Practical aspects of route planning for magazine and newspaper wholesalers

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER

European Journal of Operational Research 90 (1996) 1-12

Invited Review

Practical aspects of route planning for magazine and newspaper wholesalers Roland

Dillmann

*, B u r k h a r d

Becker, Volker Beckefeld

Bergische Universitiit GH Wuppertal, Gauflstrafle 20, 42097 Wuppertal, Germany

PROSY GmbH, Wuppertal, Germany Received 1 June 1995

Abstract

The complex distribution problem of magazine and newspaper wholesalers is an ideal field for the application of mathematical methods, though the success of consultancy projects in this sector does not depend on the quality of the methods alone. Before building models to describe the distribution problem, it is necessary to identify the specific characteristics of routing plans in the press wholesaling sector, and to choose an adequate and acceptable approach to the problem, both from a theoretical and a practical perspective. Thereafter, models for the road system, time structure and delivery volume are built and integrated into a consultancy process. The consultancy approach underlines that optimisation of routing plans in this sector is more than just a pure mathematical problem, but that a project's success depends essentially on the constructional team work of client and consultant. Keywords: Route planning; Vehicle routing; Time windows; Heuristics; Data structure; Empirical data models; Press

distribution

1. Introduction

This p a p e r deals with the experience m a d e during a significant n u m b e r of consultancy projects centred around the application of mathematical models for distribution planning. T h e article starts with a brief introduction to the G e r m a n magazine wholesaling sector and its inherent distribution problem. Also, the impact of the applied techniques to bundle consignments for delivery (henceforth: picking systems) and the

* Corresponding author.

determination of supply quantities are shown, since both of them represent important inputs for the distribution problem. This section is followed by an overview of the typical characteristics of present routing plans used for the daily distribution of magazines and newspapers. In general, these plans are used as a starting solution for the optimisation process. The applied data concepts to model and subsequently structure the routing problem are developed. Getting hold of and reviewing the required data is the initial phase of the consultancy project, followed by a simulation of the status-quo distribution plans and the development of new, optimised routing plans. This process is covered in the fourth section, while the

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subsequent chapter deals with the mathematical instrument applied to develop optimised routing plans. In the final chapter, some conclusions are drawn from completed consultancy projects.

2. Characteristics of the routing problem for magazine and newspaper wholesalers

2.1. The general distribution problem In Germany, the magazine and newspaper distribution problem, as far as wholesaling is concerned, can be described by the following figures: Shortly before the publishing day, more than 200 publishers deliver mainly newspapers, magazines and low-price paperbacks to 96 wholesalers, who, in turn, supply about 110,000 retail outlets daily. Newspaper and magazine wholesalers are mostly regional monopolists. The status as a monopolist defines to a large extent the work of a wholesaler. He is covenanted to deal with all publishers and products (about 3,000 different titles) as well as to deliver the products to all retail outlets within his region. In addition, the wholesaler is entitled to determine the individual retailer's press assortment and has to ensure the punctual and areacovering delivery of the products. Overall sales of all German magazine and newspaper wholesalers add up to around DM 6 billion p.a., though the delivery value is about 50% higher. On the basis of a value of DM 10 per kilogram press, the daily delivery volume is around 3,000 tons of press, delivered by around 3,000 vehicles, covering a distance of 150 million km p.a. Overall delivery costs of the sector are estimated to be about 150 million DM p.a. (Becker et al., 1995). The distribution problem is more than just a delivery problem. The wholesaler has not only to deliver the products to the outlets, but also to pick up (unsold) return copies on a daily or weekly basis. In effect, about one third of the distributed copies are returned to the wholesaler. This aspect emphasises the strong interdependencies between the distribution problem and the forecasting problem referring to supply quantities. Supply quantity influences several areas of

press logistics: it enhances the required time for bundling, subsequently influences the departure times of the routes, and increases the required unloading time, hence influences the arrival time at the individual outlets. Since the (geographic) positions of the supplied retail outlets are rather stable, the routing problem for press wholesalers can be described as a more strategic problem, in contrast to a tactical routing problem that considers the daily changing logistics problem. The general problem is complicated by volume peaks at the end of the week, month and year, leading to a discussion about flexible, i.e. alternating, route sequences, or fixed, i.e. stable, route sequences.

2.2. Flexible versus fixed routing plans The differentiation between flexible and fixed routing plans can be reduced to the distinction between: • routing plans adjusted to fluctuating supply quantities, leading to flexible plans, or • routing plans adjusted to stable opening times and geographic positions of the outlets, leading to fixed routing plans. Practical aspects and constraints of the distribution problem lead to the necessity of fixed routing plans, since: • A typical route serves between 40 and 50 retail outlets. • Time pressure during distribution is severe. This is mainly caused by ever increasing late copy deadlines and early opening hours of retail outlets. Since most retail outlets can be supplied in an unlimited time period before opening hours, the problem is that of a onesided time window. However, this leads to a situation where the driver has to be familiar with the surrounding of the retail outlets, since the driver has to lock in the consignment at an agreed place. • Due to the above mentioned time pressure, the driver has to get accustomed to his route - the whole daily delivery process must become routine work. • The required routine and the required knowl-

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edge of the outlets' premises cause fixed routing plans. In fact, present routing plans in the magazine and wholesaling business are fixed plans, underlining the importance of punctual delivery. 2.3. Pre-routing concept

However, an adaptation to fluctuating volumes is still necessary. The need for an adaptation is stressed by the fact, that e.g. only 12% of the overall weekly volume has to be delivered on a Tuesday, in contrast to about 23% on a Thursday. The adaptation is done through a pre-routing concept. The application of this concept, where part of the overall volume is delivered right after midnight of the respective day, is possible since only a small proportion of the daily volume is really time-critical: the daily newspapers. Other press products, mainly weekly or monthly magazines, can be delivered hours in advance of the daily newspapers. These might be limited to serve only a few high-volume outlets, depending on the overall volume of the respective delivery day. Additionally, on extremely strong days, more than one pre-route might be necessary. This means, pre-routing is an ideal tool for adapting fixed plans to fluctuating volumes, although it can only be applied in short-distance areas. 2.4. Geographic characteristics

The typical German region served by a single regional monopolist is a mixed municipal and rural area, where the wholesaler is normally located in, close to or between municipal areas, so that a large part of the daily volume has to be delivered within a short-distance area, where pre-routing can be applied. In the long-distance area, pre-routing is often impossible for reasons of distance. Two alternatives are possible: Reducing the number of outlets - and thus the volume - per route, so that pre-routes are not required. Alternatively, it might be possible to build additional routes on strong days, such as Thursday. In fact, this is practised by some drivers, though the first alternative is predominant.

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3. Current s i t u a t i o n in m a g a z i n e distribution

3.1. Typical aspects o f conventional routing plans

Routing plans for press delivery have been built and modified over a rather long time. The following aspects are the consequence of the regular attention that is spent to adapting the plans to changes in the retail structure and at the same time a constant road system. • The overall quality of the plans is already better than results of applied conventional automatic heuristics. This is mainly due to the huge amount of (human) resources spent on the modification of plans and the thorough knowledge of the delivery region. • Most plans are based on regional clustering, i.e. one region is served by one route. Overlaps of different routes are exceptions. • Since some wholesalers are not bundling all routes at the same time, the departure times of the routes vary between 3am and 6am. In consequence, early openers in regions not served by early departing routes cannot be delivered before opening hours. • This situation is complicated by the general tendency towards earlier opening times. • Referring to the applied regional clustering principle and considering the early opening hours, the number of routes in present routing plans is far too low. Thus, present routing plans are characterised by delays, strong variations with respect to the number of delivery points and delivery volume, and a strong focus on geographical clustering. This is caused by the general opinion that adjacent delivery points must be served consecutively by the same route. 3.2. Earlier mathematical approaches to the routing problem

Most wholesalers are aware of the above mentioned problems, and mathematical planning methods to solve the problem have been applied in the late 70s/early 80s, although with limited success. In that period, the implementation of time windows was not yet common, and software

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not based on time windows had to be applied. Time-related problems were considered mainly as a by-product of distance savings based on the application of the TSP, but this approach was not adequate for a situation were time constraints are one of the core problems. Furthermore, significant cost savings can only be achieved by reducing the number of routes, but conventionally applied methods do not reduce the number of routes. Additionally, in most cases the resulting number of routes cannot be anticipated. The experience made with these early approaches led t o a serious consequence, namely strong mental reservations against the application of any mathematical planning methods in the press wholesaling sector. The reservations were contradicting: Wholesalers in municipal areas argued that the described approach is only applicable in rural areas because of the lack of early openers. At the same time, wholesalers in rural areas regarded the lack of infrastructure as prohibitive for the application of an Euclidean-distances approach.

4. Modelling the general routing problem Even if an enormous amount of time and effort is spent on sampling and testing data, a solution based on mathematical methods will always remain a model solution, and must not be regarded as an isomorphic picture of reality. In the following three chapters, the principles of modelling the road system (Section 4.1), time structure (Section 4.2) and delivery volume (Section 4.3), are described.

4.1. Modelling the road system Current mathematical approaches model the road system on the basis of vector maps. At the beginning of the first consultancy project, working with vector maps was rather disadvantageous: • The only vector maps available were produced for long-distance routing problems, i.e. only a few main roads and junctions were vectorized. However, this degree of vectorization was not sufficient for the press distribution problem.

• Most available vector maps were (and still are) distorted. • Most available vector maps did not contain any information about one-way roads and traffic regulations. Today, most of the above mentioned problems are solved. However, the remaining problem is that of keeping the plans up-to-date. • The Digraph model is, even from a theoretical perspective, inadequate for an inner-city routing problem, since traffic regulations cannot be modelled adequately. This is caused by data storage problems as well as by the required modification of shortest-path methods, since the combination of a route may depend not only on the reached node but also on the predecessor arc. Efficient implementations of these modifications, applicable on problems of the described size are possible, though efficient commercial implementations are not known to the authors. Aware of the theoretical reservations, we assumed a complete graph as model, where the nodes represent the retail outlets (delivery points). The distance between two nodes is determined on an Euclidean basis. (For the practical relevance of this approach: Holt and Watts, 1988; Stokx and Tilanus, 1991; for alternative estimates of distances: Love and Morris, 1972, 1988; Brimberg and Love, 1991; Berens and K6rling, 1985; Christoffides and Eilon, 1969; Ward and Wendell, 1985. These sources refer to location problems.) Here, the p-norm leads to the best results, modelling distances by

d(x,y) ---a(Ix 1 -yllP +lx2-Y2lP) I/p.

(1)

Often applied specialisations are: the Euclidean norm with p = 2 and the rectangle norm with p -- 1. A generalisation of the problem is d ( x , y ) = a(lx 1 - rxl p +

Ix2 -

y21P) q

(2)

The advantage of the case with p -- 1 is its fast computing time, while the Euclidean norm is closely associated to bee-line distances. According to Berens and K6rling (1985), even the results of the p = 1 norm can be justified by the ease of computation. In a rare application of elaborate

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networks, Solot et al. (1990) applied a network of Switzerland due to the extreme geographic conditions and divided the road network into ten categories to describe average speed adequately. Past applications of the model lead to the impression that the main criticism against the model are not based on the inner-city problem of one-way routes, but on inadequate road systems caused by an inadequate number of bridges crossing rivers, valleys, forests, and industrial areas. In our projects, we divided heterogeneous areas into smaller, more homogeneous areas, assumed a complete graph for these areas, and connected these complete sub-graphs on the basis of bridges, etc. (Holt and Watts, 1988; Stokx and Tilanus, 1991). Since the durability of routing plans is rather long in the press wholesaling sector, it can be justified - and proved to be advantageous - to test the practical applicability of the plans before they are practically applied. These practical tests identify mistakes in the planning process which can be corrected before the implementation. Hence, the discussion about the data basis has to take into account the possibility of practical pre-tests of the theoretical plans. These tests are possible and can be justified if the planning problem is strategic and not tactical, where the number of new (often daily) plans forbid intensive pre-tests. If plans cannot be tested practically before their practical implementation, a much higher planning security is required. These practical tests outweigh the advantages of the application of more elaborate databases, especially under planning security aspects. A special case is the integration of critical e.g. isolated - nodes. By far more effective than the application of mathematical methods are organisational measures to transform the critical node into a non-critical one. The idea of the saving principle, namely to create new routes for the critical nodes, is based on a misinterpretation of the definitiveness of present organisational constraints - especially constraints due to local absurdities - and should not be applied. The applied method to model the road system is a connected graph which is associated with a relatively simple data provision. This leads to the

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requirement of an intensive test of the practical applicability of the model solution and in turn to the modification of the connected graph with respect to the results of the practical tests. This means, the final model of the detailed problem is formulated only at the end of the consultancy project and cannot be formulated precisely at the beginning. 4.2. Modelling the time structure

The second problem is that of modelling the time structure. To model the time structure, we focus on traffic and driving times, latest delivery times, departure times, and unloading times. When modelling driving times, it is possible to refer to road classifications, that duster the road system on the basis of different average driving times, including information about traffic congestions (Solot et al., 1990). However, this information is helpful for long distance routing, but cannot be applied for inner city routing problems. In the latter case, we work mainly with information about the starting time of the rush hour. Modelling the time structure in inner city areas is only possible if the delivery is completed before the rush hour starts, because an unpredictable rush hour traffic situation makes sufficiently precise time estimates impossible. Here, the appropriate tool to model this problem is to choose a pre-rush-hour opening time for the respective retailers. Latest delivery times represent the second aspect when modelling the time structure. If latest delivery times are solely determined by the retailer, these times often turn out to be unrealistically early. Hence, these times have to be verified by comparing them with the actual official opening times and - in case of significant differences investigate the reasons. In addition, if early delivery times are justified but difficult to keep by the wholesaler due to the retailer's geographical location, often individual agreements about alternative delivery locations can be found, making the mathematical problem easier to solve because of constraint relaxations. This is often applied where the retailer's home address is different but more central than the business address. -

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Furthermore, departure times for the individual routes have to be determined. Departure times depend mainly on picking methods applied by the wholesaler, and delivery times of the daily newspapers. The time structure is calculated based on driving times (depending on air distances) and unloading times (depending on delivery volumes) and a fixed unloading allowances (for empirical applications, see e.g. Kolesar et al., 1975). These parameters are identical for mixed municipal/rural regions and are estimated for each wholesaler. Obvious differences in driving times depending on the region or the Euclidean distance between two delivery points are taken into account through the fixed unloading allowances which serves as a kind of buffer (Assad, 1988). All parameters are verified during the statusquo simulation where the theoretical duration of any route is compared to its actual duration. In cases, where significant differences can be identified, the reasons for these differences are investigated. Differences are mainly caused by rush hours or single delivery points where unloading is difficult and time intensive, e.g. within pedestrian areas. These sub-routes have to be planned with different parameters and are regarded as one node in the global route plan (cf. shrinking techniques of the matching problem). These estimates turned out to be very reliable. The optimisation problem is not solely a mathematical problem, but also an organisational problem. Often, the whole logistics is under investigation during a project that started as a simple route optimisation project. Hence, we are not only talking about a mathematical preprocessing, but also about the necessity of an organisational pre-processing. 4.3. Modelling the delivery volume

Delivery volume is modelled on the basis of a parameter, connecting delivery volumes in monetary terms to weights. This parameter differs only slightly between the delivery areas and differences between single delivery points level out. The value/volume relation is slightly higher in municipal areas than in rural areas.

5. The consuitancy process The approach described in this paper results in a fundamentally new structure of routing plans in the press wholesaling sector. It transforms the regional cluster of present routing plans into time-structured routing plans. Therefore, route planning in the press wholesaling sector must be integrated in a consultancy project. Theoretically, it would be possible to sample the required data and subsequently, without any further discussions, present an optimised routing plan to the client. However, it would not make sense, because it is the client, who has to implement the new plan. Thus, it is essential that the client's objectives, ideas and experience are taken into consideration and become part of the plan, and that the client's logistics department is convinced of the quality and practicability of the new plan. This is only possible if the logistics management has understood the basics of the optimisation approach, can handle the gap between a theoretical and practical routing plan, and can discover own inputs in the resulting plan. The persistent inclusion of practicians' experience at each stage of the project makes the application of a rather simple data basis possible. This leads to a situation, where the main task during the consultancy project - next to pure mathematics - is to adapt the established approach to the client's specific situation, and at a later stage, to adapt the theoretical solution to a practicable one. 5.1. Analysing the present delivery situation

Starting point of any consultancy project is the analysis of the present delivery situation. For every delivery point, the following data is required to model the present delivery situation: • identification number; • geographical co-ordinates of the retail outlets,; • latest delivery time; • earliest delivery time, considered only if unavoidable, e.g. due to restricted access to certain areas during the night or early morning;

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• • • •

route number; route sequence number; average weekly sales; average return c o p y / s a l e s ratio. The result of the analysis is presented in a matrix with the following columns: identification number; route number; calculated delivery time; calculated buffer time; calculated push-backward time; geographical co-ordinates of the retail outlet; accumulated delivery volume; Euclidean distance to predecessor. Additionally, it is possible to derive the following information from the matrix: • latest delivery time, as sum of the calculated delivery time and buffer time; • delivery volume of the individual retail outlet; • most time-critical retail outlets within each route. The parameters 'driving time', 'unloading time' and the 'fixed unloading allowance' are estimated so that, on average, calculated and real duration of the whole delivery are about equal. It is not the objective to estimate every single parameter as realistic as possible. E.g. the driving time parameter is generally underestimated, since the overall Euclidean distance of a routing plan is generally reduced during the optimisation. In case of an overestimation of the driving time parameter, a reduction of the Euclidean distance would lead to overestimated reduction in the required overall driving time. Hence, underestimation of this parameter generates an additional time buffer for following retail outlets. The whole process of the status-quo simulation can already be regarded as the first indicator of the project's outcome. If it is possible to generate a realistic simulation, the logistics department of the client has proved its knowledge of the delivery problem and its ability to co-operate with the consultant. A difference of 10% between calculated and real time must be regarded as good result. This corresponds to a R 2 of 0.99. Better results can hardly be achieved due to daily volume fluctuations and weather conditions and their effects on time estimates.

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5.2. Determining objectives for an optimised rout° ing plan Every project has its own objectives, and that is why finding and specifying the objectives is an important step of any project. Possible objectives are manifold and often contradictory: • punctuality, i.e. delivery at least before opening hours or, even better, with a time buffer; • reduction in delivery costs by reducing the number of routes or the overall delivery distance; • saleableness of routes to sub-contractors, based on homogeneous sizes of the routes. Especially in the last case, reaching the objective depends very much on the delivery region. E.g., it is hardly useful to design routes with similar length in regions with strong purchasing power in municipal areas close to the depot, and the opposite situation in rural areas far away from the depot. In these cases, an alternative objective is the single delivery of distant retailers, hence to adhere to strict capacity restrictions for the respective routes or to design routes serving close high-volume retailers and distant retailers. Obviously, the three above mentioned objectives are contradictory. In general, there are different possibilities to model contradicting objectives: • multiple objective optimisation; • weighted objective functions; • determining objectives to be included in the objective function and those to be considered as restrictions. The last approach leads to two different kinds of restrictions. Violating the above mentioned restrictions does not result into an inadmissible solution, while this is the case with respect to usual restrictions. Then, the solution is inadmissible only if the result is not practicable. Hence, it is necessary to evaluate violations of the above mentioned restrictions. In mathematical terms, a penalty approach is applied. The practical delimitations between considering a restriction as constraint or as part of the objective function are fluently. We decided to consider opening hours and delivery volumes as binding (but not too binding)

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constraints, but at the same time to simulate the consequences of variations of the constraints. Only the overall driving distance enters the objective function, while all other objectives are formulated as constraints. Choosing the overall distance as objective is not based on costs savings through distance savings, but on its mathematical simplicity. However, it must be considered that adhering to buffer times and one-sided time windows is closely related to distances. Although respecting one-sided time windows leads to circuitous routes in a few cases, the overall distance has to be minimised. Otherwise, time constraints become relevant even for late openers because of the limited number of routes. From a mathematical perspective it is unproblematic to consider minimal buffer times as restriction, since they can be realised by modifying latest delivery times or departure times. However, if objectives are formulated as lower or upper bounds (e.g. lower or upper bounds for the number of delivery points per route), the formulation has to be done with respect to the solution procedure. Since we apply solution procedures based on route building procedures, it is easier to consider upper bounds than lower bounds. Only if the departure times can be altered into earlier times, it is possible to work with lower bounds. The reason is that routes with only few delivery points usually serve large retail outlets, or leave the depot rather late and must be punctual. Often, in these cases there is a shortage of late openers at the end of the route.

5.3. Developing an optimised routing plan Having defined the objectives, mathematical methods are applied to develop a new routing plan. The applied methods are described in more detail in Section 6. The core characteristics are described in the following. • In practice, only heuristics can be applied, since even the smallest practical problems are far too big for algorithmic solutions. The benchmark for algorithmic solutions is set at 100 delivery points, while the average wholesaler serves about 1,000 retailers (Desrochers et al., 1992). Furthermore, partitioning and covering

models can only be chosen in the case of routes with a small number of delivery points. • From previous experience, routing problems with time windows are characterised by various local optima (Savelsbergh, 1985). Here, the expression 'local optimum' is related to conventional exchange concepts. Examples for these exchange concepts are 2-opt, 3-opt and Or-ropt concepts and 1-to-1 exchanges (van Breedam, 1994). Heuristics that aim only on solution improvements lead to bad sub-optimal solutions. • Hence, strategies are required to leave local optima, and these strategies might lead to a temporary loss of solution quality. This could be a starting point for simulated annealing and tabu search (van Breedam, 1994). However, both SA and tabu search are too unspecific for the problem size we encounter, so that more problem-specific and intelligent concepts are required to leave local optima and avoid returns (for new aspects: Golden, 1993). We have just encountered first hopeful approaches for the successful implementation of heuristics (Derigs and Grabenbauer, 1993), although we follow a dialogue-based software-assisted approach to find solutions. This means, that the way to leave a local optimum depends on the latest solution. Starting points are the • visualisation of routes with a geographic information system and in matrix form, so that possible alternatives can be identified and checked with respect to the road system, time and volume constraints; • modification of the minimum degrees of goal accomplishment, where an improvement of the degree of goal accomplishment - despite a deterioration of distance values - is kept. Keeping the improvements plays the role of the tabu list and hence avoids cycles; • exchange of sub-routes within different routes to improve minimal buffer times of sub-routes. Obviously, this approach is closely connected to the neighbourhood definition and the goal conflict that an enlargement of the neighbourhood leads to a significant increase of numerical complexity when determining local optima. However, a numerically simple neighbourhood

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definition can be applied if a dialogue-based search strategy is followed. Dialogue-based approaches seem to be an adequate method to solve planning problems of the described size, especially if the plans can be visualised with a geographic information system. One of the main advantages of a dialogue-based approach is its precise target orientation. However, the success of this approach depends to a large extent on the experience of the problem solver, since the results reflect evidently what the problem solver regards as a good routing plan. The philosophy of this approach is characterised by an unconventional application of models. The approach allows for problem-specific analysis, in contrast to non-dialogue implementations of models.

5.4. Implementing an optimised routing plan The result of the optimisation process, the new but still theoretical routing plan, is then handed over to the client's logistics department, where the plan is checked, considering its ability to be implemented. During this process, the practicability of the plan and its time structure are evaluated, and, most important, the experience of the client's logistics management is integrated into the plan. A complete redesign of routing plans with the described complexity, will not be successful if the parties involved in the project do not work as a team. Modifications in this stage are mainly caused by a specific road system, retail structure or contractor structure which are not part of the model. Even during the first days after the implementation of the new plan, modifications of the plan have to be made. These are caused nearly exclusively by wrong opening times of the retailers and their effect on delivery times of the new plan.

6. Applied mathematical concepts for the routing problem

6.1. A brief overview of concepts for the general routing problem The results of the mathematical complexity theory about routing problems with time windows

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and capacity constraints justify the focus on heuristics, due to the size of the analysed problems. Hence, we do not describe algorithmic approaches in the following. Bodin et al. (1983), Desrochers et al. (1988), Laporte and Nobert (1987), Haouari et al. (1990), and Laporte (1992) give overviews of algorithmic ideas. The implemented heuristic is based on two stages, an initialisation stage to identify an admissible starting solution, and an improvement stage. Mathematicians suggested various procedures to such a solution, which have been supplemented by aspects of time-window and capacity constraints. (For overviews of these ideas and their adaptation to time windows and capacity constraints, see Solomon (1986, 1987), Desrochers et al. (1988), and van Breedam (1994).) • the savings principle (Clarke and Wright, 1964; Yellow, 1970; for software implementation: Golden et al., 1977); • the nearest-neighbour principle (Rosenkrantz et al., 1977); • general insertion methods (Karg and Thompson, 1964; Rosenkrantz et al., 1977); • sweeping strategies (Gillett and Miller, 1974; for the use of both polar co-ordinates: Thangiah et al., 1993); • clustering first, routing second (Fisher and Jaikumar, 1981). All approaches have in common that the number of routes is determined during the procedure, and no further attempts are made to limit the number of routes. For the routing problem encountered in the magazine and wholesaling sector, such an approach is not adequate, since cost savings can only be realised by reducing the number of routes and hardly by distance savings. Hence, our heuristic starts with an a-priori determined upper bound for an acceptable number of routes.

6.2. The initialisation stage In the initialisation stage of the heuristic, we apply parallel route building procedures to develop an admissible routing plan. Step by step, delivery points are inserted into the routing plan in ascending order of the their opening times. Inserting following retailers is done with respect

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to the constraint that previously inserted retailers' delivery remains punctual. Under this constraint, the retailer is inserted at a position where the additional distance is minimised. If no admissible insertion is possible, an improvement phase for the previously created routes is applied. Subsequently, another insertion attempt is undertaken. If the insertion is still inadmissible due to time constraints, the retailer is inserted under geographic aspects, and it is tried to comply with time restrictions by rearranging the respective route. Alternatively, departure times of the routes might be interchanged. Starting point for the initialisation stage is the present number of routes. If no of the described procedures is successful, the number of routes has to be increased, so that an admissible starting solution can be identified. An alternative procedure is to consider the present routing plan as (in most cases inadmissible) initial solution, and to transform the plan step by step into an admissible plan, either by interchanging the used vehicles, rearranging the route sequences, creating additional routes, or by interchanging departure times. In general, the first step when creating an admissible solution is to focus on capacities, and in a subsequent step, to consider time constraints. Creating an admissible initial solution on the basis of the present (real) routing plan has the advantage that the initial solution is practicable. Thus, it is an ideal start for a plan based on Euclidean distances. Furthermore, sub-routes are not changed during the following improvement stage, as long as they comply to the constraints. Well established sub-routes, built with the logistics management's experience, remain part of the new plan. 6.3. I m p r o v e m e n t stage o f the heuristic

The improvement stage is the repeated attempt to remove sub-routes with a maximal number of delivery points from a route, and to insert these sub-routes into other routes while maintaining the routing sequence. This concept is based on a specific 3-change reflecting the time structure. The basic optimisation concept is the Or-r

optimality of order r, with r being the maximum number of delivery points per sub-route that can be inserted into the route leading to an improvement without violating the constraints (Or, 1976). A solution is O R - r - e optimal, when no sub-route with the maximum of r delivery points can be inserted and reduce the overall distance of all routes by e, complying with the constraints. An important part of the improvement stage is the reduction of the number of routes, since this is where cost savings can be realised. Reducing the number of routes is often possible because conventional route planning adheres to geographical principles. This means, routes are assigned to delimited regions. However, the demand within these regions is very heterogeneous, so that delivery volumes of the routes vary extremely. Taking pre-routing concepts into account, the available capacities are generally more than sufficient and can be reduced. The possibility to reduce routes is achieved by creating larger clusters and allowing for more than one route (up to three or four routes) in one cluster. Time constraints are far more problematic, and are solved as described in the previous chapter. In most cases, reductions of the numbers of routes can only be achieved by temporarily relaxing the restrictions on volume and time. Subsequently, the solution has to be made feasible again. In this process, the acceptance of temporary losses of solution quality is very helpful. Simulated annealing and tabu search have to be replaced by dialogue-based approaches. (For many ideas usefully implemented in a dialoguebased system, see Wren and Holliday, 1972; Brendel, 1987.) The inherent goal conflict between the reduction of routes and punctual delivery remains. In general, alternative routing plans are developed, and it is finally the decision of the client, which plan he chooses.

7. Results and conclusion

The objectives and practical constraints of each individual project differ too much as to present the results of the projects in various charts or

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tables. A reduction in the number of routes of 5% might be an excellent result in one project, while a 10% reduction might be insufficient in another project. In the following, we present and comment the fundamental results and conclusions of completed projects: • Logistics is an area where huge cost savings can be realised if grown structures are consequently analysed and modified with an appropriate set of mathematical tools and methods; on average, completed projects resulted in a 10% reduction in route number, 10-15% (after adaptation to the road system) reduction in Euclidean distances, and sufficient buffer times. • Unfortunately, mathematics are not all in a project as complex as the logistics optimisation in the press wholesaling sector. Even if the mathematical problems are solved, the organisational and managerial problems remain. • The problem with the data: It has to be complete, it has to be verified and in most cases, parts of the data have to be gathered again, and data changes during the optimisation process have to be integrated. The use of GIS software is far more elegant than older manual approaches, but requires intensive training. Under no circumstances, the data problem becomes trivial. • Back to the managerial aspects: Most important are the people who are involved in the project and who have to implement and to work with the new plan. If these people are not convinced of the necessity to change the present situation, and the quality of the new approach, making the project a success is rather difficult, if not hopeless. The most important success factor is that the people have felt that they have a severe delivery problem before the project started. Even though the approach is dominated by mathematics, it would be fatal to neglect the human factor. • The newly developed plan might be of an excellent quality. However, it will not be successfully implemented if the client's logistics management is not fully convinced of its practical quality. From this perspective, the regional clustering principle sent the wrong signals. • One further problem should not be underes-

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timated: The application of regional clustering principles led to a situation where neighbouring retailers were served consecutively by the same route, regardless of their opening hours. One of the core principles of our routing plans is to deliver late openers on late departing routes or on the return way of early departing routes. However, this means that late openers feel unfairly treated although they are still served before opening times. • And last but not least: the consultant moves on to the next project, the logistics management stays.

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