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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

A vehicle routing cost evaluation algorithm for the strategic analysis of radial distribution networks Rachel Bachmann, André Langevin * Ecole Polytechnique de Montréal and CIRRELT, C.P. 6079, Succ. Centre-ville Montréal, Québec, Canada H3C 3A7

a r t i c l e

i n f o

Article history: Received 17 January 2006 Received in revised form 15 November 2007 Accepted 14 February 2008

a b s t r a c t This article presents a new algorithm, based on bin packing algorithms, that determines vehicle routing costs on networks with certain geographic characteristics. Requiring very little data and computation time, the algorithm is used to test the impact of adding ﬂexibility to a retailer’s Canadian distribution operations. It is shown that it is possible to save 5–10% of vehicle routing costs by modifying shipping schedules. Ó 2008 Elsevier Ltd. All rights reserved.

Keywords: Retail distribution Vehicle routing Load building and routing

1. Introduction Reducing distribution costs has become an increasingly important challenge for many companies, especially in retail distribution, where a company’s competitive position can depend on its ability to do so effectively and where each percent of savings can represent millions of dollars. In retail, margins are low, volumes are high and geographic coverage is large. Regardless of whether stores are franchises, independently owned or owned by the parent company, a certain level of customer service is crucial to the good operation of the company as a whole. Therefore retailing companies face the challenge of ﬁnding a balance between distribution costs and customer service. This study stems from the wish of a retail company to determine the ﬁnancial impact of adding ﬂexibility to the shipping schedule of its Canadian distribution operations. More speciﬁcally, the company wanted to evaluate the cost savings attainable by shifting from a situation where stores always receive their deliveries on the same day of the week, to a much more ﬂexible delivery schedule. Some of the questions that will be answered in this study are Would the company achieve savings by being able to split loads over several days? Which loads should be split, while trying to minimize disruptions to the delivery schedule? Should certain stores be shipped separately (full load only)? Using the current software package to produce this cost analysis was not possible due to the large amount of data manipulation required and the unavailability of resources trained in the use of the software. As a result, a vehicle routing cost evaluation technique was developed based on bin packing algorithms. This new technique uses very little data and is applicable to a hub-and-spoke network. Using 2 weeks of real data, when compared to the actual routes built for the same period, the

* Corresponding author. E-mail address: [email protected] (A. Langevin). 1366-5545/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2008.02.009

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algorithm produced results within 1.1% of the actual routing solution, an excellent result considering the combinatorial nature of the problem. The new algorithm was then used to compare a variety of shipping scenarios and the company was presented with the results of the study, enabling it to implement changes to its scheduling strategy. This article is organized in the following way. Firstly, a description of the load building and routing problem is presented, along with the current shipping practices and customer service requirements. In the next section, related works on vehicle routing, continuous approximation and bin packing are presented, as well as an explanation of the decision to use bin packing algorithms to solve the vehicle routing problem. Section 4 details the cost evaluation algorithm, followed by a section on the validation of the algorithm. Finally, the results of the scenarios are presented in Section 6, followed by a conclusion including possible future directions of work in this domain.

2. Problem description The company has a network of more than 400 stores spread across Canada. Each store is assigned to a geographic region and each region is assigned a day of the week for load building and shipping from the distribution center (DC). All stores in a region are shipped on the same day of the week, and this day does not vary from one week to the next. All deliveries are made directly from the DC. Regions closer to the DC are serviced by road while further regions are serviced by a combination of rail and road. The day of the week that the loads can be built for a store at the DC and shipped is static and is known as its ‘‘mandatory” day. The load is always sent on the same day that it is ﬁlled, generally arriving at the store n days later, n being the lead time for delivery. In addition to having a ‘‘mandatory” day, larger stores may also be sent a second delivery on their ‘‘optional” day, another day of the week that is pre-negotiated with the store. The objective of the study was to determine the cost savings attainable if ﬂexibility into the load building and routing schedule was introduced, by making some or all of the stores’ ‘‘mandatory” and ‘‘optional” days dynamic. For the stores chosen to be on a dynamic schedule, the orders can be shipped on any day of the week and can be grouped with any other orders to the same region. The company would prefer for the new schedule to respect two principal customer service commitments which are that (1) the receiving store be informed of a delivery at least 7 days in advance and that (2) all items be shipped no later than 7 days after they are ordered. Consequently, the store has a maximum wait of 7 days plus the lead time for delivery, for any given order. This also means that the DC must know in advance which day an order will be shipped for the stores having a lead time of less than 7 days, which is true for all but the most isolated stores. Although orders can be split to be shipped over several days, customer service level commitments must be respected for all items in the order. The volume that an order takes up on the vehicle depends on how the vehicle is loaded. More precisely, the capacity of the vehicle decreases each time another store’s order is loaded onto the vehicle. For each additional store loaded, the capacity of the vehicle decreases by approximately 100 cubic feet overall. This is due to the space needed to separate the different stores’ orders and the reduction of combinations for loading the vehicle. Generally, load building and routing software applications deal with this by making the assumption that the capacity of the vehicle is actually slightly less than the actual volume. Then planners and warehouse staff deal with any exceptions manually by adding or removing volume to make the schedule feasible with respect to real volumes. In some cases, as well as adjusting for volume, a planner can also improve the routes manually, since the software does not necessarily produce the optimal solution. Any transportation cost analysis presented to the company will be compared to the planner’s solution after this ﬁne-tuning is done, and not to the initial plan produced by the software. If the orders for a store exceed the capacity of one vehicle, full loads are ﬁrst built for these stores and the software assigns any remaining volume to another vehicle, possibly combined with other stores, in order to minimize cost. The delivery dates of these full loads for the individual stores are scheduled separately, usually in such a way that the receipt of the load does not conﬂict with other scheduled deliveries. The vehicle routing problem faced by the company can be deﬁned as follows. Each vehicle has the same ﬁxed capacity. Each order is a ﬁxed portion of vehicle capacity and consists of all the products ordered by an individual store. Orders can be placed up to the night before the load is to be built in the distribution center. Therefore, demand is known when the loads are built and does not ﬂuctuate with time during the load building process. Given the orders to be loaded on a speciﬁc day, the objective is to determine simultaneously how many vehicles are needed to make all the deliveries and which deliveries each vehicle will make, in order to minimize total delivery costs. This is in fact the classic vehicle routing problem (VRP), with the objective of minimizing total cost, subject to the following conditions: each vehicle must begin and end at the depot, every store is assigned to one (only) vehicle, and the vehicle’s capacity can not be exceeded. Several possibilities exist for solving this VRP, including using the commercial software package which uses a heuristic, the exact method based on integer linear programming, or alternative approximation techniques. It is important to remember that the method of resolution chosen will be used to test a variety of scenarios and compare the results. Although the commercial software package is used for the daily load building and routing, it was not possible to use it to compare different shipping scenarios because of data manipulation involved. It is also important to note that although the goal is to solve the load building and routing problem in each scenario, we are not looking for an algorithm that will improve on the solution

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given by the commercial scheduling software nor give the same routes, but rather approximate, after ﬁne-tuning, the cost of that solution, as closely as possible. This is due to the fact that the company wished to continue using the scheduling software which is well adapted to their daily load building and routing practices. By taking a closer look at the cost structure and geographic characteristics, an alternate way of solving the problem can be explored. 2.1. Cost structure One particularity of Canadian geography is that ‘‘customers often lie along radial corridors corresponding to major thoroughfares” (Fisher and Jaikumar, 1981) that service major cities, resulting in a geographical layout referred to as ‘‘hub-andspoke” (Fig. 1). Often, there are no major thoroughfares connecting these radial corridors that are suitable for semi-trailer use. This forces the drivers to return to the major city before doing a second delivery along another corridor, therefore making a detour of up to several hundred miles. Consequently, placing the orders for two stores on different radial corridors in the same vehicle can increase costs dramatically, usually to the point where it is beneﬁcial to send two loads, regardless of the unused volume in these vehicles. In regions serviced by rail and then road, the company sending the load is charged a ﬁxed rate corresponding to the city furthest from the railway station and a stop cost for every stop along the way. The stops must be along the way on the same spoke, or in the central city (hub), since a detour can increase cost signiﬁcantly as explained in the previous paragraph. The cost of the trip can be calculated as shown in the following example:

+ =

Cost to send a load to the furthest city Cost to make 2 stops on the way Total cost of the trip

$100 2

$2000 $200 $2200

In this cost structure the cost to send a load from the DC to each store is known. This data is readily available from the company and is based on what third-party transport companies have charged in the past. These costs are not necessarily linear with respect to distance traveled since they can be inﬂuenced by other factors such as backhaul opportunities and road conditions. The cost of sending a load from the DC to a city situated on one radial corridor (for example, A in Fig. 2) and then to another on a different radial corridor (B in Fig. 2) is not known, since this is rarely done in practice. In order to get a cost estimate for such a trip, a request for quote needs to be sent to a transport company. Consequently, obtaining point-to-point costs for all combinations of stores in a region requires getting hundreds of quotes. Doing this was not considered to be a worthwhile use of the resources dedicated to this study, since it is already known that shipping to different radial corridors is not cost effective.

z

Radial corridor (spoke)

Minor city or town with at least 1 store

Hub of 2-10 stores Fig. 1. Graphical representation of a region.

b

B

A Fig. 2. Stores on different radial corridors.

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This lack of point-to-point costs rules out the possibility of using either integer linear programming or VRP heuristics to solve the problem. Moreover, these methods also cannot deal with the diminishing capacity of the vehicles as a function of the number or stores loaded.

3. Related literature In order to test different shipping scenarios, it is therefore necessary to ﬁnd or develop an algorithm that evaluates vehicle routing costs on networks demonstrating a hub-and-spoke pattern, all while taking into account the decreasing capacity of the vehicles as stores are added as well as the customer service requirements. Two groups of algorithms commonly used for solving load building and routing problems are vehicle routing heuristics and continuous approximation approaches. Classic vehicle routing heuristic algorithms, such as Clarke and Wright (1964) savings algorithm, were ruled out because of the lack of point-to-point costs as previously explained. In some cases, continuous approximation methods as detailed by Daganzo (1995) in his book Logistics Systems Analysis, can be used to solve load building and routing problems. However, this method requires that the cost between two stores be a function of their location in two-dimensional space, which is not the case in a hub-and-spoke network. Hall (1989) combines bin packing and vehicle routing to solve a distribution problem. He notes that the cost to send a group of shipments from one origin to multiple destinations depends on two factors: (i) the number of loads built; and (ii) the average length of the routes (distance). With the use of continuous approximation techniques (see Langevin et al., 1996 for a review of continuous approximation techniques), Hall constructs an algorithm called ‘‘best-ﬁt-save-V” that divides the total region into zones and builds the loads for several destinations simultaneously. He concludes that the best balance between the number of loads and the average length of the routes is a function of the square root of the size of the regions and of the average distance between the origin and the destinations, where destinations are distributed uniformly within the region. In Hall’s algorithm, any store can be loaded with any other store in the same region. Because of this, the application of this algorithm to this study was not feasible given the cost structure described earlier. If the region size was reduced until any store can be loaded with any other in the region, there would no longer exist enough combinations to ﬁll the vehicles, given that a store’s orders cannot be divided between several vehicles. Also, the fact that central stores can be assigned to any vehicle at little extra cost, but that the same is not true for outer lying stores, would not be taken into account. Despite not being able to apply Hall (1989) work directly to this study, a bin packing based solution may still be applicable since, as Hall (1993) notes ‘‘if splitting shipments among routes is not allowed, then solving the VRP depends in parts on the solutions to a bin packing problem”. This is especially true when the cost of sending an additional vehicle increases the cost of the solution signiﬁcantly, as is the case in regions serviced by rail, where the cost of getting the load to the central city represents approximately 75% of the total cost of the trip. If the company reduces unused volume in vehicles traveling to these regions, and therefore the number of vehicles used, without signiﬁcantly increasing vehicle routing costs, overall transportation costs would be greatly impacted. The classic bin packing problem (BPP) can be deﬁned as follows: Given a bin capacity C > 0 and a list of objects {p1, p2, . . ., pn}, what is the smallest number of bins needed to accommodate all of the objects? (Each pi has size si, such that 0 6 si 6 C, i.e. none of the objects is too big to ﬁt into a bin.) The bin packing problem is NP-difﬁcult (Karp 1972). Because of this complexity, most researchers have focused on ﬁnding effective heuristics. Some of the best known bin packing heuristics are next-ﬁt (NF), ﬁrst-ﬁt (FF), best-ﬁt (BF), ﬁrst-ﬁtdecreasing (FFD), best-ﬁt-decreasing (BFD) and worst ﬁt (WF). The statistical evaluation of the ﬁrst ﬁve can be found in Ong et al. (1984). Worst-case performance of all six is documented in Coffman et al. (1984) and a survey of approximate algorithms is presented by the same authors in Coffman et al. (1997). The exact solution of the bin packing problem can be consulted in Martello and Toth (1990). When the products loaded are relatively small compared to the size of the ‘‘bin”, the three-dimensional bin packing problem becomes a one-dimensional one, with volume in cubic feet being the unit of measure. Consequently, it is only necessary to research existing one-dimensional BP heuristics. Two types of BP sub-problems are potentially applicable to the problem presented in this article: bin packing with priority relationships and bin packing with conﬂicts. Several authors have concentrated on solving bin packing problems with priority relationships in which a partial order is necessary. This partial order is deﬁned by another list of objects or tasks, with certain priorities having been predeﬁned. For example, the relationship si sj means that the task si has to be started before sj and, if assigned to two different bins, si has to be assigned to a bin with an index smaller than the bin to which sj is assigned. Bin packing with priority relationships has been applied in at least two different cases. Garey et al. (1976) include this concept in a multiprocessor task ordering problem and Wee and Magazine (1982) in an assembly line balancing problem. In the ﬁrst case, the tasks si and sj having a priority relationship had to be assigned to different bins, while respecting the

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fact that si is in a bin with a smaller index than sj. In the second case, the same is true except that si and sj can be assigned to the same bin. A comparison of the results of both can be found in the second article. Another bin packing sub-problem is bin packing with conﬂicts (BPC). The problem is to assign all items to a minimum number of bins, while respecting the capacity of each bin and while ensuring that no two conﬂicting items are placed in the same bin. The applications of the BPC include examination scheduling, machine process assignment, load balancing in parallel computing, as well as delivery problems involving ﬂammable, explosive or toxic substances. The BPC is strongly NP-difﬁcult since it generalizes the BPP where all items are mutually non-conﬂicting. Gendreau et al. (2004) propose several heuristics to solve the BPC including a heuristic based on ﬁrst-ﬁt decreasing in which the items are arranged in decreasing order and placed in the ﬁrst bin with enough capacity and not creating a conﬂict. Several other heuristics based on graph coloring concepts are also presented in the article. In order for an algorithm to be employed to evaluate transportation costs in a hub-and-spoke network with the given cost structure, it would need to integrate the two following concepts: (1) Stores lying on different radial corridors are forbidden from being assigned to the same vehicle, and (2) A store cannot be assigned to a route until all stores lying farther along the same radial corridor are already assigned to routes, therefore minimizing the number or vehicles going to the outer lying stores. The resulting algorithm would therefore need to be a combination of bin packing with priority relationships and bin packing with conﬂicts. In the next section, a new bin packing algorithm, called radial bin packing, is presented combining the two afore-mentioned sub-problems, with the objective of minimizing the total number of loads. To our knowledge this is the ﬁrst time that these two problems have been combined. In this study, the regions serviced only by road did not clearly demonstrate the hub-and-spoke pattern, mostly because they are situated closer to the DC, which in turn is situated in a densely populated area. In these regions, the problem resembles a vehicle routing problem more than a bin packing problem. Consequently, the algorithm was initially tested on railserviced regions. The application of the algorithm to regions serviced only by road is discussed later in this article. 4. The algorithm For a region composed of 25 stores, only 26 data points are needed: the cost of delivering a load from the DC to each store and the cost per stop in that region, which is the same for all stores. Therefore, the data collection step is reduced to a minimum. Each radial corridor is deﬁned as a separate ‘‘subregion” (Fig. 3) in such a way that the orders from two stores from different subregions cannot be placed in the same vehicle. At the same time, the orders destined for the central major city can be placed in a vehicle destined for any of the subregions of that city or for the city itself at little additional cost since the load must initially be sent by rail to the central city. The task of grouping stores into subregions is done manually and in most cases, can be done visually, as shown in Fig. 4. In the algorithm, stores should always be on the way to the furthest store in any given subregion. However, this rule can be modiﬁed if the distance traveled between two stores not on the same spoke is negligible when compared to the cost of the overall transportation solution, as can be seen for subregion A in Fig. 3. If the two outer stores are assigned to the same route in this example, the vehicle will return to the city at the fork in order to make the second delivery. If the distance is too great, separate subregions must be created, as can be seen with subregions B and C in Fig. 4. Sometimes the decision to assign a store to one subregion or another, or the decision to create a larger subregion or several smaller ones can be quite difﬁcult. In these cases, the grouping into subregions is done with the help of the planner, based on the most frequent routes in the past. Several different well-known BP algorithms (best-ﬁt, best-ﬁt-decreasing, worst-ﬁt, etc.) were tried as well as combinations of these. Using 2 weeks of data from one region, the one that gave the results closest to the real solution provided

b

Subregion Fig. 3. Graphical representation of a region and a sample subregion.

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b

B

C A Fig. 4. Graphical representation of a region and its subregions.

by the planner was a combination of best-ﬁt (BF) applied to each subregion once the stores had been sorted from furthest to closest, followed by best-ﬁt-decreasing (BFD) for the stores situated in the central city, this time allowing the stores to be added to the bins from any subregion. Both heuristics were modiﬁed to take into account the decreasing capacity of the vehicles as a function of the number of stores loaded into the vehicle. The algorithm, named radial bin packing (RBP) is as follows: A For each subregion 1. Sort the stores i in decreasing distance from the central city. 2. Apply the best-ﬁt algorithm: Take the load si and place it in the bin in which it ﬁts where the residual volume is the least. If no such bin exists, the load si is placed in a new bin. The residual volume of the bin in which si is placed is recalculated to take into account the diminishing capacity. Next i. B For the central city 1. Sort the stores i in decreasing order with respect to volume. 2. Apply the best-ﬁt-decreasing algorithm, which is identical to the best-ﬁt algorithm except that the loads are being processed from largest to smallest, to the bins already packed in step A, adding a new bin when deemed necessary by the algorithm. If a store has more than a full load, all full loads are ﬁrst taken out of the problem, as is the current practice. The heuristic algorithm was programmed in Microsoft Access in the Visual Basic for Applications (VBA) programming language because the interaction with a database was found to be useful. 5. Validation of the algorithm The algorithm was validated by comparing the results with those of the software, after ﬁne-tuning by the planner, over a 2-week period in a sample rail-serviced region. The results are expressed as a percentage of savings over the planner’s solution:

Result ¼ ð1 C i =C P Þ 100%

ð1Þ

where Ci is the cost of our solution and CP is the cost of the planner’s solution. Positive percentage savings indicate routing costs inferior to the planner’s solution while negative percentage savings indicate superior routing costs. For a region containing approximately 25 stores and receiving approximately 10 loads per week, the results are: Week 1 Week 2

1.1% 0.3%

The result is in fact marginally better than the solution found by the planner, this being possible because the software used initially to construct the routes also uses heuristics. However, since the objective was to approximate the planner’s schedule, the results are excellent. 6. Development and evaluation of the scenarios Three regions were selected to be used in the next step of evaluating different scenarios, each one having different geographical and transportation characteristics. Regions 1 and 2 are both serviced by rail and clearly demonstrate hub-and-

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spoke patterns with a major city situated in the center. However, the vehicle size is smaller for Region 2. For both regions the number of stores is between 12 and 30 and the stop cost represents less than 5% of the cost of the trip. In some cases, the scenarios were also tested on an enlarged version of Region 1, the goal being to study the impact of modifying the size of the region on the cost of the solution. This enlarged version included two extras stores situated further from the central city than the others. For both regions, the routes planned in the past gave an average number of stops per vehicle greater than one and less than three. The data used were the real shipping volumes for each store for the previous year. In each of the seven scenarios created and tested, all or some of the stores of the region are moved from a static delivery schedule (same day of the week) to a dynamic delivery schedule for peak periods, off-peak periods or both. Scenario A, is the status quo, meaning that all orders are shipped to the region on the same day of the week, and the routes vary from 1 week to the next. The orders cannot be separated into several loads, unless the capacity of one vehicle is exceeded. In this case, the full load is taken out of the problem and the cost is added back in at the end. The stores are informed of a delivery at least 7 days in advance because the delivery date is simply a function of the lead time for delivery and a store is always loaded on the same day of the week. Scenario B focuses on reducing the empty space on vehicles destined to outer lying stores by sending a load to each subregion every time the orders of that subregion reach the capacity of one vehicle, assuming that the orders are placed evenly over the week. This means that each vehicle will visit every store in the subregion. The remaining volume for the stores in the central region and the subregions not having enough volume to justify an independent load per week is scheduled using the RBP algorithm once a week. This scenario does not respect the commitment of being able to inform the store of the delivery date at least 7 days in advance, since it is not known in advance which day the orders will reach the capacity of one vehicle. In this scenario, if the weekly orders for a store outside the central region exceed the capacity of one vehicle, the volume is not scheduled separately but used as described above. It was expected that there would be signiﬁcant cost savings obtainable by decreasing unused space, and therefore, the number of vehicles going to outer lying regions. In scenario C, orders from the larger central city stores are used to ﬁll unused volume in vehicles going to subregions. In order to do so, subregions and the smaller central city stores are assigned to any one of three days of the week, with approximately equal volumes being shipped each day, and the routes being built using RBP. The next step is to add the dynamic central city stores’ orders to the routes already planned with RBP. Because of the previously mentioned customer service requirements, all orders become mandatory on the 7th day after they were placed. In the 6 days prior, they can be loaded as ‘‘optional” volume. It is assumed that orders are placed evenly over the week so that 1/7th of weekly volume is ordered each day. The central region store with the largest amount of ‘‘mandatory” volume is chosen and only this mandatory volume is loaded into the vehicle with the most space. This corresponds to the worst ﬁt decreasing bin packing algorithm. However, the algorithm was modiﬁed to give priority to vehicles that do not already contain orders for one of the larger central region stores, in order to facilitate the loading of ‘‘optional” volume during the next step. If it is not possible to assign a mandatory order to any of the partially full vehicles, a new vehicle is started. Empty space in the vehicles is then ﬁlled with the optional volume from the stores already loaded in the same vehicle, until full, giving priority again to those with the largest amount of ‘‘mandatory” volume in the previous step. And ﬁnally, the orders from stores that had only ‘‘optional” orders are loaded, these stores not being already present in a vehicle. Again, the worst ﬁt decreasing algorithm is used. It was considered that a store having only optional volume must have at least 300 cubic feet of volume to be added to a vehicle in order to justify the handling costs. The number of central dynamic stores is between 3 and 5 in this scenario, depending on the size of the region. Scenario C was expected to give good results because of the increase in routing combinations when central stores become more ﬂexible in the reception of their deliveries. In general, vehicles leaving the DC should be closer to capacity, and therefore a decrease in the total number of loads should be observed. Scenario D is the same as scenario C except that all subregions and non-dynamic central city stores are all scheduled on the same day. This means that the dynamic central city stores can only be added to loads destined for a subregion once a week. Any remaining volume for the central city is planned separately on one or two other days of the week, all while respecting the 7-day maximum wait for an order to be shipped. It was expected that scenario D would give better results than the status quo, but poorer results than scenario C because fewer routing combinations exist when subregions are all shipped on the same day of the week. Scenario E is a combination of scenarios B and C. During off-peak periods, scenario C is used and during the peak periods, scenario B is used. The change between scenarios is made when the volume of the orders for most subregions exceeds one load. The objective of scenario E was to test if loading the subregions separately can be a source of savings if only done during peak periods, since this practice may not be cost effective during off-peak periods. If this is true, the results should be better than both scenario B and C, since the most cost effective scenario would be applied to each period. Scenario F is also a combination of scenarios B and C, but unlike scenario E, the central stores remain dynamic during the entire 11 weeks. This means that during the peaks, the loads being shipped to the individual subregions are built separately, as in scenario B, but the remaining stores, meaning subregions not having enough volume to justify an independent load and non-dynamic central city stores, are scheduled on different days, and volume from the dynamic central city stores is used to ﬁll the empty space. During off-peak periods scenario C is used. Scenario F should give superior savings to scenario E because of the increased routing combinations stemming from the dynamic central stores during peak periods.

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Scenario G is the same as scenario C, but with all the stores dynamic in the central region. The objective of this scenario is to judge the impact of the number of dynamic stores on the cost of the solution. It was therefore expected that G would give better results than C. Table 1 shows a comparison of the seven scenarios. 6.1. Results Cost savings of each scenario are expressed relative to scenario A, the status quo, and calculated as follows: savings = 1 (Ci/CA) where Ci is the cost of the scenario i, and CA is the cost of the scenario A. The comparisons were made on 11 weeks of data and are presented in Table 2. Positive% savings indicate routing costs lower than the status quo while negative% savings indicate higher routing costs. The comparative results are graphed in Fig. 5. As expected for scenario B, the overall number of vehicles used did decrease. However, the negative or negligible savings show that the reduction in number of vehicles is cancelled out by increased stop costs, despite these being less than 5% of the trip cost. Also, this solution does not respect the constraint of being able to inform the store of the delivery date at least 7 days in advance. This solution is not acceptable to the company because the smaller outer lying stores do not have enough ﬂexibility in their receiving staff schedules to allow their delivery day to vary from 1 week to the next. It was noted that signiﬁcant savings are possible using scenario C, as expected, due to increased ﬂexibility. The results in Region 2 are particularly good and show that it is possible to attain a cost savings of 10% with relatively few dynamic stores. These savings are obtainable in practice, since the routes were actually built for the comparison. The stops increased by 10– 20% but the greater stop costs were counterbalanced by a decrease in the total number of loads shipped. Lower savings were expected with scenario D than with scenario C because the combinations for loading the dynamic stores are fewer and this was found to be true for Region 1. For the smaller Region 2, the two scenarios gave similar results because the central dynamic stores were almost always assigned to vehicles the same day. Although scenario C generally gives better results, D can be used if it is necessary to simplify either the scheduling process or internal communication, since all stores in a region, with the exception of a few large stores, are scheduled on the same day. The objective of scenario E was to see if different scenarios should be used during peak and off-peak periods. If the negative results produced with scenario B are due to performance during off-peak periods only, this scenario should give better results than scenario C. However, the results are better than scenario B but worse than scenario C. In fact the same problems as in scenario B arise, where the increased stop costs eliminate the savings due to a decreased number of loads. And by shipping subregions separately, the possibilities of combining with central stores are eliminated. These results indicate that scenario C is always a better option than B even during peak periods, when demand to subregions is high. Also, as with scenario B, this solution does not respect the constraint of being able to inform the store of the delivery date at least 7 days in advance during peak periods. It was concluded that being able to combine subregion loads with central region loads gives better results. The results of scenario F are very similar to those of scenario C. This means that by adding enough ﬂexibility, the practice of shipping full loads to the subregions separately can become cost effective. However, if the company has to choose between scenarios F and C, they will choose C because of their customer service commitments. The additional dynamic central stores of scenario G were expected to increase savings because of the increase in the possible routing combinations. This was found to be true except for Region 2. In this case, there was only one non-dynamic store in scenario C that became dynamic. The method being heuristic, the subsequent loads built were actually more costly.

Table 1 Scenario comparison Scenario and regiona

A B C D E F G a

Subregions Central Subregions Central Subregions Central Subregions Central Subregions Central Subregions Central Subregions Central

Off-peak

Peak

Delivery day

Routing

Delivery day

Routing

Static Static Dynamic Static Static – different days Dynamic Static – same day Dynamic Static Dynamic Static Dynamic Static Dynamic

Dynamic Dynamic Static Dynamic Dynamic Dynamic Dynamic Dynamic Dynamic Dynamic Dynamic Dynamic Dynamic Dynamic

Static Static Dynamic Static Static – different days Dynamic Static – same day Dynamic Dynamic Static Dynamic Dynamic Static Dynamic

Dynamic Dynamic Static Dynamic Dynamic Dynamic Dynamic Dynamic Static Dynamic Static Dynamic Dynamic Dynamic

Indicated in the table are the delivery parameters for the majority of the stores, not necessarily all the stores, in either the subregions or the central region. For more details, refer to the individual scenarios in the text.

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Table 2 Cost comparison Region

Region 1 (%)

Region 1 enlarged (%)

Region 2 (%)

B Savings C Savings D Savings E Savings F Savings G Savings

0.05 4.47 3.18 1.28 4.66 4.65

1.18 6.47 5.34 0.83 n/a 6.60

1.22 11.17 11.22 6.77 10.98 9.84

Comparison of scenarios

0.35 0.3

1 1 Enlarged 2

Savings (%)

0.25 0.2 0.15 0.1 0.05 0

B

C

D

E

F

G

Upper bounds

-0.05

Scenarios Fig. 5. Graph showing the savings corresponding to the different scenarios.

However the results remain excellent. Increased ﬂexibility can lower costs, but the results will depend on the efﬁciency of the algorithms used. As expected, the effect of having two extra stores in the enlarged version of Region 1 nearly always gives increased savings opportunities (the only exception being scenario B). The more stores in the region, the more combinations there are in the loads built, especially when there are dynamic stores. 6.2. Upper bounds on savings The obvious upper bound on savings can be computed as

savings ¼

X V j j

C

Q j:

ð2Þ

where Vj is the total volume to be shipped to store j for a given period, C is the capacity of a vehicle and Qj is the cost of sending a load to the store j. It corresponds to the cost of sending a load to each store every time the orders reach the capacity of one vehicle, therefore ignoring all customer service level commitments. Table 3 presents the results. 6.3. Results – other regions As previously mentioned, in densely populated areas located closer to the DC and serviced only by road, the problem resembles a classic vehicle routing problem more than a bin packing problem. This is due partly to the absence of a huband-spoke pattern and partly to the different cost structure which is linear with respect to distance covered, with a ﬁxed additional stop cost for all stops made along the way. It was also felt that the potential for savings was lower in these regions because of the already existing routing combinations. Despite this, a third road-serviced and densely populated region was tested with the algorithm.

Table 3 Results of upper bounds Region

Region 1 (%)

Region 1 enlarged (%)

Region 2 (%)

Region 3 (%)

Savings

29.85

32.24

26.29

25.84

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R. Bachmann, A. Langevin / Transportation Research Part E 45 (2009) 50–60 Table 4 Results for region 3 Region

Region 3 (%)

B Savings C Savings D Savings E Savings F Savings G Savings

1.07 0.65 0.45 3.03 1.47 0.57

To be used, the region’s linear cost structure must ﬁrst be converted to a structure similar to the railway system. In order to do this, the ﬁxed cost to the ﬁnal destination is simply the distance to the destination multiplied by the cost per distance unit. Considering that the other stores that can be assigned to the same vehicle should be on the way, the cost of sending a load by road can be calculated as shown in the following example:

+ =

Cost to send a load to the furthest city: Cost to make 2 stops on the way: Total cost of the trip

500 km $4/ km 100$ 2

$2000 $200 $2200

The capacity of the vehicles is smaller than that of the vehicles sent to Region 1 but larger than those sent to Region 2. The stop cost represents approximately 10% of the cost of the trip compared to 5% in rail-serviced regions. For regions serviced uniquely by road, none of the scenarios tested represent signiﬁcant opportunities for decreasing transportation cost as demonstrated in Table 4. In every case, this is due to increased stop costs. Because of the characteristics of these regions, the difference between the optimal solution and the existing solution, and therefore opportunities for savings, is less than in regions serviced by rail, given the possible combinations that already exist. 7. Conclusions and applications The contribution to literature is a new algorithm, based on bin packing, which can be used as a tool with very little data to rapidly evaluate transportation costs in hub-and-spoke networks. This algorithm has the advantage of always giving a feasible solution and takes into account the decreasing capacity of trailers if necessary. When compared to the actual routes built over a 2-week period, the algorithm produced results within 1.1% of the actual transportation solution. The algorithm presented can be used when the following three conditions are true: 1. The geographic layout corresponds to a hub-and-spoke network, with possible division into subregions. 2. The cost structure can be modeled as a ﬁxed cost to the ﬁnal destination plus a stop cost for all stops along the way. 3. Speed of results is more important than precision of results, in order to evaluate the impact of certain changes to the network and to be able to compare several scenarios against each other. The algorithm presented in this article was used to compare seven different scenarios for scheduling with different stores being on static and dynamic schedules at different times. The objective was to determine the impact of these changes to scheduling on the overall transportation costs. The recommendations made to the company were to implement as many dynamic central region stores as possible in regions serviced by rail. Because of the high weekly volume, these stores were likely to accept the receipt of loads on a dynamic schedule, by spreading their orders over several days of the week. In some cases, the volume to these stores was already justifying upwards of 3 deliveries per week, and they were therefore staffed to receive large volumes frequently. Also, since these stores are situated far from the DC, the lead times still allow them to be notiﬁed of deliveries several days in advance. Scheduling the remaining static stores on different days gives better results, but scheduling all stores on the same day gives reasonably good results as well, and facilitates the planning process as well as internal communication. Shipping full loads separately reduces the load building and routing possibilities and was therefore not recommended, except during peak periods for speciﬁc stores when their weekly volume exceeds two full loads. The company was shown that by following these recommendations, a savings of 5–10% is attainable with the implementation of a relatively few dynamic central city stores. 7.1. Other applications The algorithm could also be used to evaluate the possibility of opening or closing a distribution center, to decide where a warehouse should be located, to determine a minimum ﬂeet size for a private ﬂeet, or to compare the use of two different

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modes of transport for a given region. Moreover, since it can be programmed and run quickly, it can be used on very large problems. The algorithm can also be applied to industries other than retail. Any industry where goods are delivered by rail and road from a factory, warehouse, or distribution center could be analyzed in the same way. Even industries where the longer transportation segments are made by air are subject to the same geographical constraints when getting the goods from the closest airport to the outer lying cities. However, if goods are only delivered to major centers where airports are located, the applicability of the algorithm would need to ﬁrst be validated using real data. The results for the regions serviced uniquely by road showed that savings were negligible for the six scenarios tested. However the algorithm could still be used to approximate vehicle routing costs in these regions in order to compare different scenarios, as long as it is possible to subdivide the region into clear subregions and major cities. In this paper, a new cost evaluation algorithm was presented in the context of a strategic analysis of savings opportunities for a large retail company. It was shown that savings of 5–10% were easily attainable in regions serviced by rail without signiﬁcantly impacting service levels to stores. Following the study, the recommendations noted here were followed and changes were made to load building and routing practices, and therefore delivery schedules, network-wide. References Clarke, G., Wright, J.W., 1964. Scheduling of vehicles from a central depot to a number of delivery points. Operations Research 12, 568–581. Coffman Jr., E.G., Garey, M.R., Johnson, D.S., 1984. Approximation algorithms for bin-packing – an updated survey. In: Auseillo, Lucertini (Eds.), Algorithm Design for Computer System Design. Springer, pp. 49–106. Coffman Jr., E.G., Garey, M.R., Johnson, D.S., 1997. Approximation algorithms for bin packing: a survey. In: Hochbaum, D. (Ed.), Approximation Algorithms for NP-Hard Problems. PWS Publishing, Boston, pp. 46–93. Daganzo, C.F., 1995. Logistics systems analysis, second ed. Springer. Fisher, M.L., Jaikumar, R., 1981. A generalized assignment heuristic for vehicle routing. Networks 11, 109–124. Garey, M.R., Graham, R.L., Johnson, D.S., Yao, A.C., 1976. Resource constrained scheduling as generalized bin packing. Journal of Combinatorial Theory Series A 21, 257–298. Gendreau, M., Laporte, G., Semet, F., 2004. Heuristics and lower bounds for the bin packing problem with conﬂicts. Computers & Operations Research 31, 347–358. Hall, R.W., 1989. Vehicle packing. Transportation Research B 23, 103–121. Hall, R.W., 1993. Properties of vehicle routes with variable shipment sizes in Euclidean plane. Transportation Research Record 1413, 122–129. Karp, R.M., 1972. Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (Eds.), Complexity of Computer Computations. Plenum Press, New York, pp. 85–103. Langevin, A., Mbaraga, P., Campbell, J.F., 1996. Continuous approximation models in Freight distribution: an overview. Transportation Research B 30, 163– 188. Martello, S., Toth, P., 1990. Lower bounds and reduction procedures for the bin packing problem. Discrete Applied Mathematics 28, 59–70. Ong, H.L., Magazine, M.J., Wee, T.S., 1984. Probabilistic analysis of bin packing heuristics. Operations Research 32, 983–998. Wee, T.S., Magazine, M.J., 1982. Assembly line balancing as generalized bin packing. Operations Research Letters 1, 56–58.