uncertain environments. The tradeoff between solution robustness and model robustness is also analyzed.     

A two-dimensional strip cutting problem with sequencing constraint

We study a strip cutting problem that arises in the production of corrugated cardboard. In this context, rectangular items of different sizes are obtained by machines, called corrugators, that cut strips of large dimensions according to particular schemes containing at most two types of items. Because of buffer restrictions, these schemes have to be sequenced in such a way that, at any moment, at most two types of items are in production and not completed yet (sequencing constraint). We show that the problem of finding a set of schemes of minimum trim loss that satisfies an assigned demand for each item size is strongly NP-hard, even if the sequencing constraint is relaxed. Then, we present two heuristics for the problem with the sequencing constraint, both based on a graph characterization of the feasible solutions. The first heuristic is a two-phase procedure based on a mixed integer linear programming model. The second heuristic follows a completely combinatorial approach and consists of solving a suitable sequence of minimum cost matching problems. For both procedures, an upper bound on the number of schemes (setups) is found. Finally, a computational study comparing the quality of the heuristic solutions with respect to an LP lower bound is reported.     

One-Dimensional Cutting Stock Problem with a Given Number of Setups: A Hybrid Approach of Metaheuristics and Linear Programming

One-dimensional cutting stock problem (1D-CSP) is one of the representative combinatorial optimization problems, which arises in many industrial applications. Since the setup costs for switching different cutting patterns become more dominant in recent cutting industry, we consider a variant of 1D-CSP, called the pattern restricted problem (PRP), to minimize the number of stock rolls while constraining the number of different cutting patterns within a bound given by users. For this problem, we propose a local search algorithm that alternately uses two types of local search processes with the 1-add neighborhood and the shift neighborhood, respectively. To improve the performance of local search, we incorporate it with linear programming (LP) techniques, to reduce the number of solutions in each neighborhood. A sensitivity analysis technique is introduced to solve a large number of associated LP problems quickly. Through computational experiments, we observe that the new algorithm obtains solutions of better quality than those obtained by other existing approaches.     

Experiments in integer programming

A hybrid algorithm to solve large scale zero–one integer programming problems has been developed. The algorithm combines branch-and-bound, enumeration and cutting plane techniques. Mixed-integer cuts are generated in the initial phase of the algorithm and added to the L.P. Benders cuts are derived and used implicitly but, except for the cut from the initial LP, are not stored. The algorithm has been implemented on an experimental basis in MPSX/370 using its Extended Control Language and Algorithmic Tools. A computational study based on five well-known difficult test problems and on three practical problems with up to 2000 zer–one variables shows that the hybrid code compares favorably with MIP/370 and with results published for other algorithms.     

A mathematical programming model and solution for scheduling production orders in Shanghai Baoshan Iron and Steel Complex

In this paper, we investigate the production order scheduling problem derived from the production of steel sheets in Shanghai Baoshan Iron and Steel Complex (Baosteel). A deterministic mixed integer programming (MIP) model for scheduling production orders on some critical and bottleneck operations in Baosteel is presented in which practical technological constraints have been considered. The objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders. Due to large numbers of variables and constraints in the model, a decomposition solution methodology based on a synergistic combination of Lagrangian relaxation, linear programming and heuristics is developed. Unlike the commonly used method of relaxing capacity constraints, this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by Lagrangian multipliers. The Lagrangian relaxed problem can be decomposed into two sub-problems by separating continuous variables from integer ones. The sub-problem that relates to continuous variables is a linear programming problem which can be solved using standard software package OSL, while the other sub-problem is an integer programming problem which can be solved optimally by further decomposition. The subgradient optimization method is used to update Lagrangian multipliers. A production order scheduling simulation system for Baosteel is developed by embedding the above Lagrangian heuristics. Computational results for problems with up to 100 orders show that the proposed Lagrangian relaxation method is stable and can find good solutions within a reasonable time.     

Rollout Algorithms for Stochastic Scheduling Problems

Stochastic scheduling problems are difficult stochastic control problems with combinatorial decision spaces. In this paper we focus on a class of stochastic scheduling problems, the quiz problem and its variations. We discuss the use of heuristics for their solution, and we propose rollout algorithms based on these heuristics which approximate the stochastic dynamic programming algorithm. We show how the rollout algorithms can be implemented efficiently, with considerable savings in computation over optimal algorithms. We delineate circumstances under which the rollout algorithms are guaranteed to perform better than the heuristics on which they are based. We also show computational results which suggest that the performance of the rollout policies is near-optimal, and is substantially better than the performance of their underlying heuristics.     

Large neighborhood search for LNG inventory routing

Liquefied Natural Gas (LNG) is steadily becoming a common mode for commercializing natural gas. Due to the capital intensive nature of LNG projects, the optimal design of LNG supply chains is extremely important from a profitability perspective. Motivated by the need for a model that can assist in the design analysis of LNG supply chains, we address an LNG inventory routing problem where optimized ship schedules have to be developed for an LNG project. In this paper, we present an arc-flow formulation based on the MIP model of Song and Furman (Comput. Oper. Res., 2010). We also present a set of construction and improvement heuristics to solve this model efficiently. The heuristics are evaluated based on a set of realistic test instances that are very large relative to the problem instances seen in recent literature related to this problem. Extensive computational results indicate that the proposed methods are computationally efficient in finding optimal or near optimal solutions and are substantially faster than state-of-the-art commercial optimization software.     

Twenty years of linear programming based portfolio optimization

Markowitz formulated the portfolio optimization problem through two criteria: the expected return and the risk, as a measure of the variability of the return. The classical Markowitz model uses the variance as the risk measure and is a quadratic programming problem. Many attempts have been made to linearize the portfolio optimization problem. Several different risk measures have been proposed which are computationally attractive as (for discrete random variables) they give rise to linear programming (LP) problems. About twenty years ago, the mean absolute deviation (MAD) model drew a lot of attention resulting in much research and speeding up development of other LP models. Further, the LP models based on the conditional value at risk (CVaR) have a great impact on new developments in portfolio optimization during the first decade of the 21st century. The LP solvability may become relevant for real-life decisions when portfolios have to meet side constraints and take into account transaction costs or when large size instances have to be solved. In this paper we review the variety of LP solvable portfolio optimization models presented in the literature, the real features that have been modeled and the solution approaches to the resulting models, in most of the cases mixed integer linear programming (MILP) models. We also discuss the impact of the inclusion of the real features.     

A multi-period network design problem for cellular telecommunication systems

Mathematical Programming models for multi-period network design problems, which arise in cellular telecommunication systems are presented. The underlying network topologies range from a simple star to complex multi-layer Steiner-like networks. Linear programming, Lagrangian relaxation, and branch-and-cut heuristics are proposed and a polynomial-bounded heuristic based on an interior point linear programming implementation is described. Extensive computational results are presented on a number of randomly generated problem sets and the performance of the heuristic(s) are compared with an optimal branch-and-bound algorithm.     

Scheduling a hybrid assembly-differentiation flowshop to minimize total flow time

This study considers a hybrid assembly-differentiation flowshop scheduling problem (HADFSP), in which there are three production stages, including components manufacturing, assembly, and differentiation. All the components of a job are processed on different machines at the first stage. Subsequently, they are assembled together on a common single machine at the second stage. At the third stage, each job of a particular type is processed on a dedicated machine. The objective is to find a job schedule to minimize total flow time (TFT). At first, a mixed integer programming (MIP) model is formulated and then some properties of the optimal solution are presented. Since the NP-hardness of the problem, two fast heuristics (SPT-based heuristic and NEH-based heuristic) and three hybrid meta-heuristics (HGA-VNS, HDDE-VNS and HEDA-VNS) are developed for solving medium- and large-size problems. In order to evaluate the performances of the proposed algorithms, a lower bound for the HADFSP with TFT criteria (HADFSP-TFT) is established. The MIP model and the proposed algorithms are compared on randomly generated problems. Computational results show the effectiveness of the MIP model and the proposed algorithms. The computational analysis indicates that, in average, the HDDE-VNS performs better and more robustly than the other two meta-heuristics, whereas the NEH heuristic consume little time and could reach reasonable solutions.     

New convergent heuristics for 0–1 mixed integer programming

Several hybrid methods have recently been proposed for solving 0–1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0–1 multidimensional Knapsack problem. Other encouraging results obtained for 0–1 MIP problems are also presented.     

Four-Day Workweek Scheduling with Two or Three Consecutive Days Off

A four-day workweek days-off scheduling problem is considered. Out of the three days off per week for each employee, either two or three days must be consecutive. An optimization algorithm is presented which starts by utilizing the problem's special structure to determine the minimum workforce size. Subsequently, workers are assigned to different days-off work patterns in order to minimize either the total number or the total cost of the workforce. Different procedures must be followed in assigning days-off patterns, depending on the characteristics of labor demands. In some cases, optimum days-off assignments are determined without requiring mathematical programming. In other cases, a workforce size constraint is added to the integer programming model, greatly improving computational performance.