Integrated multiproduct batch production and truck shipment scheduling under different shipping policies

Accepted Manuscript

Integrated Multiproduct Batch Production and Truck Shipment Scheduling under Different Shipping Policies ¨ Umit Saglam , Avijit Banerjee ˘ PII: DOI: Reference:

S0305-0483(17)30070-1 10.1016/j.omega.2017.01.007 OME 1749

To appear in:

Omega

Received date: Revised date: Accepted date:

23 August 2015 16 October 2016 20 January 2017

¨ Please cite this article as: Umit Saglam , Avijit Banerjee , Integrated Multiproduct Batch Pro˘ duction and Truck Shipment Scheduling under Different Shipping Policies, Omega (2017), doi: 10.1016/j.omega.2017.01.007

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Highlights

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The common cycle solution for the classical deterministic economic lot-scheduling problem (ELSP) is integrated with two distinct shipping policies. The transportation scenarios involve both full truckload and less than truckload shipping modes, with different cost structures. Mixed integer, non-linear, constrained optimization models are developed to depict the respective optimization problems. Comparison between the two integrated planning models and comparisons across corresponding uncoordinated approaches are made via numerical experiments.

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Integrated Multiproduct Batch Production and Truck Shipment Scheduling under Different Shipping Policies Ümit Sağlam1,*, Avijit Banerjee2 1

Department of Management and Marketing, College of Business & Technology, East Tennessee State University, Johnson City, TN, 37614. 2 Department of Decision Science & MIS, LeBow College of Business, Drexel University, Philadelphia, PA, 19104.

Corresponding author. Tel.: +1-423-439-4576; Fax: +1-423-439-5661. E-mail address: [email protected]

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Abstract

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The classical economic lot-scheduling problem (ELSP) involves the batch sizing and scheduling of multiple products in a single facility under deterministic conditions over an infinite planning horizon. It is assumed that the products are delivered to customers at continuous rates. In today’s supply chains, however, often employing complex delivery networks, the finished goods are usually transported in discrete lots to succeeding stages along the distribution process, in order to take advantage of economies of scale in transportation. In this paper, we formulate mathematical models that attempt to integrate the production lot scheduling problem with outbound shipment decisions. The optimization objective is to minimize the total relevant costs of a manufacturer, which distributes a set of products to multiple retailers. In making the production/distribution decisions, the common cycle approach is employed to solve the ELSP, for simplicity. Two different shipping scenarios, i.e. periodic full truckload (TL) peddling shipments and less than truckload (LTL) direct shipping, are integrated with and linked to the multiproduct batching decisions. We consider these two shipment policies for both coordinated and uncoordinated scenarios. The resulting mixed-integer, non-linear programming models (MINLPs) are solved by the BONMIN solver. Finally, a set of numerical examples illustrate and evaluate the relative efficacies of these policies.

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Keywords: Lot scheduling, supply chain shipment policies, integrated inventory models

1 Introduction

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Over the past several years, with advances in designing efficient supply chains, relationships between customers, manufacturers and suppliers have undergone numerous notable changes by removing non-value added activities in production and distribution. Today’s supply chains are impacted by increasing complexity, unpredictable economic conditions, operational risks, environmental regulations, globalization and rising fuel and commodity costs. Historically, optimization projects within the supply chain have been cumbersome, time-consuming undertakings. Many companies find themselves in a constant struggle to maintain efficiency at every stage along the supply chain, attempting to reduce costs and increase productivity within their procurement-production-distribution networks, in the face of intense competitive pressures. In this context, holistic integration of decisions involving serial stages of activities has received considerable attention from researchers in recent years. 2

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In a typical supply chain, production planning and distribution decisions are of critical importance from the standpoint of operational efficiency and achieving competitive advantage. The existing literature suggests two approaches for making these operational decisions, viz. sequential and simultaneous optimization. As stated by Chen (2010), traditional supply chains consider these two decision problems sequentially, as a bi-level programming problem. The sequential approach attempts to derive the optimal production plan first, independent of the distribution schedule. In the second planning phase, this optimal production plan provides the input for determining the optimal distribution schedule. This methodology does not focus on integrating these two operational activities of the supply chain and, thus, leads to sub-optimality. In contrast, simultaneous optimization attempts to integrate the production and distribution plans, as a whole which tends to yield significant economic benefits (Chen and Vairaktarakis (2005)). In short, coordinated or integrated production/distribution planning incorporate the latter approach, involving simultaneous (or joint, in the case of multistage supply chains) optimization.

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Coordination of production scheduling and outbound distribution is a necessity for manufacturing firms which employ just-in-time production/distribution policies under the lean operations paradigm, emphasizing minimal or no waste. Production and transportation inefficiencies, coupled with excess inventories, are major sources of waste in a supply chain. Consequently, coordination of production and distribution activities, towards developing an integrated planning process, plays a key role in eliminating these wastes. Although just-in-time production focuses on lowering inventory levels, there is a tradeoff between a firm’s inventory investment and the resultant service level. From this perspective, a coordinated or integrated production and distribution planning approach can result in inventory reduction, while maintaining a desirable level of customer service. In fact, Hall and Pots (2005) show that coordination of production scheduling and batch deliveries tends to improve a firm’s service level. In addition, there is also a tradeoff between finished goods inventory levels and transportation costs. Needless to say, that integrated planning processes seek a balance in terms of this tradeoff, for achieving greater efficiencies in supply chain operations (see, for example, Herrmann and Lee (1993)).

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Integrated production and distribution planning can also be advantageous in make-to-order and assemble-to-order production environments. Relatively low levels of finished products inventory under such scenarios, tend to increase supply lead times. The extant literature points out that integration of production and distribution scheduling can shorten lead times, as well as provide satisfactory customer service. Li et al. (2005, 2006) study integrated assembly production and air-transportation planning in a consumer electronics manufacturing firm within a supply chain framework. Along similar lines, Chen and Vairaktarakis (2005), in their study of the food catering industry, contend that coordinated production/distribution scheduling is a necessity, rather than a possibility, for survival in a fiercely competitive business environment such as this. These application studies, among others, highlight the importance and the potential benefits of integrating the production and distribution activities in manufacturing firms. According to Chen (2010), joint scheduling of production and outbound distribution is also critical for time-sensitive products. Their contention is reinforced by Garcia and Lozano (2005) who study the joint production and vehicle scheduling problem for ready-mix concrete 3

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manufacturing that requires immediate delivery. In a similar vein, van Buer et al. (1999) develop a heuristic algorithm to solve the newspaper production and distribution problem, towards deriving low-cost solutions. In both of these application studies, the products have relatively short life span (perishable items) and finished goods inventories are of little or no value. It is noteworthy that integration of production and distribution decisions plays a crucial role in achieving higher operational efficiencies in both of these examples.

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Generally speaking, coordination of operational activities can often contribute significantly towards cost reduction efforts. For instance, King and Love (1980) show that the KellySpringfield Tire Company reduced its average inventory cost by 19% and saved $7.9 million with the integration of production planning, inventory control and distribution decisions. A number of other studies in a variety of industries report similar results. Among them, Martin et al. (1993) develop a large-scale linear programming model for integrated production, inventory planning, and distribution for a company in the flat glass business. Their proposed model yielded more than $2 million in annual saving. Metters (1996) explores the integrated production and transportation problem for the United States Postal Service (USPS). He finds that his integrated production and transportation model yielded a 5% savings. Similarly, Brown et al. (2001) develop a multiperiod linear programming model for integrated production and distribution decisions, reporting a cost reduction of about $4.5 million in 1995, as a result of such coordination. Hall and Pots (2003) consider the joint scheduling, batching and delivery problem in supply chains. They show that cooperation between a supplier and a retailer can reduce total relevant costs by at least 20%. These works cited above constitute a substantive body of literature pointing to the efficacy of employing integrated planning models in coordinating the operational activities of manufacturing supply chains.

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This study focuses on a specific supply chain scenario, where a single manufacturing plant produces multiple products for satisfying customer demands that occur at several retail outlets. The production facility can produce only one product at a time, but shipments can be made either directly to each individual retailer via relatively small, less than truckload (LTL) quantities or via larger full truckload (TL) quantities, where deliveries are made to all the retailers according to a peddling arrangement. In the TL transportation mode, a full truckload represents the aggregate retail demand during a common delivery cycle. The appropriate lot sizes are then dropped off at the respective retail locations from the same transport vehicle, which incurs a fixed shipping charge. In the case of LTL shipping, the delivery cycle times for the various products may be different, but any given item has the same inventory cycle time at all retail locations. The shipments are made directly from the supplier to the various retailers individually and the respective shipping costs depend on the amount of load delivered, based on a variable transportation charge. For either shipment policy, the transportation schedule is directly linked to the batch production schedule for the multiple items at the manufacturing facility. It is to be noted that, from the manufacturer’s perspective, the production batch sizing issue here is represented by the wellknown economic lot scheduling problem (ELSP). Our analysis differs from existing work in this area in two important ways. First, the inventories of the different products are depleted at uniform market demand rates in the traditional treatment of the ELSP, whereas in this paper, we allow such depletions to occur in discrete lot sizes, depending on the transportation policy in 4

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effect. Secondly, we make an attempt to integrate the production plan with either the TL or LTL shipment schedule, as the case may be. It is well known that the ELSP addresses the lot sizing issue for multiple products with deterministic demands produced in a single facility, over an infinite planning horizon. In this paper, we recast this problem in a way that ties the production and shipping schedules together with the objective of minimizing the sum of all the relevant costs, including setup and other fixed costs, as well as inventory holding and other variable costs, while satisfying the market demands for all products at the various retail locations. The production schedule is obtained for all products to meet the significant requirements (Chatfield, 2007). Since the ELSP has been shown to be NP-hard, the focus of most research efforts has been to generate near-optimal cyclic schedules with three well-known policies, viz. the common cycle, basic period (or multiple cycles) and time-varying lot size approaches (Torabi et al. 2005). Although all these three approaches provide a feasible schedule for the ELSP, the common cycle (CC) procedure is the easiest to implement. Bomberger (1966) shows that the basic period (BP) approach provides a better solution than the CC technique, but obtaining a feasible basic period schedule is an NP-hard problem. Hence, this method requires a heuristic procedure to get a feasible solution, which increases the computational effort significantly. Dobson (1987) points out that the time-varying lot size methodology always generates feasible schedules and tends to result in even better quality solutions. Nevertheless, the computational burden associated with this procedure is significantly higher than those for the other two approaches. Thus, in order to keep the computational complexity to a minimum, as well for the sake of simplicity of implementation, we adopt the common cycle approach in our integrated analysis for addressing the production batch scheduling question.

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The remainder of this paper is organized as follows: the next section presents a brief review of the relevant research literature. Section 3 outlines the assumptions made and the notation used in our models and in the following section, our proposed integrated (coordinated) models, and existing uncoordinated production/distribution approaches are described in detail. A numerical example and some selected sensitivity analyses are presented in Section 5. Finally, Section 6 provides a summary and some concluding remarks.

2 Literature Review

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Over the past several decades, there have been a variety of papers published on the ELSP. Earliest works in this area attempt to solve such problems optimally by using mathematical programming techniques. Maxwell (1964) constructs a linear programming model to solve ELSP by using a lower bounding technique. Bomberger (1966) and Elmaghraby (1978) develop dynamic programming approaches, whereas Delporte, and Thomas (1977) formulate a mixedinteger programming model for finding the optimal solution. However, these methods become computationally inefficient and increasingly complex, as the problem size increases. Therefore, over the last three decades or so, many researchers have worked on developing efficient optimum-seeking heuristic algorithms to generate near-optimal schedules for the ELSP. Earlier efforts along these lines include Doll and Whybark (1973), and Elmaghraby (1978). It should be noted, however, that recent theoretical and technological advances in global optimization and the development of more efficient solver software have enhanced the scope for solving these problems efficiently without resorting to heuristics. 5

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In today’s competitive business environment, the ELSP can play a key role in coordinating all the necessary activities of the various participants at each stage of a supply network. However, as pointed out earlier, one characteristic of much of the research on ELSP is that the finished products are consumed at continuous rates. This implies that retail market demands for these commodities are satisfied directly from the manufacturing facility. In today’s supply chains, however, employing complex distribution networks, involving production plants, vehicle terminals, airline hubs, warehouses, distribution centers, retail outlets, etc., finished goods inventories from manufacturing plants are usually shipped in bulk to succeeding stages along the distribution process. Moreover, existing transport economies often tend to employ full truckload or less than truckload shipments (depending on existing cost structures), in discrete, sizeable lots, for efficient movement of such goods. Thus, it becomes necessary to re-examine the ELSP, with a focus on coordination and integration of the production schedule of a manufacturing process with the transportation function, towards achieving greater supply chain efficiency.

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Some of the noteworthy early studies concerning the integrated analysis of production, inventory and distribution planning, include Blumenfeld et al. (1985) and Benjamin (1989). Blumenfeld et al. (1985) study the trade-offs between production setup, inventory holding, and transportation costs to determine optimal shipping policies, including optimal routes and shipment sizes in a supply chain context. These authors do not take into account vehicle capacity and availability but focus on obtaining the optimal policy that minimizes the sum of all the relevant costs. Benjamin (1989), on the other hand, develops a simultaneously reduced gradient based heuristic algorithm in order to embellish the economic order quantity concept and integrate it with transportation decisions. Besides these works, Chandra and Fisher (1994) consider production scheduling simultaneously with the vehicle routing problem, for exploring the value of coordinating the production and distribution planning decisions. Their computational experience indicates that such coordination can decrease the total operating costs by about 3% to 20%, depending on the existing operating conditions.

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Generally speaking, coordination in production, inventory, and delivery has been well addressed in the recent literature. There are existing models on integrated production, inventory and delivery decisions. For example, Jonrinaldi and Zhang (2013) propose an integrated production and inventory model in an entire supply chain system, which consists of several raw materials, parts, suppliers, distributors, retailers and a manufacturer. They propose a methodology for determining integrated production and inventory cycles for multiple raw materials, parts, and products in a supply chain involving reverse logistics. A significant amount of recent research has focused on the linking of production scheduling and delivery activities, under various assumptions and objectives. For example, Lee and Yoon (2010) propose a coordinated production scheduling and delivery batching model, where different inventory holding costs are accounted for in the production and delivery stages. Georgios et al. (2012) formulate a mixed integer programming (MIP) model for addressing the simultaneous food processing and logistics planning problems for multiple products at various sites. In addition to finding a feasible and optimal schedule, his proposed model help all the participants in a collaborative supply chain environment for obtaining the best balance between production, inventory, and distribution costs. In addition to these efforts, there are more recent studies that examine this problem in a variety of mathematical programming model based frameworks. For instance, Liu and Papageorgiou (2013) develop a multi-objective mixed-integer linear programming (MOMILP) model with the 6

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objectives of minimizing total cost, total flow time, and total lost sales, towards making optimal decisions with respect to production, distribution, and capacity expansion. Relvas et al. (2013) develop two alternative MILP models for integrated the production scheduling and inventory decisions for multiple items in a petroleum products distribution system. In addition, Pan and Nagi (2013) develop a model for multi-echelon supply chain network design in the agile manufacturing context; and Farahani et al. (2014) provide a thorough literature review on competitive supply chain network design models, solution techniques, and their applications. Also, Seyedhosseini and Ghoreyshi (2014) develop a heuristic algorithm to solve an integrated production and distribution planning model for perishable products, incorporating inventory and routing decisions. Eskandarpour et al. (2015) provide a comprehensive literature survey on optimization oriented sustainable supply chain network design models. Priyan and Uthayakumar (2015) propose an algorithm for a coordinated production, inventory delivery system in the presence of defective products in terms of quality. More recently, Jonrinaldi and Zhang (in press) develop a mathematical model for obtaining integrated production, inventory and distribution decisions in a green manufacturing environment. Wei et al. (2016) propose a heuristic algorithm, which is based on a mixed-integer linear programming (MILP) approach, to address the integrated production, inventory, and distribution problem. Finally, Devapriya et al. (2016) consider an integrated production and distribution scheduling problem with vehicle routing problem for a perishable product, and they develop a heuristic algorithm to solve MIP model.

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This paper attempts to extend the classical ELSP model by incorporating the transportation decision, accounting for finished goods inventories in discrete, sizeable lots. It may be beneficial to deliver quantities of the various products, using either the full truckload (TL) or less than truckload (LTL) shipment policies. In the case of the TL policy, each truckload consists of a mix of all the items. As mentioned earlier, we also adopt, for simplicity, the common production cycle (CC) approach (see, for example, Maxwell, 1964), with a production cycle that is common to all the individual items. For coordination purposes, we adopt the notion that the delivery cycle time is also the same for all items and the overall production cycle is an integer multiple of this delivery cycle. Under the LTL shipping policy, the different items may have different delivery cycles, where individual products are shipped directly to the retailers. Nevertheless, for each product, its delivery frequency is an integer multiple of the overall production frequency, which, once again, is common to all the products in question.

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This work extends the multiproduct model presented by Banerjee (2009). He formulates an analytical model to align the production schedule of multiple products, under an ELSP scenario, with a full truckload delivery plan and develops a heuristic solution methodology. We propose generalized mixed integer, non-linear programming (MINLP) models for developing a multiproduct batch production schedule, which coordinates finished goods availabilities with their outbound TL or LTL shipment plans. In our study, a TL shipment incurs a fixed cost, independent of the shipment size, whereas the LTL shipment costs are based on a variable shipping charge alone, without any fixed cost component. In practice, LTL freight rates often stipulate a minimum shipment quantity, implying a fixed shipping cost, albeit, relatively minor. This fixed cost applies for any shipment less than or equal to the threshold amount. If, however, the quantity shipped exceeds this minimum requirement, the implied fixed cost becomes irrelevant, and the total shipment cost of such a load is based on a unit variable shipping charge alone. For typical commercial shipments, the quantities tend to be large enough, such that the minimum load requirement is seldom of any concern and is of little or no significance in LTL 7

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shipping decisions. This argument provides the rationale for not incorporating any fixed cost component for such shipments. In order to evaluate the efficacy of our integrated models, we compare their performance with the traditional uncoordinated production and distribution approach under each of the two transportation regimes (TL and LTL shipping). The concepts and the models developed here integrating the production and shipment decisions, as well as their potential benefits, in comparison with uncoordinated planning, are illustrated and analyzed through a numerical example.

3 Assumptions and Notation

In this Section, we present the assumptions that are necessary for the formulation of our proposed models and the basic notational scheme used throughout the paper.

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3.1.1 Assumptions

The following assumptions are made in describing the manufacturing-distribution scenario adopted in this paper and for formulating our models that follow:

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1. Market demands for the various products are deterministic and stationary. 2. A set of products are manufactured in a single capacitated batch production facility, with different production rates for the various items. 3. Only a single product may be produced at any given time. 4. Stockouts are not permitted. 5. The common cycle (CC) approach is deployed to solve the ELSP, where each product is produced exactly once in every production cycle. 6. Each of the products is transported via truck and is delivered to one or more given demand locations, depending on one of two shipping policies in effect. 7. Under a full truckload (TL) shipping policy, a mix of all products, constituting a full load, is delivered to all retail locations on the basis of a peddling arrangement. The LTL transportation mode, on the other hand, implies direct shipment of each product to each retailer. 8. These two scenarios impose different transportation cost structures. The TL model involves a capacitated vehicle, incurring only a fixed cost for all the peddling shipments made in a single delivery, while for LTL shipments, each direct shipment cost is based on a load-based variable cost. Furthermore, for simplicity and without loss of generality, we assume that the unit variable shipping rate for a given product is the same for all retail locations. 9. Under the TL policy, an integer number, K, of deliveries are made at equal intervals of time over a production cycle. 10. For the LTL case, the number of deliveries made per common production cycle may vary for the different products but are still integer multiples, K1, K2, etc., of the production cycle. 11. At each of the various demand locations, stocks are replenished via a periodic review, order-up-to level inventory control system, when TL shipping is in effect. For

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coordination purposes, all the items at all the demand locations share a common fixed review period. 12. Under the LTL shipment policy, although the review periods for the various items may be different, for any given product, all retail locations share a common review period, for coordination purposes. 3.2 Notation

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The following notational scheme is adopted in the formulation of our models: Objective Function: TRC

Total relevant cost ($) per time unit

Parameters:

An index used to denote a specific product, i=1, 2, …, n The demand rate for product i (units/time unit) The production rate for product i (units/time unit) Manufacturing setup cost per production batch for product i ($/batch) Inventory holding (carrying) cost for product i ($/unit/time unit) The FTL capacity, i.e. maximum total load (or volume) allowable per truckload Weight (or volume) of each unit of product i ( ) Fixed cost of initiating one truck dispatch ($/shipment) for TL policy Unit shipment cost of product ( ) for less than truckload (LTL) amounts($/unit).

Decision Variables:

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Average inventory level (units) of product i ( ) Amount of product i contained in each TL shipment (units) A positive integer, representing the number of shipments per production cycle The shipment interval in time units (common to all products and locations) The production lot size (in units) for product i ( ) Production cycle length in time units

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4 Model development

In this section, we present the details of the two shipment policies adopted in this paper, based on direct shipment and peddling shipment modes. These distribution policies are depicted in Figure 1. ***Insert Figure 1 about here***

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4.1 Shipment Policies Our analyses are based on extensions of the multiproduct model presented in Banerjee (2009). We develop an analytical model and methods to minimize total inventory and transportation related costs when a supplier distributes a set of different items to several retailers or customers. This paper evaluates and compares two different distribution policies: direct shipping and peddling.

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The direct shipping distribution policy involves shipping separate loads from the supplier directly to each customer, whereas peddling shipping dispatches a fully loaded truck in each distribution cycle, that deliver items to all of the customers, based on each location’s demand during this cycle. The inventory-time plots for the latter scenario is depicted in Figure 2 and those for a coordinated LTL distribution situation is shown in Figure 3.

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4.2 TL policy model formulation 4.2.1 Coordinated Full Truckload (CTL) Policy

For illustrative purposes, the inventory-time plots for a CTL distribution scenario are shown in Figure 2. This plot illustrates a situation that involves three products (n=3) with negligibly small set up times and transit times and three full CTL shipments for each production cycle.

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***Insert Figure 2 about here***

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Figure 2 shows that there is a common delivery cycle time of T. Each truck with a limited capacity, C, contains Qi units of product i, (i = 1, 2, 3). The products should be sequenced to minimize the total set up, inventory holding and transportation costs (see Banerjee, 2009, for an explanation of this). The batches of the various products are produced in this order and held in inventory till the last item’s production quantity reaches its retail market demand during a delivery cycle, T. Then a truck is fully loaded with T time unit’s demand for all the items and dispatched for a peddling distribution to all the retail location.

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For the coordinated full truckload (CTL) shipment case, we obtain the following the average inventory values for the three items illustrated in Figure 2 (see Banerjee (2009) for detailed derivation): *

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Banerjee (2009) shows that extending the above results for the general n products case, we obtain the following expression for depicting the average inventory level:

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where the first term represents the collective average inventory level for the first (n-1) items and the second term captures the average inventory of the nth item. 4.2.1.1 Objective function

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Generalized these results for n products, we formulate the minimization objective function (the total relevant cost per time unit), as shown below. This expression includes the production setup, inventory holding, and transportation costs per time unit. Note that the cost function below is non-linear, with an integrality requirement. The decision variables are the amounts of all the products, Qi (for all i), contained in each CTL shipment and the number of shipments per production cycle, which is denoted by K, an integer.

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The first term above represents the total manufacturing setup cost per time unit for the n products. The next two terms capture the total inventory holding cost, and the final term represents the total transportation cost that is obtained by the multiplication of the fixed cost of initiating one truck dispatch and the total number of CTL shipment cycles per unit of time. 4.2.1.2 Constraints 1. The delivery cycle time is common to all products:

2. The production capacity of the facility must be sufficient to produce all products, i.e. 11

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∑ 3. The production schedule should be feasible, i.e. the total production time for all products should be less than the manufacturing cycle time (without loss of generality, we assume that the manufacturing setup times are negligibly small): ∑

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4. The load capacity of a truck is limited by the total weight (or volume) of the products, i.e. ∑

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5. At least one CTL shipment must be made over a production cycle: K ≥ 1, where

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The CTL policy, i.e. a mixed integer non-linear programming (MINLP), model formulated above may be solved using one of the several computer-based solvers available. We employ the BONMIN solver, which was developed by Bonami et al. (2008), for this purpose and obtain the optimal solution. From the optimal solution returning the items’ delivery lot sizes Qi, ( ) and K, the number of CTL shipments per production cycle, the delivery cycle time can be obtained easily. 4.2.2 Uncoordinated Full Truckload (UTL) Policy

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For comparison purposes, we formulate the minimization objective function (the total relevant cost per time unit) as shown below, for the uncoordinated full truckload (UTL) policy, based on the traditional common cycle solution technique for the ELSP. This expression includes the production setup, inventory holding, and transportation costs per time unit and is independent of the shipment decision.

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from which the optimum common cycle time, T* is based on the first order optimality condition, as follows: √∑

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It can be easily shown that TRC (T) is strictly convex in T. Note that the transportation cost is computed for each product individually and is based on the minimum integer number of trucks required for transporting the entire production batch immediately upon completion, without the need for waiting to fill up a truck. It is clear that this UTL policy attempts to reduce the finished goods holding cost per time unit, albeit at the expense of higher transportation costs. Note that under this policy, the common cycle based production schedule is derived first and, subsequently, 12

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for each product, the entire lot produced is transported via the minimum number of trucks necessary in order to avail of the TL rates. Note that in this shipment mode; all trucks may not be fully loaded, due to the uncoordinated production and shipment plans.

4.3

LTL policy model formulation

4.3.1 Coordinated Less-than-Truckload (CLTL) Policy

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For illustrative purposes, the inventory-time plots for a direct shipment based CLTL distribution policy are shown in Figure 3, which represents a scenario involving three products (n=3) with negligible setup and transit times. Note that CLTL shipment allows for different delivery frequencies for the various items under a common production cycle. ***Insert Figure 3 about here***

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Under this production/shipment regime, there is a common production cycle time of τ, such that for product i, τDi units of it are produced in a single batch, which are delivered in equal lots of Qi = τDi/Ki units to all the retail locations through CLTL shipments. For any product i, each retail location, receives its appropriate demand that occurs over the delivery cycle of Qi/Di time units, via an independent and individual CLTL shipment. Since, during the manufacturing process, the various products are not held in inventory, waiting for a truck to be fully loaded, as in the CTL shipment case, the sequence in which the items are manufactured is not relevant and is arbitrary. Due to this phenomenon of decoupling of the products, their delivery frequencies can be different, as mentioned above. 4.3.1.1 Objective function

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As before, the objective is to minimize the total relevant cost per time unit, which includes the setup, inventory holding and transportation costs. Also, the decision variables consist of the common cycle time, τ and the lot size multiplier values Ki, for i = 1, 2,…, n. Also, the number of shipments per production batch for product i is Ki, which is restricted to a positive integer value for all i. The objective function of the CLTL model then can be expressed as (

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The first term above represents the total manufacturing setup cost per production batch for the n products, the second term represents the total inventory holding cost for all the items and the last term denotes the total transportation cost per unit of time. Note that as a result of adopting a common production cycle, 4.3.1.2 Constraints 1. The production capacity should be sufficient to produce all products, i.e. ∑ 2. The following condition is necessary for ensuring production schedule feasibility: 13

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∑

, which simplifies to

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3. At least one delivery shipment per production batch should be made within a production cycle for each item: Ki ≥ 1, where .

the Qi, for i = 1, 2,…, n, can be calculated easily.

4.3.2 Uncoordinated Less-than-Truckload (ULTL) Policy

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Once again, the BONMIN solver is utilized so find the optimal solution to the MINLP CLTL policy model formulated above. From the optimal values of values returned by the solver,

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Once again, for comparison purposes, we formulate the minimization objective function (the total relevant cost per time unit), as shown below, pertaining to the uncoordinated less-thantruckload (ULTL) policy. This expression includes the production setup, inventory holding, and transportation costs per time unit. ∑

∑

∑

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from which, as before, the optimum value of T, based on the first order optimality condition, can be expressed as

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Under this ULTL policy, the transportation cost for all the products are based on the LTL shipment mode utilizing the applicable LTL shipping rate. As in the case of the UTL policy, the production schedule is not linked directly to the shipment decisions and, as a production batch of each product is completed, it is shipped out immediately.

5 Numerical Example

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This section presents an illustrative example involving three products. The relevant data pertaining to the problem are shown in Table 1. ***Insert Table 1 about here***

The truck capacity is varied from 10,000 lbs. to 70,000 lbs. in 5,000 lbs. increments, for full truckload shipments. In addition, the unit variable shipment cost of products for the less than truckload mode is varied from $0.15 to $0.24 per lb. in increments of $0.01. As mentioned before, we obtain the optimal solutions to the mixed integer nonlinear optimization problems (MINLPs) for both CTL and CLTL shipment policies using the BONMIN 14

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solver, which provides global optimal solutions for the MINLPs. Table 2 presents the summary of the computational results for CTL shipments with varying truck capacities and Table 3 shows the summary results for the CLTL shipping policy, incorporating different unit variable shipping costs. ***Insert Tables 2 and 3 here***

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Table 2 indicates that the CTL delivery cycle time varies from 0.00735 years (2.68 days) to 0.05147 years (18.79 days). This cycle length increases as the truck capacity goes up. The number of CTL deliveries per production cycle (K) tends to decrease with increasing truck capacity. These results are not unexpected since the fixed cost per shipment tends to increase with larger vehicle capacities. To compensate for this phenomenon, the production cycle time is increased, together with fewer deliveries per manufacturing cycle. Needless to say that delivery lot sizes also increase with higher truck capacities. Interestingly, the total relevant cost function value tends to exhibit a convex behavior with respect to vehicle capacity. Clearly, as can be seen in Figures 4 – 8, due to the effects of economies of scale, under a variety of parametric conditions, the TRC initially decreases with increasing vehicle size. Nevertheless, after a certain threshold truck capacity, the initial cost advantage of scale seems to be more than offset by the higher annual truck dispatching costs, as well as higher inventory holding costs, resulting from the need to hold more output in stock before a larger vehicle can be fully loaded. For the given problem parameters, it appears that under a peddling distribution policy, CTL shipments with a 50,000 lbs. truck capacity yields the lowest total relevant cost per year of $301,668, with the original problem parameter values in effect (see Figure 4). However, it is also sensitive to the cost parameters. For example, when fixed cost of initiating one truck dispatch is decreased by 50%, CTL shipments with a 40,000 lbs. truck capacity yields the lowest total relevant cost per year of $180,706 (see Figure 6).

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The results for the CLTL direct shipment policy, as shown in Table 3 and also in Figures 4 – 7, lead to some interesting observations. First, in the absence of a fixed shipping cost, a common production cycle leads to a lot-for-lot (with respect to aggregate market demand) delivery policy for each of the products concerned, i.e. Ki = 1, Although in the real world, LTL shipping rate schedules may sometimes include a relatively small fixed cost or a minimum required shipment size or load, we deliberately do not consider such features in our analyses to draw a starker distinction between the two types of shipment cost structures. From the minimization of the objective function for the CLTL policy model, it is clear that for any given production cycle time, the second term, representing the total holding cost, is minimal when all Ki values are set to 1. Thus, if a feasible solution (i.e. sufficient production capacity) exists for this model, the optimization task in this case is to determine the appropriate value of Once this is accomplished, the problem is essentially solved. Hence, we observe that regardless of the value of the unit variable shipping charge, the production and delivery cycles remain the same, although, the annual total relevant cost increases with increasing variable transportation cost. For the example chosen, the optimal cycle time remains fixed at 0.14798 year (54.01 days), with the same lot-for-lot product deliveries for differing variable shipping charges. Figure 4, comparing the TRC values for the CTL policies with varying truck capacities and the ***Insert Figure 4 about here*** 15

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CLTL policy with changing variable shipping costs, further indicates that when the unit shipping cost is sufficiently low, the latter policy is always superior from a cost perspective. Otherwise, there is clearly a beak-even point between these two policies, with respect to vehicle capacity. For relatively small-sized trucks, the CLTL policy is likely to be more desirable, whereas the CTL shipping policy tends to yield lower TRC values, beyond the break-even truck capacity level, before the effect of diseconomies of scale take effect. This is not surprising since with larger trucks the fixed charge structure is based on a decreasing prorated cost per unit shipped.

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For the purpose of sensitivity analyses, we vary the manufacturing setup cost, Ai, and the fixed CTL shipping cost, and holding cost values. The results of these analyses are summarized in Figures 5, 6, and 7, which indicate that with increasing fixed CTL shipping charges, the CLTL ***Insert Figures 5, 6 and 7 about here***

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policy tends to become more dominant. By the same token, if this cost decreases, the CTL policy tends to be superior to the CLTL shipping mode. Additionally, increasing the production setup cost appears to have a similar effect with respect to the two distribution policies examined here. In other words, all else being equal, higher setup costs tend to render the CLTL policy a better alternative to CTL distribution.

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In order to explore the relative merits of our proposed coordinated production/distribution policies under the two shipment regimes, we examine the corresponding uncoordinated policies in conjunction with both TL and LTL shipments. The results of these analyses are summarized in Tables 4, and 5, and the relative performances of the uncoordinated approach in comparison to our coordinated approach for both TL and LTL shipping are depicted in Figures 8 and 9, respectively. ***Insert Tables 4 and 5 and Figures 8 and 9 about here***

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Comparing Tables 2 and 4 and examining Figure 8 reveals that the CTL policy always outperforms the UTL policy, in terms of total relevant cost, regardless of truck capacity. The annual TRC resulting from the CTL policy is, on the average, $75,949 lower, compared to the UTL policy.

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Table 4 and Figure 8 indicate an interesting phenomenon concerning the UTL policy. It is clear that under this policy, the annual total relevant cost (TRC) varies non-monotonically with increasing truck capacity. An explanation of this behavior lies in two important characteristics of the shipping policy in effect. First, as mentioned earlier, as a production batch of a product is completed, it is shipped out via the minimum number of trucks (an integer, depending on the load capacity per truck), necessary for transporting this load, since for TL shipping, the total transportation cost is based on the number of trucks used. Note that, since the production schedule is derived independent of shipping considerations, all the trucks deployed may not be fully loaded, although the full fixed shipping cost for each such truck is incurred. Secondly, in our analysis, we vary the capacity per truck from 10,000 to 70,000 lbs. in discrete increments of 5,000 lbs. The fixed shipping charge per truck, similarly, increases with the size of the truck in a stepwise manner, although the prorated cost per lb. for a fully loaded vehicle decreases with 16

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increasing truck capacity. A simple illustration, based on the results obtained, will suffice to explain the non-monotonic behavior of the annual total relevant cost (TRC) with respect to truck size.

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Table 4 indicates that the CC solution calls for an average of about 12.29 deliveries per year (i.e. 1/T = 1/0.08137) for each of the three products. Consider, for example, the second product, which is produced in batches of about 977 units, translating to a load of 48,850 lbs., regardless of the truck size deployed. If 10000 lb. trucks (with a TL rate of $3,000/truck) are used to transport this load, the theoretical number of trucks needed is 48850/10000, i.e. 4.85 trucks. Thus, at a minimum, 5 trucks are needed for which the average annual shipping charges would amount to about $34,350 (i.e. 5 x $3,000 per truck x 12.29 annual deliveries). If, on the other hand, 15,000 lb. trucks are used, for which the TL rate is $3,300/truck, 4 such trucks are needed now for transporting the batch, and the average annual shipping cost would be about $30,228. Similarly for 20,000 and 25,000 lb. trucks (requiring, respectively, 3 and 2 trucks), the corresponding average annual transportation costs would be about $24,938 and $18,228, respectively. So far, the annual shipping cost declines with increasing truck size, largely due to a decrease in the number of vehicles needed. Since the costs pertaining to the production schedule are unaffected by shipping policy and remain the same, the overall behavior of TRC under the UTL policy, shown in Table 4 and Figure 8 reflects these changes due to the transportation cost contributions of one of the products. However, if 30,000 lb. trucks are utilized to transport the 48,850 lbs. of product 2, two trucks are still necessary at the higher cost of $4,392/truck, yielding an annual shipping cost of about $20,115, which is now higher than that resulting from the deployment of 25,000 lb. trucks. Note that using 30,000 lb., as opposed to 25,000 lb., trucks, reduces the load capacity utilization factor from 97.7% to 81.4%. Thus, for the given cost parameters, the cost per lb. of unused truck capacity, resulting from the integrality requirement, appears to outweigh the prorated lower shipping charge per lb. of capacity available, when the truck capacity increases from 25,000 to 30,000 lbs. This pattern of increasing shipping cost holds as the truck size is increased further, for up to 45,000 lbs. If the truck capacity is 50,000 lbs., only a single vehicle is required, and the load capacity utilization factor increases sharply, compared to the case of using two 45,000 lb. trucks, yielding a lower shipment cost (as well as a decline in the TRC value). For truck sizes larger than 50,000 lbs, a single truck would still be needed, but its load capacity utilization factor will decrease with truck size, as the per lb. cost of available capacity remains constant, resulting in increasing annual shipping costs and the resulting contributions to the TRC values. Similar arguments can be made for the other products involved, in order to explain the non-monotonic behavior of the overall TRC with respect to increasing truck size.

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Finally, Tables 3 and 5, as well as Figure 9 indicate the CLTL policy’s superiority over the ULTL policy, in terms of TRC, regardless of the variable shipment cost parameter values. The average annual savings resulting from coordination under LTL shipping is $17,178. These results confirm the contention that coordination of production and distribution activities in a supply chain can lead to a significant economic advantage.

6 Summary and Conclusions

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In this paper, we have made an attempt to integrate the lot scheduling decisions for multiple products produced in a single facility operating in an ELSP environment, with their shipment schedules involving two different types of shipment modes, involving different transportation cost structures under deterministic conditions. One common type of shipping rate regime found in the real world involves full truckload (TL) or carload movement of goods, where only a fixed cost is incurred depending on the points of origin and destination, as well as the type of commodity or product moved. An alternative transportation mode adopted in this study is the less than truckload (LTL) or less than carload shipping, where there is no fixed cost or is a relatively minor factor, which can be ignored. In this case, the cost of a specific shipment is based on a variable cost per unit moved from an origin to a destination. In our analyses, we incorporate both of these two transportation scenarios for a single manufacturer and several retailers. Furthermore, under a TL shipping policy, we employ a peddling type of distribution arrangement, where a fully loaded vehicle containing a mix of all the products is dispatched to all the retail locations for simultaneous delivery. In the case of LTL shipments, each product has its own delivery cycle, and shipments are made directly and individually to each of the retailers when a batch of the item is completed.

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We construct constrained (MINLP) models for linking the production and distribution decisions under both of the distribution policies described above and employ a widely available solver software for finding globally optimal solutions. Through a set of numerical experiments, we show that the respective magnitudes of the various cost parameters play a crucial in selecting either a CTL or CLTL distribution method. An important finding of this work is that when transportation involves no fixed cost, but only a variable charge per unit shipped, the optimal shipment schedule is essentially lot-for-lot with respect to aggregate retail demand. We have observed that under CTL distribution, the production, as well as the delivery cycle lengths tend to go up as vehicles of larger capacities are employed. Also, we have attempted to outline the parametric conditions under which either of the two transportation modes will dominate over the other from a cost perspective. In comparing the performance of our integrated planning approach with traditional uncoordinated production and distribution planning, we find that the former yields substantial economic benefits under the various adopted parametric conditions, with either of the shipment modes in effect. We also observe that the cost difference between the two planning procedures tend to be more pronounced in the case of TL shipping.

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It is hoped that this study will provide a helpful tool for supply chain practitioners, in terms of integrating a relatively straightforward rotational cycle based production schedule with transportation planning and selecting an appropriate method of distribution. We also hope that future research endeavors in this area will find some value in this work and will extend our findings under more complex and realistic supply chain environments, such as those involving stochastic market demands, specifically incorporating minimum shipment constraint under the LTL policy, or integrate the truck routing problem with the overall production and distribution planning process in the case of the TL shipping mode. Furthermore, moving away from the common production cycle approach, employing the notion of unequal production and delivery cycle times in future research endeavors is likely to result in more efficient, albeit involving higher complexity, solutions that link the production and shipment decisions in supply chains in effective ways.

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Blumenfeld, D.E., Burns, L.D., Diltz, J.D. and Daganzo, C.F., 1985. Analyzing trade-offs between transportation, inventory and production costs on freight networks. Transportation Research Part B: Methodological, 19(5), pp.361-380. Bomberger, E.E., 1966. A dynamic programming approach to a lot size scheduling problem. Management Science, 12(11), pp.778-784.

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Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N. and Wächter, A., 2008. An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optimization, 5(2), pp.186-204. Brown, G., Keegan, J., Vigus, B. and Wood, K., 2001. The Kellogg company optimizes production, inventory, and distribution. Interfaces, 31(6), pp.1-15.

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Chandra, P. and Fisher, M.L., 1994. Coordination of production and distribution planning. European Journal of Operational Research, 72(3), pp.503-517.

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Chatfield, D.C., 2007. The economic lot scheduling problem: A pure genetic search approach. Computers & Operations Research, 34(10), pp.2865-2881.

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Chen, Z.L., 2010. Integrated production and outbound distribution scheduling: review and extensions. Operations Research, 58(1), pp.130-148.

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Chen, Z.L. and Vairaktarakis, G.L., 2005. Integrated scheduling of production and distribution operations. Management Science, 51(4), pp.614-628.

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Delporte, C.M. and Thomas, L.J., 1977. Lot sizing and sequencing for N products on one facility. Management Science, 23(10), pp.1070-1079. Devapriya, P., Ferrell, W. and Geismar, N., 2016. Integrated Production and Distribution Scheduling with a Perishable Product. European Journal of Operational Research. Dobson, G., 1987. The economic lot-scheduling problem: achieving feasibility using timevarying lot sizes. Operations research, 35(5), pp.764-771. Doll, C.L. and Whybark, D.C., 1973. An iterative procedure for the single-machine multiproduct lot scheduling problem. Management Science, 20(1), pp.50-55. 19

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Elmaghraby, S.E., 1978. The economic lot scheduling problem (ELSP): review and extensions. Management Science, 24(6), pp.587-598. Eskandarpour, M., Dejax, P., Miemczyk, J. and Péton, O., 2015. Sustainable supply chain network design: An optimization-oriented review. Omega, 54, pp.11-32.

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Farahani, R.Z., Rezapour, S., Drezner, T. and Fallah, S., 2014. Competitive supply chain network design: An overview of classifications, models, solution techniques and applications. Omega, 45, pp.92-118. Garcia, J.M. and Lozano, S., 2004. Production and vehicle scheduling for ready-mix operations. Computers & Industrial Engineering, 46(4), pp.803-816.

Hall, N.G. and Potts, C.N., 2003. Supply chain scheduling: Batching and delivery. Operations Research, 51(4), pp.566-584.

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Hall, N.G. and Potts, C.N., 2005. The coordination of scheduling and batch deliveries. Annals of operations research, 135(1), pp.41-64. Herrmann, J.W. and Lee, C.Y., 1993. On scheduling to minimize earliness-tardiness and batch delivery costs with a common due date. European Journal of Operational Research, 70(3), pp.272-288.

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Jonrinaldi and Zhang, D.Z., 2013. An integrated production and inventory model for a whole manufacturing supply chain involving reverse logistics with finite horizon period. Omega, 41(3), pp.598-620.

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Jonrinaldi and Zhang, D.Z., (in press). An integrated production, inventory and transportation decision in a whole green manufacturing supply chain. International Journal of Industrial and Systems Engineering.

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King, R.H. and Love Jr, R.R., 1980. Coordinating decisions for increased profits. Interfaces, 10(6), pp.4-19.

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Lee, I.S. and Yoon, S.H., 2010. Coordinated scheduling of production and delivery stages with stage-dependent inventory holding costs. Omega, 38(6), pp.509-521. Li, K.P., Ganesan, V.K. and Sivakumar*, A.I., 2005. Synchronized scheduling of assembly and multi-destination air-transportation in a consumer electronics supply chain. International Journal of Production Research, 43(13), pp.2671-2685. Li, K., Ganesan, V.K. and Sivakumar, A.I., 2006. Scheduling of single stage assembly with air transportation in a consumer electronic supply chain. Computers & Industrial Engineering, 51(2), pp.264-278.

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Martin, C.H., Dent, D.C. and Eckhart, J.C., 1993. Integrated production, distribution, and inventory planning at Libbey-Owens-Ford. Interfaces, 23(3), pp.68-78. Maxwell, W.L., 1964. The scheduling of economic lot sizes. Naval Research Logistics Quarterly, 11(2), pp.89-124.

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Metters, R.D., 1996. Interdependent transportation and production activity at the United States postal service. Journal of the Operational Research Society, pp.27-37. Pan, F. and Nagi, R., 2013. Multi-echelon supply chain network design in agile manufacturing. Omega, 41(6), pp.969-983.

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Priyan, S. and Uthayakumar, R., 2015. An integrated production–distribution inventory system involving probabilistic defective and errors in quality inspection under variable setup cost. International Transactions in Operational Research. Relvas, S., Magatão, S.N.B., Barbosa-Póvoa, A.P.F. and Neves, F., 2013. Integrated scheduling and inventory management of an oil products distribution system. Omega, 41(6), pp.955-968.

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Seyedhosseini, S.M. and Ghoreyshi, S.M., 2014. An integrated model for production and distribution planning of perishable products with inventory and routing considerations. Mathematical Problems in Engineering, 2014.

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Torabi, S.A., Karimi, B. and Ghomi, S.F., 2005. The common cycle economic lot scheduling in flexible job shops: The finite horizon case. International Journal of Production Economics, 97(1), pp.52-65.

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Wei, W., Guimarães, L., Amorim, P. and Almada-Lobo, B., 2016. Tactical production and distribution planning with dependency issues on the production process. Omega.

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van Buer, M.G., Woodruff, D.L. and Olson, R.T., 1999. Solving the medium newspaper production/distribution problem. European Journal of Operational Research, 115(2), pp.237253.

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Peddling Shipping (TL) Policy

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Direct Shipping (LTL) Policy

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Central Facility

Retailer I

Retailer I

Retailer II

Retailer III

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Central Facility

Retailer III

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Retailer II

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Figure 1: Direct Shipping (LTL) vs. Peddling Shipping (TL) Policies

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Figure 2: Inventory-Time plots for a Peddling Shipment (CTL) Policy (n=3, K=3)

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Figure 3: Inventory-Time Plots for a Direct Shipment (CLTL) Policy (n=3, K 1=4, K2=3, K3=1)

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Product (i) 1 2 3

Di (units/year)

Pi (units/year)

Ai ($/setup)

hi ($/unit/year)

wi (lbs./unit)

8,000 12,000 15,000

30,000 50,000 40,000

1500 3000 2400

40 72 60

20 50 40

T (year/days)

K

0.007353/2.68 0.011029/4.03 0.014706/5.37 0.018382/6.71 0.022059/8.05 0.025735/9.39 0.029412/10.74 0.033088/12.08 0.036765/13.42 0.040441/14.76 0.044118/16.10 0.047794/17.44 0.051471/18.79

17 11 8 7 6 5 4 4 3 3 3 3 2

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3000 3300 3630 3993 4392 4832 5315 5846 6431 7074 7781 8559 9415

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10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000 70000

Q1(units) Q2(units) Q3(units) 58.82 88.24 117.65 147.06 176.47 205.88 235.29 264.71 294.12 323.53 352.94 382.35 411.77

88.24 132.35 176.47 220.59 264.71 308.82 352.94 397.06 441.18 485.29 529.41 573.53 617.65

TRC($)

110.29 165.44 220.59 275.74 330.88 386.03 441.18 496.32 551.47 606.62 661.77 716.91 772.06

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Capacity(lbs.) Gamma($)

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Table 1: Example Problem Parameters

T (year/days)

Ki

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Table 2: Numerical Results for CTL Shipment Policy with Varying Truck Capacities

1,1,1 1,1,1 1,1,1 1,1,1 1,1,1 1,1,1 1,1,1 1,1,1 1,1,1 1,1,1

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0.14798/54.01 0.14798/54.01 0.14798/54.01 0.14798/54.01 0.14798/54.01 0.14798/54.01 0.14798/54.01 0.14798/54.01 0.14798/54.01 0.14798/54.01

Q1(units) Q2(units) 1183.84 1183.84 1183.84 1183.84 1183.84 1183.84 1183.84 1183.84 1183.84 1183.84

1775.76 1775.76 1775.76 1775.76 1775.76 1775.76 1775.76 1775.76 1775.76 1775.76

Q3(units) V($/lb.) TRC($) 2219.7 2219.7 2219.7 2219.7 2219.7 2219.7 2219.7 2219.7 2219.7 2219.7

Table 3: Numerical Results for CLTL Shipment Policy

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0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15

419,656 406,056 392,456 378,856 365,256 351,656 338,056 324,456 310,856 297,256

524,079 416,535 365,524 337,349 320,766 310,463 304,509 302,186 301,668 302,462 305,668 310,877 315,569

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T (year/days)

10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000 70000

3000 3300 3630 3993 4392 4832 5315 5846 6431 7074 7781 8559 9415

0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70

Q1(units) Q2(units) Q3(units) 651.00 651.00 651.00 651.00 651.00 651.00 651.00 651.00 651.00 651.00 651.00 651.00 651.00

976.50 976.50 976.50 976.50 976.50 976.50 976.50 976.50 976.50 976.50 976.50 976.50 976.50

1220.62 1220.62 1220.62 1220.62 1220.62 1220.62 1220.62 1220.62 1220.62 1220.62 1220.62 1220.62 1220.62

Q1(units) Q2(units) Q3(units) V ($/lb.) TRC ($) 651.00

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Table 4: Numerical Results for UTL Shipment Policy with Varying Truck Capacities

T (year/days)

976.50

1220.62

0.24

436,834

651.00

976.50

1220.62

0.23

423,234

0.08137/29.70

651.00

976.50

1220.62

0.22

409,634

0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70 0.08137/29.70

651.00 651.00 651.00 651.00 651.00 651.00 651.00

976.50 976.50 976.50 976.50 976.50 976.50 976.50

1220.62 1220.62 1220.62 1220.62 1220.62 1220.62 1220.62

0.21 0.20 0.19 0.18 0.17 0.16 0.15

396,034 382,434 368,834 355,234 341,634 328,034 314,434

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Table 5: Numerical Results for ULTL Shipment Policy

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TRC($) 542,434 466,834 415,354 350,014 373,954 400,354 429,334 461,194 341,950 365,098 390,550 418,558 449,374

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TRC of Coordinated TL & LTL Shipment Policy 6,00,000 5,00,000

Coordinated TL Policy

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4,00,000 3,00,000

Coordinated LTL Policy (v=$0.24) Coordinated LTL Policy (v=$0.21)

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Coordinated LTL Policy (v=$0.18) Coordinated LTL Policy (v=$0.15)

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Figure 4: Comparison of Coordinated CTL and CLTL Shipment Policies

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CTL Policy (Double Setup Cost) CLTL Policy (Double Setup Cost)

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4,00,000

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3,00,000

CTL Policy (Original Setup Cost)

2,00,000

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1,00,000

CLTL Policy (Half Setup Cost)

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Truck Capacity (lbs.)

Figure 5: Policy Comparison with Different Setup Cost Values

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Sensitivity of TRC to Different Fixed Cost Values 10,00,000 9,00,000 8,00,000

6,00,000

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TRC ($)

7,00,000

CTL Policy (Double Gamma)

5,00,000

CTL Policy (Original Gamma)

4,00,000

CLTL Policy (Original Gamma)

3,00,000

CTL Policy (Half Gamma)

2,00,000 1,00,000

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Truck Capacity (lbs.)

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Figure 6: Policy Comparison with Different Fixed Cost Values

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Sensitivity of TRC to Different Holding Cost Values 7,00,000

CTL Policy (Double Holding Cost) CLTL Policy (Double Holding Cost)

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4,00,000 3,00,000

CTL Policy (Original Holding Cost) CLTL Policy (Original Holding Cost)

2,00,000

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5,00,000

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CTL Policy (Half Holding Cost) CLTL Policy (Half Holding Cost)

1,00,000

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Truck Capacity (lbs.)

Figure 7: Policy Comparison with Different Holding Cost Values

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Coordinated vs. Uncoordinated TL Policy 6,00,000 5,00,000

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4,00,000 3,00,000

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2,00,000

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1,00,000

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Figure 8: Policy Comparison of Coordinated and Uncoordinated TL Policy

Coordinated vs. Uncoordinated LTL Policy

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5,00,000 4,50,000

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0.21

0.22

0.23

0.24

Variable Shipping Cost ($)

Figure 9: Policy Comparison of Coordinated and Uncoordinated LTL Policy

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