Batch Delivery Costs Research Articles

A gamified teaching–learning based optimization algorithm for a three-echelon supply chain scheduling problem in a two-stage assembly flow shop environment

The supply chain scheduling problem, which is an integrated production scheduling and distribution through the batch delivery system, has been studied by researchers for more than two decades. So far, all researches in this regard have studied the two-echelon supply chain network. The first novelty of this research is that, for the first time, the three-echelon supply chain scheduling problem, including supplier, manufacturer and customer, is investigated considering two separated batch delivery systems for transportation, simultaneously. To do so, a two-stage assembly flow shop environment is used. In the first stage, there are a number of decentralized suppliers who are responsible for manufacturing components. The components are dispatched to the second stage (assembly stage) with the first batch delivery system and after assembly, the orders are dispatched to the customer through secondary batch delivery system. The purpose of this problem is to minimize the cost of total completion time plus batch delivery costs. First, a mixed integer linear programming model (MILP) is presented that is able to solve small-size instances in a logical time. The second novelty of this research is that, to solve large-size instances, a novel gamified teaching–learning based optimization (GTLBO) algorithm is developed with applying a number of dominance rules and gamification tools. The computational results indicate the superiority of the new TLBO variant over its standard form and two well-known methods, i.e. GA and PSO, especially by applying the dominance rules.

Scheduling unrelated parallel machine problem with multi-mode processing times and batch delivery cost

Scheduling unrelated parallel machine problem with multi-mode processing times and batch delivery cost

Two meta-heuristic algorithms for optimizing a multi-objective supply chain scheduling problem in an identical parallel machines environment

Optimizing the trade-off between crucial decisions has been a prominent issue to help decision-makers for synchronizing the production scheduling and distribution planning in supply chain management. In this article, a bi-objective integrated scheduling problem of production and distribution is addressed in a production environment with identical parallel machines. Besides, two objective functions are considered as measures for customer satisfaction and reduction of the manufacturer’s costs. The first objective is considered aiming at minimizing the total weighted tardiness and total operation time. The second objective is considered aiming at minimizing the total cost of the company’s reputational damage due to the number of tardy orders, total earliness penalty, and total batch delivery costs. First, a mathematical programming model is developed for the problem. Then, two well-known meta-heuristic algorithms are designed to spot near-optimal solutions since the problem is strongly NP-hard. A multi-objective particle swarm optimization (MOPSO) is designed using a mutation function, followed by a non-dominated sorting genetic algorithm (NSGA-II) with a one-point crossover operator and a heuristic mutation operator. The experiments on MOPSO and NSGA-II are carried out on small, medium, and large scale problems. Moreover, the performance of the two algorithms is compared according to some comparing criteria. The computational results reveal that the NSGA-II performs highly better than the MOPSO algorithm in small scale problems. In the case of medium and large scale problems, the efficiency of the MOPSO algorithm was significantly improved. Nevertheless, the NSGA-II performs robustly in the most important criteria.

Multi-agent supply chain scheduling problem by considering resource allocation and transportation

Today in the global market competition, integration issue in supply chain is considered as an important principle. In this study, for the first time, different requirements of the customers and different aims of the manufacturer are simultaneously addressed in an integrated problem of production scheduling, transportation, and resource allocation. The problem consists of two types of customers, considered as the agents. The first agent accepts tardiness in delivery of orders provided that the manufacturer pays the tardiness penalty; whereas, the second agent does not accept the tardy orders. The purpose is to minimize the sum of batch delivery cost, resource allocation, tardiness penalty cost, and lost sale cost (the total number of tardy orders). To solve the problem, two mathematical programming models, including a Mixed Integer Non-Linear Programming (MINLP) and a Mixed Integer Linear Programming (MILP) are proposed. Also, due to NP-hard nature of the problem, two meta-heuristic algorithms of Adaptive Genetic Algorithm (AGA) and Ant Lion Optimization (ALO), as well as a heuristic algorithm are proposed. To assess the merits of the solution methods, small and large-scale tests are designed. The results indicate the superiority of adaptive genetic algorithm in comparison with other algorithms.

Integrated production, inventory, and batch delivery scheduling with due date assignment and two competing agents

AbstractThis paper considers several integrated production, inventory, and batch delivery scheduling problems on a single machine with due date assignment and two competing agents, each of which seeks to optimize its own performance. All the jobs of the first agent have a common due date, which is a decision variable to be determined by the decision maker, whereas the due dates of the jobs of the second agent are exogenously given. The jobs of the same agent are processed sequentially in batches and the batch processing time is equal to the total processing time of the jobs in the batch, where the batch size can be either bounded or unbounded. A batch is available only when all the jobs in it are completed. An available batch is delivered to its agent immediately, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. Finished jobs are kept in inventory, if necessary, incurring an additional holding cost. The first agent wishes to find a job sequence, a common due date, and a dispatch date for each of its jobs that jointly minimize the total cost comprising the earliness, weighted number of tardy jobs, job holding, due date assignment, and batch delivery costs, while the second agent seeks to minimize one of the following criteria plus the batch delivery cost: the maximum value of a regular scheduling criterion, total completion time, and weighted number of tardy jobs. The overall objective is to minimize the objective value of the first agent, subject to the objective value of the second agent not exceeding a given threshold. For each of the problems considered, we study its computational complexity and develop exact and/or approximation solution algorithms.

Single-machine serial-batch delivery scheduling with two competing agents and due date assignment

We consider a set of single-machine batch delivery scheduling problems involving two competing agents under two due date assignment models. Belonging to one of the two agents, each job is processed and delivered in a batch to its agent, where the jobs in each batch come from the same agent. The jobs in a batch are processed sequentially and the processing time of a batch is equal to the sum of the processing times of the jobs in it. A setup time is required at the start of each batch. The dispatch date of a job equals the delivery date of the batch it is in, i.e., the completion time of the last job in the batch. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The due date of each job is a decision variable, which is to be assigned by the decision maker using one of two due date models, namely the common and unrestricted due date models. Given the due date assignment model, the overall objective is to minimize one agent’s scheduling criterion, while keeping the other agent’s criterion value from exceeding a threshold given in advance. Two kinds of scheduling criteria are involved: (i) the total cost comprising the earliness, tardiness, job holding, due date assignment, and batch delivery costs; and (ii) the total cost comprising the earliness, weighted number of tardy jobs, job holding, due date assignment, and batch delivery costs. For each of the problems considered, we show that it is $$\mathcal {NP}$$ -hard in the ordinary sense and admits a fully polynomial-time approximation scheme.

A further study on two-agent scheduling on an unbounded serial-batch machine with batch delivery cost

For the two-agent scheduling on an unbounded serial-batch machine with batch delivery cost, Yin et al. (2016) presented a comprehensive study, where the objective of each agent (A or B) is calculated by his scheduling cost plus his batch delivery cost proportional to the number of batches of this agent. Among their results, they provided a polynomial-time algorithm for minimizing the objective of agent A subject to the constraint that the objective of agent B does not exceed a given threshold value, where the criterion of agent A is the total completion time plus batch delivery cost and the criterion of agent B is the maximum lateness plus batch delivery cost. We show in this paper that their algorithm is incorrect by a counterexample and the algorithm presented in Kovalyov et al. (2015) for solving the same problem without batch delivery cost can be used to solve the problem in Yin et al. (2016) in polynomial time. We further study two corresponding Pareto scheduling problems and provide polynomial-time algorithms.

Mathematical programming and grey wolf optimiser for minimising sum of the total due date assignment, maximum tardiness and delivery costs for a supply chain scheduling problem

This paper focuses on the integration of the three most important and practical decisions in a supply chain, including due date assignment, production scheduling and outbound distribution. The issue under consideration is to minimise the summation of maximum tardiness, and due date assignment and batch delivery costs. To provide solution methods, first, two mathematical programming models including one mixed nonlinear model and one mixed linear model to solve it are presented. After these developments, since the problem is NP-hard and solving real problems on a large scale is virtually impossible, the combinational meta-heuristic algorithm based on Grey Wolf optimiser (GWO) and heuristic algorithm for solving the problem are presented. The GWO is a fast and robust algorithm in which the number of parameters that should be controlled is few. In this study, various computational tests have also been used to evaluate the effectiveness of the development methods - Taguchi method to set the parameters, design of experiments to generate the experiments, and analysis of variance to analyse the results. The computational results show that the combinational meta-heuristic method introduced in this paper puts in a high performance.

Mathematical programming and grey wolf optimiser for minimising sum of the total due date assignment, maximum tardiness and delivery costs for a supply chain scheduling problem

This paper focuses on the integration of the three most important and practical decisions in a supply chain, including due date assignment, production scheduling and outbound distribution. The issue under consideration is to minimise the summation of maximum tardiness, and due date assignment and batch delivery costs. To provide solution methods, first, two mathematical programming models including one mixed nonlinear model and one mixed linear model to solve it are presented. After these developments, since the problem is NP-hard and solving real problems on a large scale is virtually impossible, the combinational meta-heuristic algorithm based on Grey Wolf optimiser (GWO) and heuristic algorithm for solving the problem are presented. The GWO is a fast and robust algorithm in which the number of parameters that should be controlled is few. In this study, various computational tests have also been used to evaluate the effectiveness of the development methods - Taguchi method to set the parameters, design of experiments to generate the experiments, and analysis of variance to analyse the results. The computational results show that the combinational meta-heuristic method introduced in this paper puts in a high performance.

Two-agent single-machine scheduling to minimize the batch delivery cost

We consider integrated production and batch delivery scheduling in a make-to-order production system involving two competing agents, each of which having its own job set competes to process its jobs on a shared single machine. To save the delivery cost, the jobs of the same agent can be processed and delivered together batches. The completion time of each job in the same batch coincides with the batch completion time. A batch setup time is incurred before the processing of the first job in each batch. Each of the agents wants to minimize an objective function depending on the completion times of its own jobs. The goal is to determine a schedule for all the jobs of the two agents that minimizes the objective function of one agent, while keeping the objective function value of the other agent below or at a given value. For each of the problems under consideration, we either provide a polynomial-time algorithm to solve it or show that it is NP-hard. In addition, for each of the NP-hard problems, we present a mixed integer linear programming (MILP) formulation and provide a pseudo-polynomial dynamic programming algorithm, establishing that it is NP-hard in the ordinary sense only, and show that it admits an efficient fully polynomial-time approximation scheme, if viable. Finally, we compare the performance of the pseudo-polynomial dynamic programming algorithms against the corresponding MILP formulations with randomly generated instances.

Parallel-Machine Scheduling to Minimize Flowtime, Holding, and Batch Delivery Costs

This paper considers a batch delivery scheduling problem in which n independent and simultaneously available jobs are to be processed on m unrelated or uniform parallel machines. The jobs scheduled on the same machine are delivered in batches to customers and the delivery date of a batch equals the completion time of the last job in the batch. The number of jobs in each delivery batch is constrained by the batch size, and the cost of delivering a batch depends not only on the number of jobs in the batch but also on the machine on which the batch is processed. The objective is to find jointly the optimal number of batches on each machine, the optimal assignment of jobs to the batches, and the optimal job processing sequence to minimize the sum of total flowtime, total holding time, and delivery costs. When the number of batches has a fixed upper bound, we present polynomial-time algorithms to solve the problems with unrelated and uniform parallel machines. If the bound constraint on the number of batches is relaxed, we provide polynomial-time algorithms to solve two special cases of the problem with uniform parallel machines.

Pseudo-polynomial dynamic programming for an integrated due date assignment, resource allocation, production, and distribution scheduling model in supply chain scheduling

In this study, we consider an integrated due date assignment, production, and batch delivery scheduling problem with controllable processing times for multiple customers in a supply chain. The objective is to minimize the sum of the weighted number of tardy jobs as well as the due date assignment, resource allocation, and batch delivery costs. This model can also be applied when some parts of the jobs are outsourced. The problem is NP-hard. We propose a pseudo-polynomial dynamic programming algorithm to solve this problem, which shows that the problem is ordinary NP-hard. We performed computational tests to evaluate the proposed method.

A branch-and-bound algorithm for two-machine flow-shop scheduling problems with batch delivery costs

Manufacturers often dispatch jobs in batches to reduce delivery costs. However, this technique can have a negative effect on other scheduling-related objective functions such as minimising maximum tardiness. This paper aims to minimise the maximum tardiness and the sum of delivery costs of a two-machine flow-shop problem in a batch delivery system. It is proven in the literature that this problem is NP-complete. We present a mixed integer linear programming (MILP) model to solve the problem. As this is an MILP model, the commercial solver (the GAMS software using CPLEX solver) is not guaranteed to find the optimal solution for large-size problems at a logical CPU run time. Therefore, a branch-and-bound (B&B) algorithm is proposed for obtaining the optimum solution. Additionally, an upper bound (UB) heuristic with a quick processing time is presented. These methods are evaluated using randomly generated instances and the results indicate the efficiency of the B&B and UB heuristic algorithms.

Single machine batch scheduling to minimize the sum of total flow time and batch delivery cost with an unavailability interval

This paper addresses the problem of scheduling n nonresumable and simultaneously available jobs on a single machine, where the machine has a fixed unavailability interval, and the jobs are delivered in batches to the customers. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The goal is to find the optimal delivery date for each job and an optimal job sequence to minimize the sum of total flow time and batch delivery cost. The problem is shown to be NP-hard based on a reduction from the Equal-Size Partition Problem. Then a pseudo-polynomial time algorithm is developed, establishing that it is NP-hard in the ordinary sense. Finally, by applying the interval partitioning technique and the rounding technique, a fully polynomial time approximation scheme (FPTAS) and a bicriteria fully polynomial time approximation scheme are developed, which run in time On5ε2+n5logn and On6ε, respectively.

Single-machine common due-date scheduling with batch delivery costs and resource-dependent processing times

In this paper we consider the problem of single-machine batch delivery scheduling with an assignable common due date and controllable processing times. The job processing time is either a linear or a convex function of the amount of a continuously divisible and non-renewable resource allocated to the job. Finished jobs are delivered in batches and there is no capacity limit on each delivery batch. The objective is to find a job sequence, a partition of the job sequence into batches, a common due date, and resource allocation that jointly minimise a cost function based on earliness, weighted number of tardy jobs, job holding, due-date assignment, batch delivery, makespan, and resource consumption. We provide some properties of the optimal solution, and show that the problem with the linear and convex resource consumption functions can be solved in and time, respectively. We also show that some special cases of the problem can be solved by lower-order algorithms.

Supply chain scheduling at the manufacturer to minimize inventory holding and delivery costs

In a supply chain, a manufacturer receives jobs from suppliers at distinct time points, and produces and delivers final products to customers in batches. The manufacturer can be modeled as a single machine in the supply chain and the problem can thus be modelled as minimizing the sum of weighted flow time and the batch delivery costs. Since the problem with arbitrary processing times, release times and weights is strongly NP-hard, we first analyze some polynomially solvable special problems. Then we develop a heuristic algorithm to solve the general problem. We also develop a lower bound to study the performance of our heuristic algorithm and the computational experiments show that the solutions of the heuristic algorithm are close to optimal solution.

Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries for multiple customers in supply chains

In this paper, an integrated due date assignment and production and batch delivery scheduling problem for make-to-order production system and multiple customers is addressed. Consider a supply chain scheduling problem in which n orders (jobs) have to be scheduled on a single machine and delivered to K customers or to other machines for further processing in batches. A common due date is assigned to all the jobs of each customer and the number of jobs in delivery batches is constrained by the batch size. The objective is to minimize the sum of the total weighted number of tardy jobs, the total due date assignment costs and the total batch delivery costs. The problem is NP-hard. We formulate the problem as an Integer Programming (IP) model. Also, in this paper, a Heuristic Algorithm (HA) and a Branch and Bound (B&B) method for solving this problem are presented. Computational tests are used to demonstrate the efficiency of the developed methods.

Single-machine common due window assignment and scheduling to minimize the total cost

In this paper we consider a single-machine common due window assignment and scheduling problem with batch delivery cost. The starting time and size of the due window are decision variables. Finished jobs are delivered in batches. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The objective is to find a job sequence, a delivery date for each job, and a starting time and a size for the due window that jointly minimize the total cost comprising earliness, weighted number of tardy jobs, job holding, due window starting time and size, and batch delivery. We provide some properties of the optimal solution and present polynomial-time algorithms for the problem.

Single-machine batch scheduling minimizing weighted flow times and delivery costs with job release times

Article history: Received 25 November 2011 Accepted 11 January 2011 Available online 22 January 2012 This paper addresses scheduling a set of weighted jobs on a single machine in presence of release date for delivery in batches to customers or to other machines for further processing. The problem is a natural extension of minimizing the sum of weighted flow times by considering the possibility of delivering jobs in batches and introducing batch delivery costs. The classical problem is NP-hard and then the extended version of the problem is NP-hard. The objective function is that of minimizing the sum of weighted flow times and delivery costs. The extended problem arises in a real supply chain network by cooperation between two layers of chain. Structural properties of the problem are investigated and used to devise a branch-andbound solution scheme. Computational experiments show the efficiency of suggested algorithm for solving instances up to 40 jobs. © 2012 Growing Science Ltd. All rights reserved

Common due date assignment and scheduling with a rate-modifying activity to minimize the due date, earliness, tardiness, holding, and batch delivery cost

e consider a single-machine batch delivery scheduling and common due date assignment problem. In addition to making decisions on sequencing the jobs, determining the common due date, and scheduling job delivery, we consider the option of performing a rate-modifying activity on the machine. The processing time of a job scheduled after the rate-modifying activity decreases depending on a job-dependent factor. Finished jobs are delivered in batches. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The objective is to find a common due date for all the jobs, a location of the rate-modifying activity, and a delivery date for each job to minimize the sum of earliness, tardiness, holding, due date, and delivery cost. We provide some properties of the optimal schedule for the problem and present polynomial algorithms for some special cases.