A solution approach for optimizing long- and short-term production scheduling at LKAB’s Kiruna mine

We present a mixed-integer program to schedule long- and short-term production at LKAB’s Kiruna mine, an underground sublevel caving mine located in northern Sweden. The model minimizes deviations from monthly preplanned production quantities while adhering to operational constraints. Because of the mathematical structure of the model and its moderately large size, instances spanning a time horizon of more than a year or two tend to be intractable. We develop an optimization-based decomposition heuristic that, on average, obtains better solutions faster than solving the model directly. We show that for realistic data sets, we can generate solutions with deviations that comprise about 3-6% of total demand in about a third of an hour.     

Discrete and continuous time representations and mathematical models for large production scheduling problems: A case study from the pharmaceutical industry

The underlying time framework used is one of the major differences in the basic structure of mathematical programming formulations used for production scheduling problems. The models are either based on continuous or discrete time representations. In the literature there is no general agreement on which is better or more suitable for different types of production or business environments. In this paper we study a large real-world scheduling problem from a pharmaceutical company. The problem is at least NP-hard and cannot be solved with standard solution methods. We therefore decompose the problem into two parts and compare discrete and continuous time representations for solving the individual parts. Our results show pros and cons of each model. The continuous formulation can be used to solve larger test cases and it is also more accurate for the problem under consideration.     

A Multi-Period Renewal equipment problem

This paper looks at a Multi-Period Renewal equipment problem (MPR). It is inspired by a specific real-life situation where a set of hardware items is to be managed and their replacement dates determined, given a budget over a time horizon comprising a set of periods. The particular characteristic of this problem is the possibility of carrying forward any unused budget from one period to the next, which corresponds to the multi-periodicity aspect in the model. We begin with the industrial context and deduce the corresponding knapsack model that is the subject of this paper. Links to certain variants of the knapsack problem are next examined. We provide a study of complexity of the problem, for some of its special cases, and for its continuous relaxation. In particular, it is established that its continuous relaxation and a special case can be solved in (strongly) polynomial time, that three other special cases can be solved in pseudo-polynomial time, while the problem itself is strongly NP-hard when the number of periods is unbounded. Next, two heuristics are proposed for solving the MPR problem. Experimental results and comparisons with the Martello&Toth and Dantzig heuristics, adapted to our problem, are provided.     

Machine scheduling with a rate-modifying activity

Motivated by a problem commonly found in electronic assembly lines, this paper deals with the problem of scheduling jobs and a rate-modifying activity on a single machine. A rate-modifying activity is an activity that changes the production rate of the equipment under consideration. Hence the processing times of jobs vary depending on whether the job is scheduled before or after the rate-modifying activity. The decisions under consideration are when to schedule the rate-modifying activity and the sequence of jobs to optimize some performance measure.In this paper, we develop polynomial algorithms for solving problems of minimizing makespan, and total completion time respectively. We also develop pseudo-polynomial algorithms for solving problems of total weighted completion time under the agreeable ratio assumption. We prove that the problem of minimizing maximum lateness is NP-hard and also provide a pseudo-polynomial time algorithm to solve it optimally.     

Lattice preconditioning for the real relaxation branch-and-bound approach for integer least squares problems

The integer least squares problem is an important problem that arises in numerous applications. We propose a real relaxation-based branch-and-bound (RRBB) method for this problem. First, we define a quantity called the distance to integrality, propose it as a measure of the number of nodes in the RRBB enumeration tree, and provide computational evidence that the size of the RRBB tree is proportional to this distance. Since we cannot know the distance to integrality a priori, we prove that the norm of the Moore–Penrose generalized inverse of the matrix of coefficients is a key factor for bounding this distance, and then we propose a preconditioning method to reduce this norm using lattice reduction techniques. We also propose a set of valid box constraints that help accelerate the RRBB method. Our computational results show that the proposed preconditioning significantly reduces the size of the RRBB enumeration tree, that the preconditioning combined with the proposed set of box constraints can significantly reduce the computational time of RRBB, and that the resulting RRBB method can outperform the Schnorr and Eucher method, a widely used method for solving integer least squares problems, on some types of problem data.     

Joint pricing and ordering policy for an advance booking system with partial order cancellations

In this paper, an inventory model for deteriorating items with price-dependent demand is developed. The cycle is divided into two periods, where an advance sales period is followed by a spot sales period. In practice, customers with reservations may cancel their orders before receiving them. During the advance sales period, the rate of reservations which will not be cancelled is dependent on the length of the waiting time for the receiving order. During the spot sales period, all customers receive their orders at the time of the purchase. We prove the existence of the realistic relationship that the advance sales price is smaller than the spot sales price. We also develop some useful properties and provide an iterative procedure for solving the maximization problem. Numerical examples are given to demonstrate the effectiveness of the proposed approach and we conclude the paper with suggestions for possible future research.     

考虑交货期的生产运输联合优化问题研究

结合企业实际场景研究了考虑交货期的多个工厂、多条生产线、单一产品的生产与运输联合优化问题.已知客户订单需求量和交货时间窗,考虑了各条生产线在不同时段的生产能力约束,在满足交货时间窗约束的前提下,以生产、存储、运输费用之和极小化为目标建立了生产与运输联合优化问题的混合整数规划模型,通过分析模型结构证明了在不考虑固定生产成本的情况下,模型等价于最小费用最大流问题,得出了直接用Gurobi软件求解模型的可行性.最后利用具体算例进行模拟计算并与企业目前采用的顺序优化方法进行对比,发现采取联合优化策略可以明显降低总费用,并随着单位运输成本的增加,联合优化策略较顺序优化策略的优势在不断扩大.研究成果为企业制定生产与运输计划提供了理论依据.     

Production planning problem with sequence dependent setups as a bilevel programming problem

Each of n products is to be processed on two machines in order to satisfy known demands in each of T periods. Only one product can be processed on each machine at any given time. Each switch from one item to another requires sequence dependent setup time. The objective is to minimize the total setup time and the sum of the costs of production, storage and setup. We consider the problem as a bilevel mixed 0–1 integer programming problem. The objective of the leader is to assign the products to the machines in order to minimize the total sequence dependent setup time, while the objective of the follower is to minimize the production, storage and setup cost of the machine. We develop a heuristics based on tabu search for solving the problem. At the end, some computational results are presented.     

Formulating and solving a multi-mode resource-collaboration and constrained scheduling problem (MRCCSP)

The main motivation of this study is to provide, for the first time, a formulation and solution for a class of production scheduling problems (as in cluster tools) characterized mainly by resource collaboration to perform an operation and while allowing batches and considering alternative production methods. We develop a formulation for the new problem and term it a multiple mode per operation, resource collaboration, and constrained scheduling problem (MRCCSP). Some of the important new characteristics we consider are: multiple products (families); multiple orders (jobs) per family; precedence restrictions among the operations that constitute a job; alternative modes for the performance of an operation (each of which needs a set of collaborating resources) may be defined; complementary and exclusive restrictions between operation-modes; batch production is allowed; and setup times may depend on sequence and batch-size. The objective of the MRCCSP is to minimize makespan. We formulate the MRCCSP as a mixed integer linear programming model, and acknowledging the considerable size of the monolithic formulation required, we prescribe a specific method to achieve size reduction. Finally, a customized branch and bound algorithm for optimally solving this problem is proposed and examined experimentally.     

Scheduling with batch setup times and earliness-tardiness penalties

We consider a scheduling model in which several batches of jobs need to be processed by a single machine. During processing, a setup time is incurred whenever there is a switch from processing a job in one batch to a job in another batch. All the jobs in the same batch have a common due date that is either externally given as an input data or internally determined as a decision variable. Two problems are investigated. One problem is to minimize the total earliness and tardiness penalties provided that each due date is externally given. We show that this problem is NP-hard even when there are only two batches of jobs and the two due dates are unrestrictively large. The other problem is to minimize the total earliness and tardiness penalties plus the total due date penalty provided that each due date is a decision variable. We give some optimality properties for this problem with the general case and propose a polynomial dynamic programming algorithm for solving this problem with two batches of jobs. We also consider a special case for both of the problems when the common due dates for different batches are all equal. Under this special case, we give a dynamic programming algorithm for solving the first problem with an unrestrictively large due date and for solving the second problem. This algorithm has a running time polynomial in the number of jobs but exponential in the number of batches.     

A functional partial differential equation arising in a cell growth model with dispersion

In this paper we solve an initial‐boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first‐order pde; the stochastic version yields a second‐order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first‐order case, the analysis does not readily extend to the second‐order case. We develop a method for solving the second‐order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.     

Quasi-linear stochastic programming model based on expectation and variance and its application in transportation problem

The stochastic transportation problem involves in many areas such as production scheduling, facility location, resource allocation, logistics management. Constructing an operable solving method has important theoretical and practical value. In this paper, we first analyze the characteristic and deficiencies of the existing stochastic programming methods, such as higher computation complexity. We then give the concept of reliability coefficient and a quasi-linear processing pattern based on expectation and variance. We further analyze the relationship between reliability coefficient and reliability degree, also give the selecting strategy of reliability coefficient. Based on that, we establish a quasi-linear programming model for stochastic transportation problem, and we analyze its performance by a case-based example. The results indicate that this model has good interpretability and operability. It can effectively solve the transportation problem under complex stochastic environment or with incomplete information.     

Flow shop scheduling with heterogeneous workers

We propose an extension to the flow shop scheduling problem named Heterogeneous Flow Shop Scheduling Problem (Het-FSSP), where two simultaneous issues have to be resolved: finding the best worker assignment to the workstations, and solving the corresponding scheduling problem. This problem is motivated by Sheltered Work centers for Disabled, whose main objective is the labor integration of persons with disabilities, an important aim not only for these centers but for any company desiring to overcome the traditional standardized vision of the workforce. In such a scenario the goal is to maintain high productivity levels by minimizing the maximum completion time, while respecting the diverse capabilities and paces of the heterogeneous workers, which increases the complexity of finding an optimal schedule. We present a mathematical model that extends a flow shop model to admit a heterogeneous worker assignment, and propose a heuristic based on scatter search and path relinking to solve the problem. Computational results show that this approach finds good solutions within a short time, providing the production managers with practical approaches for this combined assignment and scheduling problem.     

Recursive functions on the plane and FPTASs for production planning and scheduling problems with two facilities

We consider a production model with two facilities sharing a resource during a time horizon consisting of a number of time periods. Cumulative production levels at the ends of consecutive periods are linked with constraints of a general form. This allows us to give different interpretations related to scheduling and input–output analysis. The model may arise either separately or in the structure of more general production models. In both cases it is reasonable to find an optimal or near-optimal distribution of resources between these two facilities. This helps either to develop a new production plan or to improve an existing one. The problem in question is NP-hard. We show that our approach leads to fully polynomial time approximation schemes (FPTASs).     

Some remarks on the dynamics of cell membranes

Motivated by the desire to model the entry of 1,25D into a cell by receptor mediated endocytosis, we have formulated the problem as the dynamics of a bilayer membrane. We have discussed setting the problem as a variational problem using the Helfrich modeling of the bilayer in terms of the free energy. Using a Lagrangian formulation we arrive at the Euler–Lagrange equations for the system. The model we have used depends on the amount of reagent in the neighborhood of the upper membrane. The problem thereby reduces to a moving boundary problem, which is dependent on a diffusion equation for a region changing with time. In order to solve this problem we seek the correct Neumann function for this altered. This is accomplished by deriving a Hadamard variational formula for the diffusion equation. We also offer an iterative procedure for solving this non-linear problem.     

Overlapping Schwarz waveform relaxation method for the solution of the reaction-diffusion equation

We are interested in solving time dependent problems using domain decomposition methods. In the classical approach, one discretizes first the time dimension and then one solves a sequence of steady problems by a domain decomposition method. In this article, we treat directly the time dependent problem and we study a Schwarz waveform relaxation algorithm for the convection diffusion equation. We study the convergence of the overlapping Schwarz waveform relaxation method for solving the reaction-diffusion equation over multi-overlapped subdomains. Also we will show that the method converges linearly and superlinearly over long and short time intervals, and the convergence depends on the size of overlap. Numerical results are presented from solutions of a specific model problems to demonstrate the convergence, linear and superlinear, and the role of the overlap size.     

Optimization-based heuristics for underground mine scheduling

Underground mine production scheduling possesses mathematical structure similar to and yields many of the same challenges as general scheduling problems. That is, binary variables represent the time at which various activities are scheduled. Typical objectives seek to minimize costs or some measure of production time, or to maximize net present value; two principal types of constraints exist: (i) resource constraints and (ii) precedence constraints. In our setting, we maximize “discounted metal production” for the remaining life of an underground lead and zinc mine that uses three different underground methods to extract the ore. Resource constraints limit the grade, tonnage, and backfill paste (used for structural stability) in each time period, while precedence constraints enforce the sequence in which extraction (and backfill) is performed in accordance with the underground mining methods used. We tailor exact and heuristic approaches to reduce model size, and develop an optimization-based decomposition heuristic; both of these methods transform a computationally intractable problem to one for which we obtain solutions in seconds, or, at most, hours for problem instances based on data sets from the Lisheen mine near Thurles, Ireland.     

A sliding time window heuristic for open pit mine block sequencing

The open pit mine block sequencing problem (OPBS) seeks a discretetime production schedule that maximizes the net present value of the orebody extracted from an open-pit mine. This integer program (IP) discretizes the mine’s volume into blocks, imposes precedence constraints between blocks, and limits resource consumption in each time period. We develop a “sliding time window heuristic” to solve this IP approximately. The heuristic recursively defines, solves and partially fixes an approximating model having: (i) fixed variables in early time periods, (ii) an exact submodel defined over a “window” of middle time periods, and (iii) a relaxed submodel in later time periods. The heuristic produces near-optimal solutions (typically within 2% of optimality) for model instances that standard optimization software fails to solve. Furthermore, it produces these solutions quickly, even though our OPBS model enforces standard upper-bounding constraints on resource consumption along with less standard, but important, lower-bounding constraints.     

An Evolutionary Approach for the Design and Scheduling of Electroplating Facilities

This paper tackles the Cyclic Hoists Scheduling Problem. This problem is often encountered in electroplating facilities when mass production is required. Then a repetitive sequence of moves is searched for the hoists. We more precisely deal with a global optimization problem that simultaneously considers the design and the scheduling of such production lines. It consists in studying systems integrating several transportation resources, called hoists, by minimizing the cycle time, while minimizing the number of hoists used. To achieve these goals, we use an evolutionary approach. The encoding of one solution is based on the representation of the empty moves of the hoists. To evaluate each individual, we propose a linear programming model. This one both verifies the satisfaction of constraints and provides the best cycle time for the considered number of hoists. This contribution describes a promising approach to solving a simple version of this problem, namely cyclic hoist scheduling, based on Evolutionary Algorithms (EAs), which is an optimization method inspired by biological evolution models. The issues of solution encoding and specialised genetic operators with a repair procedure of the infeasible solutions are discussed. Some results are presented with benchmark examples.      

Developing mathematical modelling skills: The role of CAS

Mathematical modelling as one component of problem solving is an important part of the mathematics curriculum and problem solving skills are often the most quoted generic skills that should be developed as an outcome of a programme of mathematics in school, college and university. Often there is a tension between mathematics seen at all levels as ‘a body of knowledge’ to be delivered at all costs and mathematics seen as a set of critical thinking and questioning skills. In this era of powerful software on hand-held and computer technologies there is an opportunity to review the procedures and rules that form the ‘body of knowledge’ that have been the central focus of the mathematics curriculum for over one hundred years. With technology we can spend less time on the traditional skills and create time for problem solving skills. We propose that mathematics software in general and CAS in particular provides opportunities for students to focus on the formulation and interpretation phases of the mathematical modelling process. Exploring the effect of parameters in a mathematical model is an important skill in mathematics and students often have difficulties in identifying the different role of variables and parameters This is an important part of validating a mathematical model formulated to describe, a real world situation. We illustrate how learning these skills can be enhanced by presenting and analysing the solution of two optimisation problems.