Coordinated Production and Delivery Operations With Parallel Machines and Multiple Vehicles

This paper investigated a coordinated optimization problem of production and delivery operations with parallel machines and multiple vehicles so that a more cost-effective and sustainable supply chain performance can be achieved. We propose an effective hybrid metaheuristic solution framework to deal with this problem, by which the investigated problem is decomposed into 3 sub-problems namely, vehicle assignment, parallel machine scheduling and traveling salesman sub-problem. This framework is established for handling the 3 sub-problems in a coordinated manner so as to simplify the optimization process and to reduce the computational complexity. To evaluate the effectiveness of the methodology, this paper integrates a genetic algorithm, the longest processing time heuristic and a tabu search under this framework to solve the investigated problem. Extensive numerical experiments have been conducted and experimental results show that the proposed solution framework can handle the investigated problem efficiently and effectively.


I. INTRODUCTION
Supply chain management is one of the most important research subjects in operation management. The increasing fierce competition forces enterprises to compete with other enterprises both on price and quality, and on the reliability and timeliness of delivery [1]. Due to the increasing market globalization, coordination between various stages of supply chain operations has received more and more attention from industry practitioners and academic researchers [2], [3]. One of critical aspects of the supply chain coordination is the integration of production and outbound logistics operations, which is important to reach a sustainable supply chain. To obtain an expected delivery performance at the minimum total cost, both production scheduling and logistics scheduling need to be considered in a coordinated and jointly manner [4]- [10], which is particularly important in some manufacturing companies producing time-sensitive products such as fast food and cement. In these companies, finished products usually need to be transported to customers shortly or immediately after the production process is finished. This The associate editor coordinating the review of this manuscript and approving it for publication was Zhiwu Li . paper deals with a coordinated optimization problem of production and delivery operations with parallel machines and multiple vehicles, called coordinated production and delivery optimization (CPDO) problem.
We first establish the mathematical model of the CPDO problem. To solve this problem, a novel solution framework is proposed based on hybrid metaheuristic, by which we decompose the investigated problem into 3 sub-problems, including an assignment sub-problem of orders to vehicles, a machine scheduling sub-problem with makespan minimization, and a travelling salesman sub-problem. This research contributes the literature by proposing the novel multi-level hybrid metaheuristic solution framework for complex CPDO problems. Comparing with other approaches, the proposed novel hybrid framework is able to simplify the optimization process and reduce the complexity of computation.

II. LITERATURE REVIEW A. COORDINATED PRODUCTION AND DELIVERY OPTIMIZATION
The coordinated production and delivery optimization problem is also called as integrated production and delivery (or distribution) scheduling (CPDO) problem. Research on VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ CPDO has attracted more and more researchers' attention in recent years. Chang and Lee (2004) investigated a CPDO problem with one vehicle and single machine [2]. Boudia et al. (2008) researched into a single-product CPDO problem on a multi-period horizon [11]. They considered a production environment that both manufactures and customers could store the finished products. Toptal et al. (2013) investigated a CPDO problem with heterogeneous vehicles [12]. The manufacturer used two types of vehicles to deliver goods. Viergutz and Knust (2014) addressed a CPDO problem with lifespan constraints in single machine and single vehicle environment [1]. Seyedhosseini and Ghoreyshi (2014) presented a CPDO model for perishable products [13]. Chen (2010) has made a comprehensive review on CPDO problems and pointed out that the CPDO problems were more challenging than other CPDO problems and deserved more research due to their practicality [4]. However, CPDO problems with vehicle routing (i.e., CPDO) have been investigated seldom in the literature although vehicle routing widely exists in production and delivery operations in real world. Chen et al. (2009) examined CPDO problem of perishable food products in a single-machine environment [14]. Ullrich (2013) addressed a CPDO problem with the total tardiness [15]. Li et al. (2016) addressed a CPDO problem in a single-machine environment with the objective of minimizing vehicle delivery and total customer waiting time [10]. However, CPDO problems with parallel machines, heterogeneous vehicles and total cost minimization objective have not been reported so far. These problems widely exist in many make-to-order manufacturing environments, such as lumber and porcelain industry [16], which are the focus of this research.

B. TECHNIQUES FOR COORDINATED PRODUCTION AND DELIVERY OPTIMIZATION PROBLEM
Two types of approaches are usually adopted to handle CPDO problems, including exact methods, heuristics and metaheuristics. The exact methods are usually used to handle small-sized CPDO problems with a limited number of machines and vehicles. Li et al. (2005) addressed a CPDO problem in a single-machine environment and presented a dynamic programming model for handling a general case [17]. They pointed out that the model's computational complexity is very high when the number of machines was greater than 1. Mazdeh et al. (2007) addressed a single-machine scheduling problem with delivery by presenting a branch-and-bound scheme for this problem [18]. Due to the NP-hard nature of most CPDO problems [15], these problems have been solved usually by metaheuristic methods. Armentano et al. (2011) presented a tabu search method with a process of path relinking to deal with the CPDO problem in an environment with single plant and homogenous vehicles [19]. Chang et al. (2014) developed an ant colony optimization process for the CPDO problem with multiple unrelated machines and capacitated vehicles [20].
With the complexity increase of CPDO problems, traditional heuristics and metaheuristics have difficulties in finding effective solutions to these problems because even a very simple CPDO problem is NP-hard. Some researchers thus aim at converting the original problem into some simpler sub-problems so as to simplify the optimum-seeking process [21]. Geismar et al. [22] proposed a two-phase metaheuristics to handle a CPDO problem for short shelf life products. They used either genetic or memetic algorithm to choose firstly a locally optimal customer sequence, and then splitted the customer sequence and used the Gilmore-Gomory approach to sort the subsequences of customers and to form the coordinated schedule. Chen et al. (2009) presented a non-linear mathematical program model for CPDO problem with a single machine and perishable food products [14]. They decomposed the integrated problem into a production sub-problem handled by the constrained Nelder-Mead approach, and a vehicle routing sub-problem handled by a metaheuristic. Ullrich (2013) adopted a genetic algorithm-based approach for the CPDO problem [15], which decomposed the problem into a machine scheduling sub-problem as well as a vehicle routing sub-problem. This approach is easy to get trapped in local optimum because of the complexity of its genetic optimization mechanism with a complicated solution representation. Following these previous studies, this research develops a novel hybrid metaheuristic solution framework with simpler solution representation for the investigated CPDO problem by effective problem decomposition.

A. PROBLEM DESCRIPTION
We investigate a coordinated production and delivery optimization problem with one manufacturing plant and multiple vehicles and many customers. The plant has multiple identical parallel machines to produce customer orders. Orders cannot be split and each order must be continuously produced on one machine. After production, orders are delivered to widely dispersed customers by heterogeneous capacitated vehicles. Finished orders should be delivered to specified customers before the specified delivery due dates, otherwise tardiness penalties are incurred. The manufacturing plant has to obtain a joint production and delivery plan, by deciding the optimal assignment of orders to machines, the optimal production sequence of customer orders, and the optimal vehicle routes of delivering finished orders to customers. The problem objective is to minimize the total cost of transport and delivery tardiness.
On the basis of real-world practices, we make the following reasonable settings in establishing the mathematical programming model of the investigated CPDO problem: (1) each customer has one order and one location; (2) each vehicle's departure time is the last completion time of all orders transported by this vehicle; (3) orders that are produced (delivered) on a same machine (vehicle) are processed consecutively on the same machine; (4) the product volumes of all orders are the same; and (5) the transport time among customers is equal to the transport distance among customers. To facilitate the understanding of the investigated problem, we consider a CPDO example with ten orders (customers), two machines and three vehicles. FIGURE 1 depicts a feasible CPDO solution. Machine 1 produces orders 1, 2, 5, 6, 10 and production sequence is 2, 10, 1, 6 and 5 in turn. Machine 2 produces the remaining five orders and production sequence is 3, 4, 8, 7 and 9 in turn. Vehicle 1 delivers orders 2, 3, 4, 10 and the delivery route is 0→3→10→2→4→0 (0 is the depot). Vehicle 2 delivers orders 1, 7, 8 and the delivery route is 0→7→1→8→0. Vehicle 3 delivers orders 5, 6, 9 and the delivery route is 0→6→5→9→0.

B. NOTATIONS
Notations that are used in this paper are presented below. Indices Subject to : j k Objective (1) aims to minimize the total cost, including transport costs and tardiness penalty of all customer orders. Constraints (2)-(7) are machine scheduling constraints. Constraint (2) stipulates that all I orders should be allocated to machines I times while constraint (3) stipulates that each order must be exactly allocated to one machine. Constraint (4) ensures that order j either succeeds another order i or is the first order to be produced on machine m, where order 0 indicates the dummy first order processed on parallel machines m. Constraint (5) indicates that the order i is either the last to be produced on machine m or the immediate predecessor of order j, where I + 1 is the dummy last order processed on parallel machines. Constraints (6)- (7) calculates order i's completion time and the completion time CT 0 of the dummy first order is set to zero. Constraint (8) indicates that, for any vehicle k , the makespan d k of orders delivered by this VOLUME 8, 2020 vehicle is the maximal completion time of these orders. Constraints (9)-(10) guarantee that each order is allocated exactly to one vehicle. Constraints (11)- (17) are flow conservation constraints of traveling salesman problem. Constraints (11) and (12) enforce that vehicle k arrives at customer i once at most and each entering vehicle leaves customer i. Constraint (13) and Constraint (14) guarantee that each vehicle route starts and returns the same plant. Constraint (15) stipulates that the load delivered by vehicle k cannot exceed the vehicle capacity. Constraint (16) calculates the delivery time a i of order i. Constraint (17) indicates that the delivery time a i of order i is equal to or greater than the transport time from the customer to the plant plus the departure time of the vehicle transports this order. Constraint (18) stipulates the value ranges of four decision variables.

D. PROBLEM COMPLEXITY
The problem formulated above can be reduced easily to a vehicle routing problem if we set all orders' production times to zero. It is well-known that the vehicle routing problem is strongly NP-hard. The integrated problem is thus strongly NP-hard as well.
The investigated CPDO problem is very hard to solve because if its huge solution space. Consider the problem instance with M machines, K vehicles and I orders, the solution space of machine scheduling is C M −1 Take a simple CPDO problem with M = 2, K = 2, and I = 10 as an example. This problem has 8.7 × 10 9 candidate solutions. The real-world CPDO problems have the much larger solution space and are intractable since they usually need to deal with more machines, orders and vehicles.

IV. METHODOLOGY A. PROBLEM DECOMPOSITION AND HYBRID METAHEURISTIC SOLUTION FRAMEWORK
We simplify the problem formulated above by effective problem decomposition and transformation. According to the actual characteristics of the CPDO, each order must be produced on one machine and transported by one vehicle, all orders are delivered by K vehicles and each vehicle leaves the plant immediately after the orders transported by the vehicle are finished. We can thus classify all orders into K groups, each of which consists of orders transported by a vehicle. Let N k denote the number of orders transported by vehicle k. We have N 1 + N 2 + ... + N K = I . As a result, we need to address a vehicle assignment sub-problem, which assign appropriate orders to each vehicle (determining the values of variable t ki ). Based on the vehicle assignment solution, we then need to address a parallel machine scheduling subproblem, which decides these orders' production sequences on appropriate parallel machines (i.e., deciding the values of two variableso mi and y ijm ). Lastly, for each vehicle, a travelling salesman sub-problem is solved, which determines the optimal travel routes of each vehicle (i.e., determining the values of x ijk ).
All the three sub-problems are classical optimization problems. The 1st sub-problem is equivalent to a generalized assignment problem, the 2nd sub-problem is a parallel machine scheduling sub-problem with makespan minimization, while the 3 rd one is a travelling salesman problem. All these problems are NP-hard problems [23], [24], solution approaches for which range from exact techniques [25]- [32], heuristic techniques [33]- [43] to various metaheuristic techniques [44]- [48]. With the increase of problem scale and complexity of these problems, metaheuristic techniques usually exhibit superior performances to exact and heuristic techniques because the latter two have difficulty in finding effective solutions to complex NP-hard problems in a reasonable computation time.
Compared with the original CPDO problem, the 3 subproblems are easier to handle. For example, the parallel machine scheduling sub-problem can be solved effectively by some heuristic rules [49], [27]. If so, the remaining solution space we need to search for is only the solution space size (C k−1 I · I !) of the vehicle assignment sub-problem times the solution space size (I !) of travelling salesman sub-problem, which is On the basis of these 3 sub-problems, we propose a novel multi-level hybrid metaheuristic solution framework to solve the problem investigated. FIGURE 2 outlines the pseudo-code of the hybrid metaheuristic solution framework. First, we initialize algorithm parameters, including population size (ρ), the total number of orders ( i ), the total number of vehicles (k), and the total number of parallel machines (m) as well as the maximal number of iterations (g max ). We initialize the number of iterations in line 2. An Initialization procedure is utilized to generate randomly some initial vehicle assignment (VA) solutions {t ki } to the vehicle assignment sub-problem in line 3. Line 4 is utilized to access the performance (objective value) of candidate VA solution {t ki }, which is performed according to Procedure 1. Then, a repeat-untilloop (from lines 5 to 10) is executed to generate the best solution I best . From the parental VA solution at the gth iteration, the new VA solution is generated by performing genetic operators in line 6. In line 7, the new VA solution is evaluated according to Procedure 1. Then, the new VA population M g+1 for the next generation I g+1 of VA solution is produced in line 8. M g+1 is composed of the best ρ VA solutions in the specified value range. After each iteration, the termination criteria of maximal iterations g max is checked in the next line. If the number of maximum iteration is satisfied, the iterative process is ended. The resulting best solution is selected from M g as the best one produced by our hybrid metaheuristic solution approach in line 11. Otherwise, the iterative process goes back to line 6 and continues generating new VA solution.
Procedure 1: Step 1: obtain the values of {o mi } and y ijm by handling the machine scheduling sub-problem; Step 2: obtain the values of x ijk by solving the traveling salesman sub-problem; Step 3: calculate the objective function values according to formulation (1) based on the values of {t ki } , {o mi } , y ijm and x ijk .

B. A GENETIC ALGORITHM-BASED HYBRID INTELLIGENT APPROACH
On the basis of the solution framework presented in the section A. PROBLEM DECOMPOSITION AND HYBRID METAHEURISTIC SOLUTION FRAMEWORK, a genetic algorithm-based hybrid intelligent (GAHI) approach is constructed to deal with the problem investigated [47], [50]. This research adopts the standard genetic algorithm, proposed by Goldberg (1989) [51], to handle the vehicle assignment sub-problem and obtain variable t ki 's value. The LPT rule, proposed by Adams et al. (1988), is adopted to handle the parallel machine scheduling sub-problem and obtain variable o mi 's value. The tabu search with intermediate-term and long-term memory, proposed by Fiechter (1994) [52], is utilized to handle the traveling salesman sub-problem and obtain variable x ijk 's value. In the genetic algorithm, the chromosome representation incidates a candidate vehicle assignment solution {t ki }, which is coded by a gene sequence, whose length is the number of orders to be produced. A gene indicates an order and the gene value implies the vehicle assigned to transport the order. A representation example of assigning 10 orders to 2 vehicles is shown in Fig. 3. The tournament selection (Goldberg 1989) [51] and the uniform-order crossover (Davis 1991) [52] are utilized as selection and crossover operations respectively. On the basis of the uniform mutation (Goldberg 1989) [51], a modified mutation operator is presented, which randomly changes the values of several randomly chosen genes.
The longest processing time (LPT) heuristic, developed by Adams et al. (1988) [54], has been proved to be very effective for the parallel machine scheduling problem with makespan objective and without preemptions [55], [56]. The tabu search, proposed by Fiechter (1994) [52], has been demonstrated that it can solve travelling salesman problems with up to 500 vertices very effectively. This method is able to find the optimal solutions to travelling salesman problems since the problem sizes of travelling salesman problem instances involved in this research are not more than 100.

V. EXPERIMENTS
This section validates the performance of proposed hybrid framework for the investigated CPDO problem based on extensive numerical experiments. We highlight 3 experiments next due to the page limit of the paper. The capacities of vehicles in these experiments are 900, 1000, 1100 respectively. The data of customers and their orders are shown in Appendix. In the first 6 columns, the values show the customer number, geographical data in x-axis and y-axis, delivery due date, order size and processing time respectively. The processing time is set as the corresponding order size. The plant is located at coordinate (40,50).
In our experiments, we set the population size to 100, the number of generations to 2000, the crossover probability to 0.7, and the mutation probability to 0.3 in GA [57], [47]. In tabu search [48], the tabu list length is set to 10, the number of neighboring solutions is set to 25, the maximum generation is 70, and ten best neighboring solutions are saved in each move. We set α = 1, β = 0.25. The experiments were conducted on a laptop with Intel Core i5-3210M CPU @2.1 GHz and 3 GB RAM using MATLAB R2013a.

B. EXPERIMENTAL RESULTS AND PERFORMANCE COMPARISON
To evaluate the performance of the developed hybrid metaheuristic solution framework, we compared the performance of GAHI approach with that of the genetic algorithm, UGA, proposed by Ullrich (2013) [15].
In GAHI approach and UGA, if no better solution is found in several consecutive iterations or the maximum number of iterations is reached, the optimization process is stopped. To make a fair comparison, some algorithm parameters are set to the same. In GAHI approach and UGA, we set the population size to 100, the crossover probability to 0.7, the mutation probability is 0.3, the maximum number of iterations to 2000, and the number of consecutive iterations is 500. Table 1 shows that the comparison results produced by the proposed GAHI approach and the UGA for the 3 CPDO experiments in 10 repetitive runs. In table 1, the rows of ''Minimum'' and ''Maximum'' mean the minimum and maximum of the objective functions, generated via GAHI and UGA in 10 repetitive runs. For each experiment, the relative gap between GAHI approach and UGA is calculated by the following formula (19).
where T UGA and T GAHI denote the minimal total cost generated by the UGA and by the GAHI approach respectively. The experimental results of table 1 show that: 1) The GAHI approach shows the better optimization performance than the UGA. The objective values of three experiments obtained by the GAHI approach are 861, 2323 and 11160 respectively, while the objective values obtained by the UGA are 1143, 3057 and 23080 respectively. These results show that the objective values obtained by the GAHI approach are much smaller than those obtained by UGA. The developed GAHI approach is capable of generating the better solutions to the CPDO problem compared to the UGA.
2) Compared to the UGA, the GAHI approach has a faster convergence rate and more difficult to trap in local optimum. Due to the difference of genetic code, the GAHI approach has a relative smaller searching space than the UGA, which leads to the UGA converges slower than GAHI approach. According to the results of experiment 1, the GAHI approach converged in the 85th generation, while the UGA converged in the 300th generation.
In summary, the proposed novel hybrid metaheuristic solution framework converts the CPDO into three sub-problems, which makes it possible to narrow down the search space of the CPDO. Compared with UGA, the GAHI approach under this framework could generate better solutions in terms of the examined experiments, and show a superior optimization performance.

C. OPTIMIZATION PERFORMANCE OF THE GAHI APPROACH
We further examine the optimum-seeking performance of the GAHI approach. The enumeration method (EM) is adopted to find the optimal solutions to small-sized CPDO problems. The comparisons were conducted based on 3 small-sized problem instances with 10 orders, 3 vehicles and 2 machines. We perform 10 trials for the GAHI approach for each instance.
The comparison results between the GAHI approach and the EM are shown in table 2. The GAHI approach generated the optimal solutions to the three instances in each trial. The GAHI approach exhibits the good optimum-seeking performance for the CPDO problem investigated. For these instances, the computation time (denoted by t_GAHI ) of GAHI approach is much shorter than that (t_EM ) of the EM. Taking the instance with 2 machines, 10 orders and 3 vehicles as an example, the CPU time (denoted by t_EM ) of the EM is 6123.012 s, while the GAHI approach only uses 143.335s, which is only 2.34% of the former. When the problem size increases, the gap in CPU time between the GAHI approach and the EM goes bigger.
Based on the comparison results described above, the GAHI approach and the proposed hybrid metaheuristic solution framework have the capability of handling the investigated CPDO problem effectively.

VI. CONCLUSION
This paper addressed a collaborated production and delivery optimization problem with parallel machines and multiple heterogeneous vehicles in manufacturing environments producing time-sensitive products. The mathematical programming model of the CPDO problem is established. A multi-level hybrid metaheuristic solution approach is developed to deal with this problem, by which the investigated problem is decomposed into 3 simple sub-problems including an assignment sub-problem of orders to vehicles, a machine scheduling sub-problem with makespan objective, and a travelling salesman sub-problem. By so doing, the candidate solution space searched by metaheuristics is reduced largely since the machine scheduling sub-problem can be solved determinedly by some classical heuristics.
To evaluate the performance of the hybrid metaheuristic solution framework, this research constructed a genetic algorithm-based hybrid intelligent approach by combining  a genetic algorithm, a LPT rule and a tabu search under this framework. Extensive experiments have been performed to validate the effectiveness of the proposed framework by comparing the GAHI approach and two other approaches. For small-sized instances, the GAHI approach showed the ability of achieving the optimal solutions within a much shorter time than the enumeration method did. The GAHI approach generated much smaller objective values for the examined three problem instances, and showed a better optimization performance than the Ullrich's genetic algorithm did [15].
To sum up, this paper demonstrates that it is possible to handle effectively the CPDO problem based on the proposed hybrid metaheuristic solution framework. As a start, this paper presented the results of integrating several commonly used metaheuristics into this framework. We do not claim that the used techniques (e.g., GA, LPT heuristic) are the best under the solution framework, under which other metaheuristics can also be used. Future work can aim to integrate more advanced metaheuristics for each sub-problem into the proposed framework, and to compare their performances with those of the GAHI approach. Another future direction can consider the effects of vehicle departure time on integrated scheduling performances. APPENDIX See Table 3. From 2010 to 2015, he was an Associate Professor with the Chengdu University of Information Technology, where he has been a Professor, since 2015. He has authored two books and more than 40 articles. He has completed more than 40 scientific research and engineering application projects in modern logistics. He has won three second prizes and two third prizes of provincial science and technological progress. His research interests include achievements transformation and data mining.
Prof. Liu was an Evaluation Expert for the National Natural Science Funds of China and for the Sichuan Provincial Science and Technology Funds. From 2010 to 2019, she was an Assistant Professor with Hohai University. From 2013 to 2014, she was a Visiting Scholar with the University of Wisconsin-Madison. She was an Evaluation Expert with the National Investment Project Evaluation Center of NDRC. Since 2019, she has been an Associate Professor with the School of Public Administration, Hohai University. She has authored three books, more than 40 articles, and more than 3 inventions. She has completed more than 40 scientific research and engineering application projects of modern logistics. Her research interest is in modern logistics. Dr. Huang has won a second prize and three third prizes of provincial and ministerial science and technological progress.
DAVID PROVERBS worked for 10 years in the construction sector as a Project Manager before joining academia as a Research Assistant in 1994, progressing quickly to become Head of Department and a Full Professor in 2004. He has led large built environment schools at two modern universities. He is currently responsible for establishing and growing a high-quality international portfolio of provision across the various disciplines in the faculty. He is currently a Professor of construction management and the Associate Dean International with the Faculty of Computing, Engineering and Built Environment, Birmingham City University.
Combining his expertise in building technology and flooding, he has pioneered the development of flood reinstatement and recovery to the benefit of many governments, agencies, charities, companies, and institutions worldwide. He has published extensively on topics linked to flood damage, flood adaptation, and resilience. He is a confident and adept communicator and has given many keynote presentations and guest lectures, and undertaken various media activities. He has excellent leadership skills, having led many teams, supervised numerous funded projects and provided mentoring to colleagues and students he has supported. He has published well over 200 refereed conference papers and journal articles, numerous consultancy reports and several leading textbooks on flood risk management. He is a Co-Editor to the International Journal of Building Pathology and Adaptation, a member of the RICS UK Education Standards Board, and a Trustee to the RICS Research Trust. He received research funding to the value of over 1.8 million sterling secured from a range of prestigious sources including EPSRC, NERC, World Bank, and DFID.
Dr. David, an Invited Eminent Fellow of the Royal Institution of Chartered Surveyors, in 2011, and a member of the RICS Education and Standards Boards, was a Chair of the Council for Heads of the Built Environment (CHOBE), from 2012 to 2016.