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An application of interdependent lot size and consolidation point choice
Mathematical and Computer Modelling 44 (2006) 65–72 www.elsevier.com/locate/mcm
An application of interdependent lot size and consolidation point choice Daniel G. Conway a,∗ , Michael F. Gorman b a Department of Management, University of Notre Dame, 348 Mendoza College of Business, Notre Dame, IN 46556, United States b University of Dayton, Department of MIS, Operations Management, and Decision Sciences, School of Business Administration,
300 College Park, Dayton, OH 45469-2130, United States Received 2 October 2004; accepted 9 December 2004
Abstract The efficiency of consolidation points in a distribution system is driven both by the total volume transported and by the lot size in which the product travels. We examine a finished goods distribution system for a major automobile manufacturer that is setting policy concurrently for consolidation points and for the lot size of transport. Through simulation of the distribution system we show the impacts of consolidation on product holding time, volume volatility and customer service. Our results show a significant reduction in total system inventory from consolidation; however, some customers’ service is adversely affected. We then develop a methodology for determining the best combination of consolidation, direct shipment and lot size for all origins and destinations in the distribution system in order to minimize inventory holding time while simplifying operations and maintaining customer service. Preliminary results are shown which demonstrate the potential for substantial reductions in the total finished vehicle holding time in the system. c 2005 Elsevier Ltd. All rights reserved.
Keywords: Consolidation points; Distribution; Lot sizing; Freight networks
1. Introduction In order for automobile manufacturers to compete in emerging e-business markets, it became necessary to better understand the roles played by their distribution systems. In particular, the simple promise of a to-ordervehicle assembled and delivered in seven days has presented some complicated distribution channel problems previously ignored by major automobile manufacturers. This paper presents one automobile manufacturer’s attempt at researching these distribution issues. Effective distribution channel management balances the direct costs of transportation such as fuel, equipment and handling with time cost of the finished good in the distribution pipeline and customer service and market presence considerations. Two choices to be made in setting up distribution channels are the number of consolidation points and the lot size of the transport. In general, increasing consolidation points and decreasing lot size have the same effect ∗ Corresponding author.
E-mail address: [email protected] (D.G. Conway). c 2005 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2004.12.015
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on the distribution channel: increasing the direct costs of product movement while increasing its velocity. Benefits from consolidation can be found in: improvement in the speed at which the product travels through the distribution channel, buffering rates of production and demand, simplifying the building of full lots at origins, and the reduction in variation of shipments out of origins. However, a sizable investment in a consolidation center must be made to realize the benefits, and additional handling of product is required as lots are broken down and rebuilt at consolidation points. Economies of scale are encountered when choosing lot size transport. However, the savings are tempered by inventory holding time, particularly in distribution channels with multiple origins and destinations, as a full lot for shipment can take a considerable time to produce, and shipping partial lots is very rarely done. Customer service considerations and finished good inventory reductions can make smaller lots more attractive at the cost of higher direct variable costs of shipment. Similar to consolidation, reducing lot size can reduce time in transit and variability in shipment levels. While both consolidation and reducing lot size may improve delivery time and variability, the two decisions cannot be made independently as the benefits of each are diminished when used in tandem. Thus, the lot size and consolidation decision must be made jointly when designing a distribution network. In our problem, we assume the candidate consolidation points are known. The network consists of numerous, heterogeneous origins and destinations. Due to the heterogeneity, the optimal lot size and consolidation strategy may be different for each origin–destination pair. For example, consolidation may negatively impact transit times for higher volume destinations that enjoy a relatively short amount of time in the distribution channel. However, the volumes of the large origins and destinations at a consolidation point help create the benefits of consolidation for the low volume destinations; thus, the consolidation strategies for all destinations cannot be made independently. In the face of numerous interdependent lot size and consolidation strategies, we develop a heuristic model for developing a lot size choice and consolidation strategy for all origins and destinations in the network, in order to shorten the delivery cycle and minimize the product inventory holding time while simplifying operations and maintaining customer service. Permitting only one lot size to service each origin and a single consolidation strategy for each origin and destination combination simplifies operations and is common in practice. Restricting variety of lot size for each origin and destination lends the model to a more strategic, rather than tactical, use. Other constraints on lot size choice exist for particular origins and destinations. Customer service requirements are ensured by constraining delivery times not to exceed current standards. High volume destinations are not willing to accept degradation in their service in order to improve overall network transit times. Thus, any consolidation strategy faces the constraint of not increasing any single destination’s delivery times. Using the simulation, we test the robustness of the model solution under a wide range of volume possibilities. For a major automotive distribution network, we show the direct interdependence between level of consolidation and lot size choice and suggest a methodology for choosing the combination of consolidation points and lot sizes which reduces overall network transit time while maintaining existing service to all destinations. The result could easily be applied to many distribution networks. The paper is organized as follows. The next section contains a review of relevant literature. The following section describes the specifics of the mixing center distribution problem. The methodology used to assist in decision-making is presented next. Finally, results are given and a conclusion is offered. 2. Mixing center distribution problem In the present form of distribution (see Fig. 1), units are moved from plants to final customers (dealerships) either directly or through a distribution center. Each customer is served by a unique distribution center. Thus, a product may move from plant to customer directly or via the distribution center. The addition of a consolidation point, or mixing center M as it is referred to in the rail industry, would allow inventory to move from plant to mixing center and from there to either the distribution centers or directly to customers in addition to all previous arcs (see Fig. 2). The new topology would thus contribute the following additional arcs to the previous model. In this scenario, the options of routes from the plants to the mixing center as well as the mixing center to all points downstream are added. As there is no requirement to use these additional routes, the research question becomes: will the additional routes lower total inventory and reduce total transit time enough to pay for the capital expense of building the infrastructure. Typically in the literature, lot sizing and consolidation point decisions have been treated independently. Lot sizes at the production level are not covered here. In our problem, production is driven by demand (pull system) and the
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Fig. 1. Present configuration of distribution network.
Fig. 2. Additional arcs created by proposed network configuration.
production is not influenced by distribution. We thus concentrate on lot sizing within finished goods transport. The Freight Transport Problem (FTP) studies the problem of choosing consolidation points in a network. Geoffrion and Graves [1] apply Bender’s Decomposition to determine the location of intermediate distribution facilities. Zangwill [2, 3] and Jordan [4] have studied single-product network flow problems. Multi-product with linear cost functions are discussed in Evans [5] and the more general case of concave costs by Klincewicz [6]. Costs are based on average flows and strategic routes are determined for special cases. Hall [7] and Blumenfeld et al. [8,9] consider the FTP for General Motors for the case of one consolidation point. Again, the study seeks strategic shipping between source and destination pairs rather than routing particular lots. Lot sizing in routing is studied in Tersine [10]. Tersine discusses the problem of determining a shipment strategy based on volume discounts from the producer. Occena and Yokota [11] considers a problem similar to ours in scheduling AGV’s with various lot sizes introduced by a JIT system. The study looks at the trade-off between the lot size and the number of trips necessary. Egbelu [12] examines how lot size effects are related to process sequencing, which has some similar components to our lot size routing problem. 3. Model and solution method We develop an iterative method to heuristically determine lot size and consolidation strategies for each origin and destination in the network. The key difficulty in developing an optimization approach to the consolidation and lot sizing problem is that the benefits gained from consolidation depend on the number of origins contributing to the consolidation point. In other words, because the savings of the consolidation point are volume-driven, the cost of each arc to a consolidation point is a function of the sum of the volumes over all arcs into the consolidation point. Thus, the problem cannot be formulated in a standard network formulation with static arc costs. When trying to minimize the total post-assembly inventory directly, the problem arises that the average waiting time at each possible location is a function of the lot size and route choice over time. Thus, we need to know the average waiting times to solve the problem, but we do not know the average waiting times until we choose the routes and lot sizes. The simulation methodology for estimating holding times is described later in this section. We first describe the strategic model used to determine routes and lot sizes. Without the lot size and delivery constraints, the mixing center distribution problem would be trivial: ship each vehicle separately to the final customer. The solution would never include the mixing center, as there would be no incentive to consolidate. Since the mixing center behaves similarly to a large plant, it would impose a minimum
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consolidation holding time and would thus be avoided. The strategic model is presented with the following notation. Let P be the set of plants, M the mixing center, D the set of distribution centers, and C the set of customers (or dealerships). The holding times at the plants, mixing center, and distribution centers are denoted by h with the appropriate subscript p, M, d, or c. These variables are determined by using a tactical simulation described in a following section. Ipc (IMc , Idc ) is the amount of inventory at plant p (mixing center, destination d) destined for customer c. S is the set of shipment types, and `s refers to the lot size of shipment s ∈ S. The binary decision variables are denoted as xpMs , xpds , xpcs , xMds , xMcs , and xdcs , and indicate the arcs used in the distribution system of shipment between locations p ∈ P, d ∈ D, c ∈ C, and mixing center M of shipment type s. The corresponding integer decision variables ypMs , ypds , ypcs , yMds , yMcs , and ydcs refer to the number of shipments of type s. Intuitively, a shipment type s ∈ S refers to a type of railcar of vehicle capacity `s . Each plant produces a unique type of vehicle. The number of vehicles demanded from customer c from plant p is represented by δpc . This demand occurs regularly over a set time period, typically one week. Thus, if plant 1 produced minivans, and dealership 11 requested five minivans this week, this demand would be represented by δ1,11 = 5. The strategic objective S(xi,j,s yi,j,s ) is to determine which routes from each source facility i to each destination facility j of shipment type s(xi,j,s ) should be used and how many of each shipment type should scheduled (yi,j,s ) over that route in order to minimize average holding time in the distribution system. The optimal solution is constrained by the requirement that no plant should increase its current average holding time in order to improve overall system performance. The model also imposes network flow constraints at each facility at the vehicle level, and constrains shipments to available arcs. In practice, many arcs would be restricted simply due to physical barriers, facility capabilities, or weight restrictions. In our case, the optimal solution was found through enumeration. The strategic model requires estimates of holding times at each location. Determining these holding times through simulation might alter the optimal network and lot size solution. The following iterative approach was used. 1. Simulate the distribution network in its current state (no consolidation) to determine average hold times at each plant. 2. Simulate the distribution network forcing all traffic to consolidate through the mixing center. This is the current best solution. Use the resulting holding time parameters in the strategic model. 3. Eliminate any routes from plants to the mixing center in the current solution which increased the holding time at that plant from the original times generated in step 1. If there were no routes eliminated, stop. 4. Solve the strategic model by enumeration to determine routes and lot sizes with the reduced routing options. Make this the current solution. 5. Simulate the resulting distribution network to determine holding times at each facility. Go to step 3. The decision variables, which routes and lot sizes to choose, can help determine whether it is cost-effective to build a mixing center at all — over a billion dollar capital expenditure. Should the reduction in transit time plus the actual vehicle production time support the manufacturer’s goal of build-to-order and deliver in seven days, then the manufacturer would have to weigh the benefits of their quick response with that of the fixed cost of the facility. We next present the details of the simulation methodology. 3.1. Simulation methodology A simulation was created in order to evaluate the various scenarios and determine which sources would benefit from a mixing center. The simulation had the following characteristics and assumptions: 1. The manufacturer’s automobile production strategy is determined independently of distribution considerations as the product holding time savings of a distribution-oriented production scheme pale in comparison to the high changeover costs that would be required of production lines. Thus, the distribution process begins upon completion of production of a unit of finished goods inventory. Thus, demand is simulated based on historical data regarding production. 2. Only full lots are shipped in all modes of conveyance. Due to the high cost of transport equipment and the rate structure governing product movements, only consolidated bulk movements are made. 3. A minimum handling time at mixing centers M, and distribution centers Di is imposed for consolidation and rebuilding lots, where applicable. 4. Sufficient equipment is available for any mode of transport.
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The simulation required the following inputs provided by a manufacturer: 1. Aggregate plant production volumes. 2. Cumulative Density Function (CDF) of demand for each plant–customer pair, i.e., which customers buy how many of each product. 3. Finished goods physical dimensions — to determine feasible lot sizes. The simulation is used to evaluate the impact of changing the lot size, consolidation and shipment frequency options on non-transient holding time. In particular, the following measures were of interest. 1. Average daily holding time at each location in the distribution network where holding occurs. 2. Product moving under each mode of transport out of each point in the distribution channel. 3. Number of full lots moving each day out of each point in the transportation network. The simulation initiates by generating finished inventory sampled from historical daily demand at each plant. State changes occur periodically at each plant, mixing center, and distribution center with scheduled departures of trains every 24 h. State changes also occur with the arrival of the rail service at each destination. Travel times were fixed based on historical data. Decisions regarding routing and bin packing were made with the heuristic rules described in the following paragraphs. Each plant and mixing center has options as regards which destination to choose to forward inventory. As a rule, we follow a lexicographic preference function for inventory shipments: move inventory as far as possible first, only then in as big a lot as possible. For instance, it is better to go to the customer in a small lot than a distribution center in a medium lot or the mixing center in a large lot. This may be short-sighted in some instances where the tactical decision to ship a larger lot to an intermediate point may accelerate other inventory sufficiently to justify slowing a few, but we found that as a rule this is not the case. Out of the mixing center, a choice had to be made as to which types of units were first into an outbound lot. The physically largest units were chosen to load first as they had the highest holding costs per physical unit and it would have the effect of moving inventory faster. These units generally had the higher marginal profit value as well. The simulation was performed using production data from 21 Midwestern manufacturing plants with annual volumes ranging from 1300 (large trucks) to 81,000 (top selling vehicle) and eight distribution centers at points south of Kansas city and west of the Mississippi river. A second study was performed with similar plants and a proposed mixing center in St. Paul, MN, serving the geographic Northwest of the United States. Daily volumes were sampled from a uniform distribution within two standard deviations from the mean. The automotive manufacturer arbitrarily chose these distribution parameters as they were sensitive regarding the sharing of actual data. We motivate the solution process with an application of the automotive distribution problem. 4. Automotive distribution problem According to Blumenfeld [13], finished goods distribution for General Motors was over $4 billion in 1986, so it is a problem where even a small percentage improvement can result in substantial savings. In the automotive distribution problem, finished automobiles and trucks are moved from the plant (P) to the final customer (C) or dealerships (see Fig. 1). In this network, a limited degree of consolidation is naturally present. The consolidation choices examined in this study are whether to add a consolidation point central to the rail network or leave the rail network unchanged. The estimated cost for construction and operation of a mixing center exceeds one billion dollars. Only full lots are sent to the successive consolidation point, which makes the lot size choice critical. Three competing rail technologies provide the lot size options. The established conventional rail equipment moves autos in the largest lots while on rail. A second option provides the same functionality in smaller, medium sized lots. Finally, the third technology option provides the smallest lot size(s), but provides the capability of traveling on both rail and road. Where small lots are not used, haulaway trucks are required to transport to and from the rail network. The actual lot sizes of the large, medium, small and haulaway modes of transport depend on the dimensions of the vehicles. From one to three vertical levels of vehicles may fit in each form of conveyance, depending on the maximum height of the vehicles in the lot and the deck clearance of the conveyance. Typically, trucks and utility vehicles allow one fewer deck than standard automobiles. Anywhere from 2 to 6 vehicles may fit per level, depending on the average length of the vehicles and the interior length of the conveyance. Thus, depending on the vehicle height and length and the conveyance type, the lot size may be anywhere from 2 to 18 vehicles.
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Fig. 3. Average total holding time at the plants.
Currently, conventional equipment moves between origination and termination rail consolidation points with no central consolidation. Using this configuration as the baseline, alternative lot size and consolidation scenarios are considered for both cost and service. The simulation was performed with and without use of the mixing center. The general relationship is shown in Fig. 3 below. The sloped line indicates the relationship between annual volume (1000’s) and average holding time at a plant if no mixing center is employed whereas the less sloped line indicates the same relationship when usage of the mixing center is forced upon all plants. Plant volumes given below are estimated volumes from a plant that would be shipped along the former Santa Fe route (primarily the Southwestern U.S.) and are not indicative of a plant’s total annual volume. Intuitively, the mixing center functions very similarly to a plant, in that the higher the volume, the less wait vehicles endure before continuing in the network. The graph indicates a fairly strong inverse relationship between volume and holding time. We would expect this to be the case — the higher volume plants experience less holding time as they tend to fill up railcars faster. Unfortunately, the intersection in the graph indicates that the inventory holding time performance of high volume plants is actually hindered by use of consolidation. The above graph indicates that this point occurs at a plant with annual production of approximately 45,000 vehicles. However, this cannot serve as a critical cutoff volume for using the consolidation. We can conclude though that plants above that volume are adversely affected by the consolidation. We cannot conclude that plants below that volume would benefit from consolidation. If high volume plants (currently defined as annual volume >45,000 vehicles per year) are excluded from consolidation, the consequence reduces the volume at the mixing center, which in turn increases the holding time at that point. This in effect shifts upward the less sloped line in the above graph and thus moves the intersection point to the left. In general, more plants now benefit from not using the mixing center. Consider the following twelve-plant example (Table 1). We assume one mixing center for simplicity. Also, assume trains leave the plants every 24 h. In this case, we assume the mixing center adds 6 h to the total holding time in the system. From this, we see that using the mixing center disadvantages plants 8–12. We then exclude them from mixing. This removes 350,000 autos annually from the mixing center, leaving it with 280,000 autos. With this reduced number, we perhaps find that the mixing center adds 9 h (through simulation) to the delay at the plant for plants 1–7. Now, plant 6 suffers a 21.1 h delay and plant 7 suffers a 21.0 h delay. Thus, it is to the advantage to plant 7 not to use the mixing center, but it remains a good option for plant 6. The process is repeated for plants 1–6 and so on. For the sake of the example, we shall conclude that only plants 1–5 benefit by using the mixing center. If this were the case, then only 28% of the vehicles would see improvement using consolidation. However, the numbers represented here are but a fraction of one automobile manufacturer’s distribution needs and such a mixing center could seamlessly handle automobiles from many different manufacturers. We next discuss the business results of the simulation applied to a major automobile manufacturer.
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Table 1 Plant sizes and holding times Plant
Annual volume (000’s)
Expected daily holding time at plant without consolidation
Expected daily holding time at plant with consolidation + 6 h delay at mixing center
1 2 3 4 5 6 7 8 9 10 11 12
25 30 35 40 45 50 55 60 65 70 75 80
36.18 32.50 29.18 26.20 23.58 21.30 19.38 17.80 16.58 15.70 15.18 15.00
18.6 18.5 18.4 18.3 18.2 18.1 18.0 18.9 18.8 18.7 18.6 18.5
4.1. Business-perspective results The business-perspective results are split into two sections: mixing center introduction and lot size variation introduction. Of major business concern in this decision were issues such as fleet appropriation and allocation, train handling issues, inter-railroad coordination, and simple and standardized service. While these and a host of other business-related issues enter into the decision of mixing center and lot size choice, the focus in this section is on inventory holding time impacts. The primary implication of the creation of a mixing center point was that total inventory holding time in the system was reduced and the variance of that time was greatly reduced. The holding time at the plant in most cases was reduced by over 50%; however, because of the introduction of holding time at the mixing center for loading, unloading, and connection time, total system holding time is reduced by just over 10%. As expected, substantial improvement was gained in the lowest volume plants and destinations, which helps to standardize delivery times among locations. Although holding time on the whole for all plants and destinations was reduced by the introduction of a consolidation point, the service time for the top 7% of the highest volume plant–destination pairs was slightly impaired. Imposing the constraint of “do no harm” by allowing these 7% to bypass the mixing center reduced the savings in system holding time to 9%. In order to maintain service to the biggest customers, the system holding time as a whole was increased. A second benefit was identified by examining the load factors (number of autos per rail car) out of the mixing center. As could be expected, the average load factors were not affected in the system. However, because of the mixing of different types of autos, the range of the load factor drops from 3.5 autos per rail car out of each plant to 0.8 out of the mixing center. The reduction in variability of autos per rail car greatly enhances the ability to set regular and achievable service standards at destination locations. A third benefit created by the mixing center is to reduce the variability of the number of rail cars (train size) out of the plants each day. Without the mixing center, there is a strong positive correlation between the volume of autos produced at a plant and the standard deviation of rail cars departing each day. The mixing center reduced train size variability out of the plant by 65% system-wide. The standard shipment size vastly improves railcar requirements forecasting and railcar utilization. However, the reduction in variability out of the plant does not carry to the destination side of the mixing center. Where aggregation of the demand locations reduces the impacts of remainder autos at plants, the disaggregation that takes place at the mixing center regenerates the variability once experienced out of the plant. Interestingly, savings from standardization of train sizes are felt in only half of the distribution network. As expected, reducing lot sizes out of plants reduced holding time at plants without the help of a mixing center. The holding times for smaller volume plants were most benefited: the remainder autos that could not fill out a large lot, but could a small, helped the smaller plants for which this remainder was significant. However, the holding time reductions were not dramatic: a 4% reduction in total system holding time. The major benefit of the small lot – the ability to ship direct to customer without a destination holding time – was offset by both the frequency that this option was viable from the plant due to volume requirements, and by the deterioration of service for those autos that were shipped to the rail destination. The odd implication for the small lots is that it vastly improved the total holding time
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of the highest volume plant–destination pairs, but significantly increased others that suffered from low volumes at rail destinations. A key constraint was to not degrade service to any plant or destination; thus, the small lot size approach on its own is insufficient. The combination of both lot size reduction and mixing center further increases the holding time savings; however, the additional holding time savings of the two strategies combined is less than the sum of the savings of the two strategies used independently. From total network holding time savings of 12%, one can see that each strategy somewhat mitigates the effect of the other. Both approaches reduce holding time at a plant by reducing the time to wait for a full lot; the smaller lot reduces the remainder autos at the plant that the mixing center serves to accelerate. The smaller remainder with small lots translates to smaller additional savings from the mixing center. The combination of mixing center and lot size choice produces many possible combinations of distribution strategies. This approach is a strategic rather than tactical one; judicious use of a combination of the two distribution strategies could net increased savings. For the objective of minimizing system inventory holding time, the strategies in tandem are fruitful; however, the combination of the two approaches yields a net present value less than either one independently. 5. Conclusion The problem of freight transport routing combined with lot sizes is discussed in this paper. We develop an iterative approach for determining strategic routing based on lot sizing issues. We show that the issues cannot be treated separately as they tend to influence each other. Independently, the problems are difficult to solve optimally, so an iterative method using simulation is devised which assists in determining along which routes finished goods should be transported. In this research, the mixing center did indeed perform variance reduction on delivery times, improving the large truck delivery time by several weeks, but slowing the delivery of the most popular models by up to a day. The large cost of the mixing center was also a factor in this application and would likely be less significant in many other tiered inventory systems. As a final note, the researchers recommended against building the mixing center, citing that the total cost savings in this application could not recover the billion dollar capital expense in any reasonable time. However, a second party whose study did not take into account a possible deterioration of performance for high volume plants but rather whose message focused on the value in moving inventory off the plant’s property was successful. This was unfortunately a predictable decision as the onsite inventory issue was pressing, visible, and the incentives and metrics of decision makers were aligned with those inventory goals. The mixing center was built and its performance has been so disruptive that it is now being circumvented. References [1] A. Geoffrion, G. Graves, Multicommodity distribution systems design by bender’s decomposition, Management Science 29 (5) (1974) 822–844. [2] W.I. Zangwill, Minimum concave costs flows in certain networks, Management Science 14 (1968) 429–450. [3] W.I. Zangwill, A backlogging model and a multi-echelon model of a dynamic economic lot size production system — a network approach, Management Science 15 (1969) 506–527. [4] W.C. Jordan, Scale economies on multi-commodity distribution networks, Report GMR-5579, General Motors Research Laboratories, Warren, Michigan, 1986. [5] J.R. Evans, A network decomposition aggregation procedure for a class of multicommodity transportation problems, Networks 13 (1983) 197–205. [6] J. Klincewicz, Solving a freight transport problem using facility location techniques, Operations Research 38 (1) (1990) 99–109. [7] R.W. Hall, Dependence between shipment size and mode in freight transportation, Transportation Science 19 (4) (1985) 436–444. [8] D.E. Blumenfeld, J. Beckmann Martin, Use of continuous space modeling to estimate freight distribution costs, Transportation Research 19A (2) (1985) 173–197. [9] D.E. Blumenfeld, L.D. Burns, C.F. Daganzo, Analyzing trade-offs between transportation, inventory and production costs on freight networks, Transportation Research 19B (5) (1985) 361–380. [10] D. Tersine, Optimal lot sizes for unit and shipping discount situations, IIE Transactions 26 (2) (1994) 97–101. [11] L. Occena, T. Yokota, Analysis of the AGV loading capacity in a JIT environment, Journal of Manufacturing Systems 12 (1) (1993) 24–35. [12] P. Egbelu, Establishment of economic production rate, production batch size, and production sequencing in manufacturing systems with flexible routing, European Journal of Operations Research 67 (3) (1993) 358–372. [13] D.E. Blumenfeld, M.C. Burns, C.F. Daganzo, R.W. Hall, Reducing logistics costs at general motors, Interfaces 17 (1987) 26–47.
An application of interdependent lot size and consolidation point choice Daniel G. Conway a,∗ , Michael F. Gorman b a Department of Management, University of Notre Dame, 348 Mendoza College of Business, Notre Dame, IN 46556, United States b University of Dayton, Department of MIS, Operations Management, and Decision Sciences, School of Business Administration,
300 College Park, Dayton, OH 45469-2130, United States Received 2 October 2004; accepted 9 December 2004
Abstract The efficiency of consolidation points in a distribution system is driven both by the total volume transported and by the lot size in which the product travels. We examine a finished goods distribution system for a major automobile manufacturer that is setting policy concurrently for consolidation points and for the lot size of transport. Through simulation of the distribution system we show the impacts of consolidation on product holding time, volume volatility and customer service. Our results show a significant reduction in total system inventory from consolidation; however, some customers’ service is adversely affected. We then develop a methodology for determining the best combination of consolidation, direct shipment and lot size for all origins and destinations in the distribution system in order to minimize inventory holding time while simplifying operations and maintaining customer service. Preliminary results are shown which demonstrate the potential for substantial reductions in the total finished vehicle holding time in the system. c 2005 Elsevier Ltd. All rights reserved.
Keywords: Consolidation points; Distribution; Lot sizing; Freight networks
1. Introduction In order for automobile manufacturers to compete in emerging e-business markets, it became necessary to better understand the roles played by their distribution systems. In particular, the simple promise of a to-ordervehicle assembled and delivered in seven days has presented some complicated distribution channel problems previously ignored by major automobile manufacturers. This paper presents one automobile manufacturer’s attempt at researching these distribution issues. Effective distribution channel management balances the direct costs of transportation such as fuel, equipment and handling with time cost of the finished good in the distribution pipeline and customer service and market presence considerations. Two choices to be made in setting up distribution channels are the number of consolidation points and the lot size of the transport. In general, increasing consolidation points and decreasing lot size have the same effect ∗ Corresponding author.
E-mail address: [email protected] (D.G. Conway). c 2005 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2004.12.015
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on the distribution channel: increasing the direct costs of product movement while increasing its velocity. Benefits from consolidation can be found in: improvement in the speed at which the product travels through the distribution channel, buffering rates of production and demand, simplifying the building of full lots at origins, and the reduction in variation of shipments out of origins. However, a sizable investment in a consolidation center must be made to realize the benefits, and additional handling of product is required as lots are broken down and rebuilt at consolidation points. Economies of scale are encountered when choosing lot size transport. However, the savings are tempered by inventory holding time, particularly in distribution channels with multiple origins and destinations, as a full lot for shipment can take a considerable time to produce, and shipping partial lots is very rarely done. Customer service considerations and finished good inventory reductions can make smaller lots more attractive at the cost of higher direct variable costs of shipment. Similar to consolidation, reducing lot size can reduce time in transit and variability in shipment levels. While both consolidation and reducing lot size may improve delivery time and variability, the two decisions cannot be made independently as the benefits of each are diminished when used in tandem. Thus, the lot size and consolidation decision must be made jointly when designing a distribution network. In our problem, we assume the candidate consolidation points are known. The network consists of numerous, heterogeneous origins and destinations. Due to the heterogeneity, the optimal lot size and consolidation strategy may be different for each origin–destination pair. For example, consolidation may negatively impact transit times for higher volume destinations that enjoy a relatively short amount of time in the distribution channel. However, the volumes of the large origins and destinations at a consolidation point help create the benefits of consolidation for the low volume destinations; thus, the consolidation strategies for all destinations cannot be made independently. In the face of numerous interdependent lot size and consolidation strategies, we develop a heuristic model for developing a lot size choice and consolidation strategy for all origins and destinations in the network, in order to shorten the delivery cycle and minimize the product inventory holding time while simplifying operations and maintaining customer service. Permitting only one lot size to service each origin and a single consolidation strategy for each origin and destination combination simplifies operations and is common in practice. Restricting variety of lot size for each origin and destination lends the model to a more strategic, rather than tactical, use. Other constraints on lot size choice exist for particular origins and destinations. Customer service requirements are ensured by constraining delivery times not to exceed current standards. High volume destinations are not willing to accept degradation in their service in order to improve overall network transit times. Thus, any consolidation strategy faces the constraint of not increasing any single destination’s delivery times. Using the simulation, we test the robustness of the model solution under a wide range of volume possibilities. For a major automotive distribution network, we show the direct interdependence between level of consolidation and lot size choice and suggest a methodology for choosing the combination of consolidation points and lot sizes which reduces overall network transit time while maintaining existing service to all destinations. The result could easily be applied to many distribution networks. The paper is organized as follows. The next section contains a review of relevant literature. The following section describes the specifics of the mixing center distribution problem. The methodology used to assist in decision-making is presented next. Finally, results are given and a conclusion is offered. 2. Mixing center distribution problem In the present form of distribution (see Fig. 1), units are moved from plants to final customers (dealerships) either directly or through a distribution center. Each customer is served by a unique distribution center. Thus, a product may move from plant to customer directly or via the distribution center. The addition of a consolidation point, or mixing center M as it is referred to in the rail industry, would allow inventory to move from plant to mixing center and from there to either the distribution centers or directly to customers in addition to all previous arcs (see Fig. 2). The new topology would thus contribute the following additional arcs to the previous model. In this scenario, the options of routes from the plants to the mixing center as well as the mixing center to all points downstream are added. As there is no requirement to use these additional routes, the research question becomes: will the additional routes lower total inventory and reduce total transit time enough to pay for the capital expense of building the infrastructure. Typically in the literature, lot sizing and consolidation point decisions have been treated independently. Lot sizes at the production level are not covered here. In our problem, production is driven by demand (pull system) and the
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Fig. 1. Present configuration of distribution network.
Fig. 2. Additional arcs created by proposed network configuration.
production is not influenced by distribution. We thus concentrate on lot sizing within finished goods transport. The Freight Transport Problem (FTP) studies the problem of choosing consolidation points in a network. Geoffrion and Graves [1] apply Bender’s Decomposition to determine the location of intermediate distribution facilities. Zangwill [2, 3] and Jordan [4] have studied single-product network flow problems. Multi-product with linear cost functions are discussed in Evans [5] and the more general case of concave costs by Klincewicz [6]. Costs are based on average flows and strategic routes are determined for special cases. Hall [7] and Blumenfeld et al. [8,9] consider the FTP for General Motors for the case of one consolidation point. Again, the study seeks strategic shipping between source and destination pairs rather than routing particular lots. Lot sizing in routing is studied in Tersine [10]. Tersine discusses the problem of determining a shipment strategy based on volume discounts from the producer. Occena and Yokota [11] considers a problem similar to ours in scheduling AGV’s with various lot sizes introduced by a JIT system. The study looks at the trade-off between the lot size and the number of trips necessary. Egbelu [12] examines how lot size effects are related to process sequencing, which has some similar components to our lot size routing problem. 3. Model and solution method We develop an iterative method to heuristically determine lot size and consolidation strategies for each origin and destination in the network. The key difficulty in developing an optimization approach to the consolidation and lot sizing problem is that the benefits gained from consolidation depend on the number of origins contributing to the consolidation point. In other words, because the savings of the consolidation point are volume-driven, the cost of each arc to a consolidation point is a function of the sum of the volumes over all arcs into the consolidation point. Thus, the problem cannot be formulated in a standard network formulation with static arc costs. When trying to minimize the total post-assembly inventory directly, the problem arises that the average waiting time at each possible location is a function of the lot size and route choice over time. Thus, we need to know the average waiting times to solve the problem, but we do not know the average waiting times until we choose the routes and lot sizes. The simulation methodology for estimating holding times is described later in this section. We first describe the strategic model used to determine routes and lot sizes. Without the lot size and delivery constraints, the mixing center distribution problem would be trivial: ship each vehicle separately to the final customer. The solution would never include the mixing center, as there would be no incentive to consolidate. Since the mixing center behaves similarly to a large plant, it would impose a minimum
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consolidation holding time and would thus be avoided. The strategic model is presented with the following notation. Let P be the set of plants, M the mixing center, D the set of distribution centers, and C the set of customers (or dealerships). The holding times at the plants, mixing center, and distribution centers are denoted by h with the appropriate subscript p, M, d, or c. These variables are determined by using a tactical simulation described in a following section. Ipc (IMc , Idc ) is the amount of inventory at plant p (mixing center, destination d) destined for customer c. S is the set of shipment types, and `s refers to the lot size of shipment s ∈ S. The binary decision variables are denoted as xpMs , xpds , xpcs , xMds , xMcs , and xdcs , and indicate the arcs used in the distribution system of shipment between locations p ∈ P, d ∈ D, c ∈ C, and mixing center M of shipment type s. The corresponding integer decision variables ypMs , ypds , ypcs , yMds , yMcs , and ydcs refer to the number of shipments of type s. Intuitively, a shipment type s ∈ S refers to a type of railcar of vehicle capacity `s . Each plant produces a unique type of vehicle. The number of vehicles demanded from customer c from plant p is represented by δpc . This demand occurs regularly over a set time period, typically one week. Thus, if plant 1 produced minivans, and dealership 11 requested five minivans this week, this demand would be represented by δ1,11 = 5. The strategic objective S(xi,j,s yi,j,s ) is to determine which routes from each source facility i to each destination facility j of shipment type s(xi,j,s ) should be used and how many of each shipment type should scheduled (yi,j,s ) over that route in order to minimize average holding time in the distribution system. The optimal solution is constrained by the requirement that no plant should increase its current average holding time in order to improve overall system performance. The model also imposes network flow constraints at each facility at the vehicle level, and constrains shipments to available arcs. In practice, many arcs would be restricted simply due to physical barriers, facility capabilities, or weight restrictions. In our case, the optimal solution was found through enumeration. The strategic model requires estimates of holding times at each location. Determining these holding times through simulation might alter the optimal network and lot size solution. The following iterative approach was used. 1. Simulate the distribution network in its current state (no consolidation) to determine average hold times at each plant. 2. Simulate the distribution network forcing all traffic to consolidate through the mixing center. This is the current best solution. Use the resulting holding time parameters in the strategic model. 3. Eliminate any routes from plants to the mixing center in the current solution which increased the holding time at that plant from the original times generated in step 1. If there were no routes eliminated, stop. 4. Solve the strategic model by enumeration to determine routes and lot sizes with the reduced routing options. Make this the current solution. 5. Simulate the resulting distribution network to determine holding times at each facility. Go to step 3. The decision variables, which routes and lot sizes to choose, can help determine whether it is cost-effective to build a mixing center at all — over a billion dollar capital expenditure. Should the reduction in transit time plus the actual vehicle production time support the manufacturer’s goal of build-to-order and deliver in seven days, then the manufacturer would have to weigh the benefits of their quick response with that of the fixed cost of the facility. We next present the details of the simulation methodology. 3.1. Simulation methodology A simulation was created in order to evaluate the various scenarios and determine which sources would benefit from a mixing center. The simulation had the following characteristics and assumptions: 1. The manufacturer’s automobile production strategy is determined independently of distribution considerations as the product holding time savings of a distribution-oriented production scheme pale in comparison to the high changeover costs that would be required of production lines. Thus, the distribution process begins upon completion of production of a unit of finished goods inventory. Thus, demand is simulated based on historical data regarding production. 2. Only full lots are shipped in all modes of conveyance. Due to the high cost of transport equipment and the rate structure governing product movements, only consolidated bulk movements are made. 3. A minimum handling time at mixing centers M, and distribution centers Di is imposed for consolidation and rebuilding lots, where applicable. 4. Sufficient equipment is available for any mode of transport.
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The simulation required the following inputs provided by a manufacturer: 1. Aggregate plant production volumes. 2. Cumulative Density Function (CDF) of demand for each plant–customer pair, i.e., which customers buy how many of each product. 3. Finished goods physical dimensions — to determine feasible lot sizes. The simulation is used to evaluate the impact of changing the lot size, consolidation and shipment frequency options on non-transient holding time. In particular, the following measures were of interest. 1. Average daily holding time at each location in the distribution network where holding occurs. 2. Product moving under each mode of transport out of each point in the distribution channel. 3. Number of full lots moving each day out of each point in the transportation network. The simulation initiates by generating finished inventory sampled from historical daily demand at each plant. State changes occur periodically at each plant, mixing center, and distribution center with scheduled departures of trains every 24 h. State changes also occur with the arrival of the rail service at each destination. Travel times were fixed based on historical data. Decisions regarding routing and bin packing were made with the heuristic rules described in the following paragraphs. Each plant and mixing center has options as regards which destination to choose to forward inventory. As a rule, we follow a lexicographic preference function for inventory shipments: move inventory as far as possible first, only then in as big a lot as possible. For instance, it is better to go to the customer in a small lot than a distribution center in a medium lot or the mixing center in a large lot. This may be short-sighted in some instances where the tactical decision to ship a larger lot to an intermediate point may accelerate other inventory sufficiently to justify slowing a few, but we found that as a rule this is not the case. Out of the mixing center, a choice had to be made as to which types of units were first into an outbound lot. The physically largest units were chosen to load first as they had the highest holding costs per physical unit and it would have the effect of moving inventory faster. These units generally had the higher marginal profit value as well. The simulation was performed using production data from 21 Midwestern manufacturing plants with annual volumes ranging from 1300 (large trucks) to 81,000 (top selling vehicle) and eight distribution centers at points south of Kansas city and west of the Mississippi river. A second study was performed with similar plants and a proposed mixing center in St. Paul, MN, serving the geographic Northwest of the United States. Daily volumes were sampled from a uniform distribution within two standard deviations from the mean. The automotive manufacturer arbitrarily chose these distribution parameters as they were sensitive regarding the sharing of actual data. We motivate the solution process with an application of the automotive distribution problem. 4. Automotive distribution problem According to Blumenfeld [13], finished goods distribution for General Motors was over $4 billion in 1986, so it is a problem where even a small percentage improvement can result in substantial savings. In the automotive distribution problem, finished automobiles and trucks are moved from the plant (P) to the final customer (C) or dealerships (see Fig. 1). In this network, a limited degree of consolidation is naturally present. The consolidation choices examined in this study are whether to add a consolidation point central to the rail network or leave the rail network unchanged. The estimated cost for construction and operation of a mixing center exceeds one billion dollars. Only full lots are sent to the successive consolidation point, which makes the lot size choice critical. Three competing rail technologies provide the lot size options. The established conventional rail equipment moves autos in the largest lots while on rail. A second option provides the same functionality in smaller, medium sized lots. Finally, the third technology option provides the smallest lot size(s), but provides the capability of traveling on both rail and road. Where small lots are not used, haulaway trucks are required to transport to and from the rail network. The actual lot sizes of the large, medium, small and haulaway modes of transport depend on the dimensions of the vehicles. From one to three vertical levels of vehicles may fit in each form of conveyance, depending on the maximum height of the vehicles in the lot and the deck clearance of the conveyance. Typically, trucks and utility vehicles allow one fewer deck than standard automobiles. Anywhere from 2 to 6 vehicles may fit per level, depending on the average length of the vehicles and the interior length of the conveyance. Thus, depending on the vehicle height and length and the conveyance type, the lot size may be anywhere from 2 to 18 vehicles.
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Fig. 3. Average total holding time at the plants.
Currently, conventional equipment moves between origination and termination rail consolidation points with no central consolidation. Using this configuration as the baseline, alternative lot size and consolidation scenarios are considered for both cost and service. The simulation was performed with and without use of the mixing center. The general relationship is shown in Fig. 3 below. The sloped line indicates the relationship between annual volume (1000’s) and average holding time at a plant if no mixing center is employed whereas the less sloped line indicates the same relationship when usage of the mixing center is forced upon all plants. Plant volumes given below are estimated volumes from a plant that would be shipped along the former Santa Fe route (primarily the Southwestern U.S.) and are not indicative of a plant’s total annual volume. Intuitively, the mixing center functions very similarly to a plant, in that the higher the volume, the less wait vehicles endure before continuing in the network. The graph indicates a fairly strong inverse relationship between volume and holding time. We would expect this to be the case — the higher volume plants experience less holding time as they tend to fill up railcars faster. Unfortunately, the intersection in the graph indicates that the inventory holding time performance of high volume plants is actually hindered by use of consolidation. The above graph indicates that this point occurs at a plant with annual production of approximately 45,000 vehicles. However, this cannot serve as a critical cutoff volume for using the consolidation. We can conclude though that plants above that volume are adversely affected by the consolidation. We cannot conclude that plants below that volume would benefit from consolidation. If high volume plants (currently defined as annual volume >45,000 vehicles per year) are excluded from consolidation, the consequence reduces the volume at the mixing center, which in turn increases the holding time at that point. This in effect shifts upward the less sloped line in the above graph and thus moves the intersection point to the left. In general, more plants now benefit from not using the mixing center. Consider the following twelve-plant example (Table 1). We assume one mixing center for simplicity. Also, assume trains leave the plants every 24 h. In this case, we assume the mixing center adds 6 h to the total holding time in the system. From this, we see that using the mixing center disadvantages plants 8–12. We then exclude them from mixing. This removes 350,000 autos annually from the mixing center, leaving it with 280,000 autos. With this reduced number, we perhaps find that the mixing center adds 9 h (through simulation) to the delay at the plant for plants 1–7. Now, plant 6 suffers a 21.1 h delay and plant 7 suffers a 21.0 h delay. Thus, it is to the advantage to plant 7 not to use the mixing center, but it remains a good option for plant 6. The process is repeated for plants 1–6 and so on. For the sake of the example, we shall conclude that only plants 1–5 benefit by using the mixing center. If this were the case, then only 28% of the vehicles would see improvement using consolidation. However, the numbers represented here are but a fraction of one automobile manufacturer’s distribution needs and such a mixing center could seamlessly handle automobiles from many different manufacturers. We next discuss the business results of the simulation applied to a major automobile manufacturer.
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Table 1 Plant sizes and holding times Plant
Annual volume (000’s)
Expected daily holding time at plant without consolidation
Expected daily holding time at plant with consolidation + 6 h delay at mixing center
1 2 3 4 5 6 7 8 9 10 11 12
25 30 35 40 45 50 55 60 65 70 75 80
36.18 32.50 29.18 26.20 23.58 21.30 19.38 17.80 16.58 15.70 15.18 15.00
18.6 18.5 18.4 18.3 18.2 18.1 18.0 18.9 18.8 18.7 18.6 18.5
4.1. Business-perspective results The business-perspective results are split into two sections: mixing center introduction and lot size variation introduction. Of major business concern in this decision were issues such as fleet appropriation and allocation, train handling issues, inter-railroad coordination, and simple and standardized service. While these and a host of other business-related issues enter into the decision of mixing center and lot size choice, the focus in this section is on inventory holding time impacts. The primary implication of the creation of a mixing center point was that total inventory holding time in the system was reduced and the variance of that time was greatly reduced. The holding time at the plant in most cases was reduced by over 50%; however, because of the introduction of holding time at the mixing center for loading, unloading, and connection time, total system holding time is reduced by just over 10%. As expected, substantial improvement was gained in the lowest volume plants and destinations, which helps to standardize delivery times among locations. Although holding time on the whole for all plants and destinations was reduced by the introduction of a consolidation point, the service time for the top 7% of the highest volume plant–destination pairs was slightly impaired. Imposing the constraint of “do no harm” by allowing these 7% to bypass the mixing center reduced the savings in system holding time to 9%. In order to maintain service to the biggest customers, the system holding time as a whole was increased. A second benefit was identified by examining the load factors (number of autos per rail car) out of the mixing center. As could be expected, the average load factors were not affected in the system. However, because of the mixing of different types of autos, the range of the load factor drops from 3.5 autos per rail car out of each plant to 0.8 out of the mixing center. The reduction in variability of autos per rail car greatly enhances the ability to set regular and achievable service standards at destination locations. A third benefit created by the mixing center is to reduce the variability of the number of rail cars (train size) out of the plants each day. Without the mixing center, there is a strong positive correlation between the volume of autos produced at a plant and the standard deviation of rail cars departing each day. The mixing center reduced train size variability out of the plant by 65% system-wide. The standard shipment size vastly improves railcar requirements forecasting and railcar utilization. However, the reduction in variability out of the plant does not carry to the destination side of the mixing center. Where aggregation of the demand locations reduces the impacts of remainder autos at plants, the disaggregation that takes place at the mixing center regenerates the variability once experienced out of the plant. Interestingly, savings from standardization of train sizes are felt in only half of the distribution network. As expected, reducing lot sizes out of plants reduced holding time at plants without the help of a mixing center. The holding times for smaller volume plants were most benefited: the remainder autos that could not fill out a large lot, but could a small, helped the smaller plants for which this remainder was significant. However, the holding time reductions were not dramatic: a 4% reduction in total system holding time. The major benefit of the small lot – the ability to ship direct to customer without a destination holding time – was offset by both the frequency that this option was viable from the plant due to volume requirements, and by the deterioration of service for those autos that were shipped to the rail destination. The odd implication for the small lots is that it vastly improved the total holding time
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of the highest volume plant–destination pairs, but significantly increased others that suffered from low volumes at rail destinations. A key constraint was to not degrade service to any plant or destination; thus, the small lot size approach on its own is insufficient. The combination of both lot size reduction and mixing center further increases the holding time savings; however, the additional holding time savings of the two strategies combined is less than the sum of the savings of the two strategies used independently. From total network holding time savings of 12%, one can see that each strategy somewhat mitigates the effect of the other. Both approaches reduce holding time at a plant by reducing the time to wait for a full lot; the smaller lot reduces the remainder autos at the plant that the mixing center serves to accelerate. The smaller remainder with small lots translates to smaller additional savings from the mixing center. The combination of mixing center and lot size choice produces many possible combinations of distribution strategies. This approach is a strategic rather than tactical one; judicious use of a combination of the two distribution strategies could net increased savings. For the objective of minimizing system inventory holding time, the strategies in tandem are fruitful; however, the combination of the two approaches yields a net present value less than either one independently. 5. Conclusion The problem of freight transport routing combined with lot sizes is discussed in this paper. We develop an iterative approach for determining strategic routing based on lot sizing issues. We show that the issues cannot be treated separately as they tend to influence each other. Independently, the problems are difficult to solve optimally, so an iterative method using simulation is devised which assists in determining along which routes finished goods should be transported. In this research, the mixing center did indeed perform variance reduction on delivery times, improving the large truck delivery time by several weeks, but slowing the delivery of the most popular models by up to a day. The large cost of the mixing center was also a factor in this application and would likely be less significant in many other tiered inventory systems. As a final note, the researchers recommended against building the mixing center, citing that the total cost savings in this application could not recover the billion dollar capital expense in any reasonable time. However, a second party whose study did not take into account a possible deterioration of performance for high volume plants but rather whose message focused on the value in moving inventory off the plant’s property was successful. This was unfortunately a predictable decision as the onsite inventory issue was pressing, visible, and the incentives and metrics of decision makers were aligned with those inventory goals. The mixing center was built and its performance has been so disruptive that it is now being circumvented. References [1] A. Geoffrion, G. Graves, Multicommodity distribution systems design by bender’s decomposition, Management Science 29 (5) (1974) 822–844. [2] W.I. Zangwill, Minimum concave costs flows in certain networks, Management Science 14 (1968) 429–450. [3] W.I. Zangwill, A backlogging model and a multi-echelon model of a dynamic economic lot size production system — a network approach, Management Science 15 (1969) 506–527. [4] W.C. Jordan, Scale economies on multi-commodity distribution networks, Report GMR-5579, General Motors Research Laboratories, Warren, Michigan, 1986. [5] J.R. Evans, A network decomposition aggregation procedure for a class of multicommodity transportation problems, Networks 13 (1983) 197–205. [6] J. Klincewicz, Solving a freight transport problem using facility location techniques, Operations Research 38 (1) (1990) 99–109. [7] R.W. Hall, Dependence between shipment size and mode in freight transportation, Transportation Science 19 (4) (1985) 436–444. [8] D.E. Blumenfeld, J. Beckmann Martin, Use of continuous space modeling to estimate freight distribution costs, Transportation Research 19A (2) (1985) 173–197. [9] D.E. Blumenfeld, L.D. Burns, C.F. Daganzo, Analyzing trade-offs between transportation, inventory and production costs on freight networks, Transportation Research 19B (5) (1985) 361–380. [10] D. Tersine, Optimal lot sizes for unit and shipping discount situations, IIE Transactions 26 (2) (1994) 97–101. [11] L. Occena, T. Yokota, Analysis of the AGV loading capacity in a JIT environment, Journal of Manufacturing Systems 12 (1) (1993) 24–35. [12] P. Egbelu, Establishment of economic production rate, production batch size, and production sequencing in manufacturing systems with flexible routing, European Journal of Operations Research 67 (3) (1993) 358–372. [13] D.E. Blumenfeld, M.C. Burns, C.F. Daganzo, R.W. Hall, Reducing logistics costs at general motors, Interfaces 17 (1987) 26–47.