This paper considers a flow shop scheduling problem with batch processing machines. Each batch processing machine has a limited capacity and can process a group of jobs, each of them having a different known capacity requirement, simultaneously. Job processing time on each machine is known and arbitrary. The processing time of a batch on each machine is the longest processing time of all jobs in the batch. We improve the only existing mixed integer linear formulation (MILF) of the problem through significant reduction in size complexity of the model. Results justify that the improved MILF is clearly more efficient in reducing the required time for obtaining optimal makespan of small-size problems, in comparison with the existing MILF. Motivated by relaxing variety of the problem assumptions, several valid lower bounds on the optimal makespan are also proposed that can furtheraccelerate obtaining optimal solution through proposed MILF. Robustness evaluation of each bound under the different problem settings is reported through computations.

This paper investigates the first attempt on the batch-processing machine scheduling problem, where the machine can process multiple jobs simultaneously, using an ant colony optimization metaheuristic. We consider the scheduling problem of a single batch-processing machine with incompatible job families and the performance measure of minimizing total weighted completion time. Jobs of a given family have an identical processing time and are characterized by arbitrary sizes and weights. Based on a number of developed heuristic approaches, we propose an ant colony framework (ACF) in two versions, which are distinguished by the type of embedded heuristic information. Each version is also investigated in two formats, that is the pure ACF and the hybridized ACF. To verify the performance of our framework, comparisons are made based on using a set of well-known existing heuristic and meta-heuristic algorithms taken from the literature, on a diverse set of artificially generated test problem instances. Computational results show the high performance of the proposed framework and signify its ability to outperform the comparator algorithms in most cases as the problem size increases.

Long-term production planning (LTPP) is a large-scale optimization problem that aims to find the block extraction sequence that produces the maximum possible net present value (NPV) whilst satisfying a variety of physical and economical constraints. The economic feasibility of a mine is highly dependent upon careful LTPP. As the mining industries extract deeper and lower grade ores, LTPP is becoming a key item that can result in ceasing operations or continuing the project. Mathematical programming models are well suited to optimizing LTPP of open pit mines. These mathematical models have been studied extensively in the literature since the 1960s. The result of this study shows that there are two approaches for dealing with LTPP problems: (1) deterministic and (2) uncertainty-based approaches. This paper first discusses the deterministic algorithms and then, after an introduction to uncertainty associated with mining projects, reviews uncertainty-based algorithms. The advantages and disadvantages of these algorithms are discussed and suggestions for future research are offered.

This paper investigates the scheduling problem of parallel identical batch processing machines in which each machine can process a group of jobs simultaneously as a batch. Each job is characterized by its size and processing time. The processing time of a batch is given by the longest processing time among all jobs in the batch. Based on developing heuristic approaches, we proposed a hybrid genetic heuristic (HGH) to minimize makespan objective. To verify the performance of our algorithm, comparisons are made through using a simulated annealing (SA) approach addressed in the literature as a comparator algorithm. Computational experiments reveal that affording the knowledge of problem through using heuristic procedures, gives HGH the ability of finding optimal or near optimal solutions in a reasonable time.

This article addresses the economic lot sizing and scheduling problem in flexible flow lines with unrelated parallel machines over a finite planning horizon. The objective is determination of a cyclic schedule that minimizes the sum of setup and inventory holding costs per unit time without any stock-out. A new mixed zero–one nonlinear mathematical programming has been developed for the problem. Due to difficulty of obtaining the optimal solution especially for medium and large-sized problems, we have also proposed two algorithms: a hybrid genetic algorithm (HGA) and a simulated annealing (SA). Two efficient constructive heuristic algorithms have been also proposed that provide some initial solutions for the algorithms. Moreover, a new local search approach has been proposed to improve solution quality of algorithms. The two proposed solution methods are compared on 180 randomly generated problems. Computational results indicate the superiority of the proposed HGA in compare to the SA with respect to the solution quality, but the proposed SA outperforms the proposed HGA with respect to the required computations time.

There are some issues which have to be addressed when designing an automated guided vehicles system (AGVS) such as flow-path layout, traffic management, the number and the location of pick-up and delivery points, vehicle routing and so on. One of the AGVS guide path configurations discussed in the previous researches includes a single-loop which is the subject of this paper. Many researchers have worked on the subject of finding the shortest single loop in a given block layout. This problem has been already proved to be NP-complete. Finding the location of pick-up/drop-off (P/D) stations on a given loop is what is typically done, but locating the loop and the P/D stations simultaneously is actually more realistic. In this paper, a genetic algorithm (GA) has been developed to determine these two simultaneously. The algorithm only reproduces the feasible solutions. The objective is to minimize the total traveled distance. A from-to chart and a block layout are the inputs and a unidirectional loop including its corresponding direction and along with the locations of the P/D stations on the loop are the outputs. To show the efficiency of the algorithm, the associated results for some sample problems have been compared with the results generated by LINGO for the equivalent mixed integer programming (MIP) problem. Computational results show the efficiency of the algorithm in solving relatively large size problem.

This paper addresses the common cycle multi-product lot sizing and scheduling problem in deterministic flexible flow shops where the planning horizon is finite and fixed by management and the production stages are in series, while separated by finite intermediate buffers. The limited intermediate buffers between the stages result in a blocking scheduling problem, where a completed part may remain on a machine and block it until a downstream machine becomes available. This prevents another part from being processed on the blocked machine. We propose a new mixed integer nonlinear program and due to difficulty of obtaining the optimal solution in medium and large-scaled problems, we use both genetic algorithm and tabu search methods to find an optimal or near-optimal solution for the problem. The two proposed solution methods are compared with optimal enumeration method (OEM) on randomly generated problems. Computational results show that the performance of both GA and TS are very promising.

In this paper, a practical and appropriate algorithm is provided for order production planning under uncertainty conditions. Producing the order production projects using different methodologies is facing uncertainty and therefore specific techniques are required to determine their production planning schedule. Network models combining fuzzy logic to define the sequence of the production stages and uncertainty are appropriate tools for order production planning modeling. In this paper, start time parameters of each activity, duration of each activity and number of loops in the products production network are defined as triangular fuzzy numbers (pessimistic, possible, optimistic) and results of the production planning are defined as fuzzy sets. In this paper, network calculations including fuzzy forward CPM are performed. Output of this algorithm is a fuzzy order production schedule. This method is more practical than existing production planning schedules determination methods and has less calculation and has upgrade and mechanization capability.

In this paper, we investigate the lot and delivery scheduling problem in a simple supply chain where a single supplier produces multiple components on a flexible flow line (FFL) and delivers them directly to an assembly facility (AF). It is assumed that all of parameters such as demand rates for the components are deterministic and constant over a finite planning horizon. The main objective is to find a lot and delivery schedule that would minimize the average of holding, setup, and transportation costs per unit time for the supply chain. We develop a new mixed integer nonlinear program (MINLP) and an optimal enumeration method to solve the problem. Due to difficulty of obtaining the optimal solution in medium and large-scaled problems, a hybrid genetic algorithm (HGA) is also developed. The proposed HGA incorporates a neighborhood search (NS) into a basic genetic algorithm that enables the algorithm to perform genetic search over the subspace of local optima. The two proposed solution methods are compared on randomly generated problems, and computational results show that the performance of HGA is very promising because it is able to find an optimal or near-optimal solution for majority of the test problems.

The paper addresses minimizing makespan by a genetic algorithm (GA) for scheduling jobs with non-identical sizes on a single-batch-processing machine. A batch-processing machine can process up to B jobs simultaneously. The processing time of a batch is equal to the longest processing time among all jobs in the batch. Two different GAs are proposed based on different encoding schemes. The first is a sequence-based GA (SGA) that generates random sequences of jobs using GA operators and applies the batch first fit heuristic to group the jobs. The second is a batch-based hybrid GA (BHGA) that generates random batches of jobs using GA operators and ensures feasibility by using knowledge of the problem based on a heuristic procedure. A greedy local search heuristic based on the problem characteristics is hybridized with a BHGA that has the ability of steering efficiently the search toward the optimal or nearoptimal schedules. The performance of proposed GAs is compared with a simulated annealing (SA) approach proposed by Melouk et al. (Melouk, S., Damodaran, P. and Chang, P.Y., Minimizing makespan for single machine batch processing with non-identical job sizes using simulated annealing. Int. J. Prod. Econ., 2004, 87, 141–147) and also against a modified lower bound proposed for the problem. Computational results show that BHGA performs considerably well compared with the modified lower bound and significantly outperforms the SGA and SA in terms of both quality of solutions and required runtimes.

In this paper,the multi-item, single-level, capacitated, dynamic lot sizing problem with set-up carry-over and backlogging,abbreviated to CLSP+,is considered. The problem is formulated as a mixed integer programming problem. A heuristic method consisting of four elements: (1) a demand shifting rule,(2) lot size determination rules,(3) checking feasibility conditions and (4) set-up carry-over determination,provides us with an initial feasible solution. The resulting feasible solution is improved by adopting the corresponding set-up and set-up carry-over schedule and re-optimizing it by solving a minimum-cost network flow problem. Then the improved solution is used as a starting solution for a tabu search procedure,with the value of moves assessed using the same minimum-cost network problem. Computational results on randomly generated problems show that the algorithm,which is coded in C++,is able to provide optimal solutions or solutions extremely close to optimal. The computational efficiency makes it possible to solve reasonably large problem instances routinely on a personal computer.

This paper attempts to model long-term production scheduling problems by chance constrained binary integer programming in a stochastic environment. This stochastic model is set up to account for ore block grade uncertainty. The probability distribution function of grade in each block is used as a stochastic input to the optimization model. This distribution function in each block should be determined using a Geostatistical Simulation approach. The deterministic equivalents of these chance constraints are then achieved, which are in the form of nonlinear in binary variables. A confidence level at which it is desired that the uncertain constraints holds, is specified in each scheduling period. Rather than the previous risk-based model, this formulation will yield schedules with a high chance of achieving planned production targets while maximizes the expectation of net present value and minimizing the variance function simultaneously. Using this method, the grade uncertainty is integrated explicitly into the optimization process.

This paper addresses the common cycle multi-product lot-scheduling problem in flexible flow lines (FFL) where the product demands are deterministic and constant over a finite planning horizon. Objective is minimizing the sum of setup costs, work-in-process and final products inventory holding costs per time unit while satisfying the demands without backlogging. This problem consists of a combinatorial part (machine assignment and sequencing sub-problems) and a continuous part (lot sizing and scheduling sub-problems). To account for these two elements, a new mixed integer nonlinear program (MINLP) is developed which simultaneously determines machine allocation, sequencing, lot-sizing and scheduling decisions. In order to reduce computational complexity, instead of solving this MINLP directly, we propose an efficient enumeration method to determine optimal solution of the model. Moreover, the performance of the proposed method is evaluated by some numerical experiments. Two other applicable cases (zero setup costs and Lot streaming) are studied and required modifications in the model formulation and the solution method are described. Finally, a case example in a PCB assembly system is presented to illustrate applicability of the mathematical model and the proposed solution method.

This paper addresses the common cycle multi-product lot-scheduling problem in deterministic flexible job shops where the planning horizon is finite and fixed by management. This problem consists of a combinatorial part (machine assignment and sequencing sub-problems) and a continuous part (lot-sizing and -scheduling sub-problems). To account for these two elements, a new mixed integer nonlinear program (MINLP) is developed which simultaneously determines machine allocation, sequencing, lot-sizing and -scheduling decisions. In order to reduce computational complexity, instead of solving this MINLP directly, we propose an efficient enumeration method to determine optimal solution for this model. The performance of the proposed method is evaluated by some numerical experiments. Two other applicable cases (zero setup costs and lot streaming) are also studied and required modifications in the model formulation and the solution procedure are described. Moreover, a numerical example is presented to illustrate applicability of the proposed mathematical model and the solution method.

Lot sizing is one of the most important and also one of the most di4cult problems in production planning. This subject has been studied extensively in the literature. In this article, we consider single-level lot sizing problems, their variants and solution approaches. After introducing factors a9ecting formulation and the complexity of production planning problems, and introducing di9erent variants of lot sizing and scheduling problems, we discuss single-level lot sizing problems, together with exact and heuristic approaches for their solution. We conclude with some suggestions for future research.

The problem addressed in this paper is the determination of lot sizes for multiple products to be produced on a single production facility with limited capacity that can vary from period to period. Demand is assumed to be deterministic and time varying and backlogging is allowed. For each product there is a fixed setup cost incurred each time the facility is set up to produce the product. Setup can be carried into the next period if the same product is produced last in a period and first in the following period. The objective is to minimize the sum of setup, inventory holding, backlogging and production costs. We abbreviate this problem as CLSPBSC. The formulation of CLSPBSC as a mixed integer programming (MIP) problem and a heuristic with four elements are presented. These elements consist of (1) demand shifting rule (2) lot size determination rules (3) feasibility conditions and (4) setup carry-over determination. Based on the heuristic algorithm a computer program in C++ is prepared. The computational results demonstrate that the heuristic algorithm is efficient to provide good quality solutions with very high speed. The computational efficiency of the heuristic introduces it as a promising technique for solving realistically large problem instances routinely on a personal computer.