applicability and the achieved speedup of the focused search. We also ... large delivery quantities allow a minimization of the routing costs. An alternative ...
Interactive approach to the Inventory Routing Problem: computational speedup through focused search Huber, Sandra * /Geiger, Martin Josef * /Sevaux, Marc § * Helmut-Schmidt-University, Logistics Management Department, Holstenhofweg 85, 22043 Hamburg, Germany, Phone: +4940 6541 2357 §
Université de Bretagne-Sud, Lab-STICC – Centre der Recherche, 2 rue de St Maudé, F56321 Lorient, France, Phone: +33 297 874 564
Abstract: We study an interactive-approach to the Inventory Routing Problem with the goal of supporting the decision maker. Combining the supply chain management aspects ‘inventory management’ and ‘transportation’ into a simultaneous model can lead to beneficial cost reductions for both the supplier and the customer. A preference model, namely the reference point, is introduced to elicit individual preference information of the experts. Then, a subsequent interactiveapproach is developed to solve the dynamic Inventory Routing Problem. The comparison of the interactive-approach with an a posteriori-approach shows the applicability and the achieved speedup of the focused search. We also consider an extended interactive-approach for the benchmark test instances that is meaningful in terms of including a reservation point as a ‘natural’ convergence criterion.
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Introduction
The Inventory Routing Problem (IRP), an extension of the classical Vehicle Routing Problem (VRP), integrates the determination of inventory levels in the routing decision. This problem arises when vendor managed inventory concepts are employed. On the one hand, this concept utilizes the supplier to determine the routing for the overall customers instead of planning them independently, and on the other hand the responsibility of the inventory is shifted from the customer to the supplier (Popović et al., 2012). According to the IRP problem setting, a considerable tradeoff exists between the inventory levels, and the transportation effort. Therefore, our multi-objective IRP minimizes simultaneously two conflicting objectives: (i) inventory levels (sum of all inventory levels at each customer at the end of each period) and (ii) routing costs (sum of all distances traveled by the vehicles in each period). The topic of simultaneously minimizing two objectives is not commonly done in the literature. For example Coelho and Laporte (2013), among numerous others, minimize the inventory and the routing costs in one objective function.
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In the literature, basically three methods are described to solve multiobjective problems which combine search and decision making (Hwang and Masud, 1979). Each method has the aim to support the decision maker (DM) in selecting a most-preferred solution (x* ∈ P) of the set of Pareto-optimal solutions P. Due to the difficulties of a priori- (‘selection before search’) and a posterioriapproaches (‘search before selection’), our focus lies in an interactive-approach to the IRP. Since we are interested to obtain high quality approximations in a limited area rather than to cover the whole Pareto-front, we present a new idea for solving this problem by integrating gradually formulated preferences and incorporating this information for the subsequent computing of solutions. The contribution of this article is twofold. First, we investigate how the preference model interacts with the local search metaheuristic. Specially, we consider the evaluation of the relative performance of the local search depending on the chosen reference point of the approximation of the Pareto-front. Second, we propose an extended-approach which additionally uses a reservation point as a termination criterion. Its applicability is analysed and tested on benchmark instances.
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Problem description of the investigated IRP
For our problem description, we may also refer to the typology of Coelho et al. (2012) and Bertazzi and Speranza (2013). Our single-item IRP is concerned with repeated deliveries in a distribution network with one depot and a geographically dispersed set of n customers over a finite planning horizon (t, t =1,…, T). Inventory costs and capacities are taken into account at the customers, but not at the depot. According to the fleet, the size is unconstrained and capacitated vehicles (homogeneous fleet) are used. Decision variables of the investigated IRP are: (i) the delivery quantities qit for each customer i, i = 1,… ,n and each period t of the planning horizon T and (ii) the VRP must be solved for each period t, t =1,…,T, including the delivery quantities qit into tours for the involved vehicles. The here considered IRP is deterministic. In this sense, the consumption patterns are known for each customer and each period. The demand dit of a customer i at each period t is served when the current inventory is insufficient. Particularly, the currently held inventory at the customers is either able to fully cover the customers’ demand or the inventory is zero. In terms of the replenishment strategy, stockout situations are avoided by shipping enough goods in advance or just in time. Note that the customers’ demand can vary from period to period, resulting in changing delivery quantities over the time horizon T (dynamic IRP). As aforementioned, we assume two objectives which are clearly in conflict to each other. While small delivery quantities lead to low inventory levels over time, large delivery quantities allow a minimization of the routing costs. An alternative
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is evaluated by these two objective function values (outcome space) in order to determine the quality of an alternative. The DM has to take this trade-off into account when searching x*. For a more detailed formulation of the IRP, we may refer to Geiger and Sevaux (2011).
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Methodology
3.1 Construction and improvement procedure Construction procedure: it is initially decided how much to deliver to the customer and at what time. In terms of the delivery strategy, alternatives are encoded by a ndimensional vector π = (π1,…,πn) of integers. Each element πi corresponds to a customer i and shows for how many periods the demand of customer i, i = 1,… ,n is covered (delivery period). We assume, as an initial solution, identical periods for all customers, starting with 1 and increasing them by steps of 1 until the alternative cannot be added to the archive of non-dominated solutions. For example, defining the ‘delivery period’ as π = 2, then the exact demand of every customer is served for the next two consecutive periods. In this context, the supplier ensures that the customer does not run out of product. Therefore, a growing demand over the time horizon results in higher delivery quantities. This is a rather direct idea of representing delivery policies. Alternatively, a ‘constant delivery quantity approach’ could be applied, which leads to changing delivery periods. A more general solution is represented in Barthélemy et al. (2012), where customers are synchronized. Improvement procedure: a run of the local search is performed on the ndimensional vector π. Particularly, a multi-point hillclimbing algorithm is used to randomly change the values within π by ± 1, but obviously, values < 1 are to be avoided. Throughout search an unbounded archive is kept which deletes solutions by dominance comparisons. Preliminary results show that the memory of a typical computer is sufficient to store the solutions.
3.2 A posteriori-approach In this section, we propose our methodology for the a posteriori-approach based on the idea of Geiger and Sevaux (2011). Assuming that the methodology exists of two decision levels: (1) determination of the delivery quantities of each period and (2) the subsequent computation of the routing for each period, taking the already determined delivery quantities as an input. According to the underlying complexity (NP-hard problem) of the Vehicle Routing Problem, often heuristic solution approaches are used for this problem (Gendreau et al., 2008). Note, the decomposition of the problem ensures a more intuitive understanding of the approach for the DM.
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The identified first, rough approximation of the Pareto-front (initial solution with identical delivery periods) is used to represent different reference points Rj = (rj1,…,rjk) with respect to the number of objectives k = 1,…,K. The reference point method, representing a goal attainment method (Eppe et al., 2011), was established by Wierzbicki (1980) with regard of satisficing decision making. A reference point characterizes desirable or acceptable values of the DM for every objective function gk. In general, any point in the objective space, no matter whether it is a feasible or infeasible point can be selected. After defining a reference point, an achievement scalarizing function uses the reference point to project towards the Pareto-front (Figueira et al., 2010). After the search, the DM can select a solution in outcome space. In principle, the a posteriori-approach simply takes all reference points at once and tries to converge to the entire Pareto-front without discriminating between particular search directions. In more detail: the local search uses the different reference points (identified through the identical delivery periods) as an input to approximate the Pareto-front. As a remark, the interactive-approach only uses one preferred reference point of the DM in order to improve this solution.
3.3 Interactive-approach Contrary to the a posteriori-approach, preference information of the DM are elicited during the search. Particularly, after the construction phase, the DM is actively involved in the intermediate decision making process. He/she selects a reference point in outcome space, thus guiding the subsequently derived search direction. The local search takes the resulting reference point Rj and minimizes the maximum distance from the computed solutions x to Rj.
{ (
)} ∑
K ⎡ ⎤ min ⎢ max wk g k (x ) − r jk + ε g k ( x) ⎥ ⎢⎣ k =1,K, K ⎥⎦ k =1
(1)
Expression (1) defines the distance of each computed solution x to the reference point Rj. The objectives are denoted with gk and a normalization of the objective functions is performed by means of wk. This is due to the fact that the objectives are measured on rather different scales: inventory levels on the one hand and traveled distances on the other hand. For performance comparisons, normalized values are better in terms of quality comparisons of different test instances. We assume the minimum and the maximum values (estimated ‘nadir point’) of the computed outcomes for defining the wk values.
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Determination of ε The additional term ε∑k gk(x) of Expression (1) avoids the identification of weakly-efficient solutions. In our experiments the local search maintains an archive of non-dominated solutions and eliminates weakly-efficient solutions in terms of dominance comparisons. While primarily minimizing Expression (1), maintaining an archive comes with another effect: search can continue after identifying a (locally) optimal alternative to (1). However, if ε can be defined for our application, the archive of nondominated solutions is not needed for further applications. The determination of ε is difficult in the general case. In this sense, ε is often described as a small positive parameter (Nikulin et al., 2012). However for our application ε in Expression (1) can be defined in more detail. Let Δmin be the smallest positive difference bek tween any solution x and x' w.r.t. gk. Then we can easily consider the smallest positive difference of the inventory management (called ‘inv’) by the following relation:
Δmin inv = 1 .
(2)
The smallest positive difference of any solution x and a neighboring solution x' is one because we assume that products cannot be splitted. Similar to the computation of the smallest positive difference between two inventory levels, we may derive an insight for the routing. In case of positive distances dij between customers i and j, the smallest positive difference between the length of two solutions x and x’ is identical to the one of the inventory levels, which is 1. Often, and also in our application, the values of dij are measured as truncated Euclidean distances (Range, 2007 and Kohl et al., 1999). If the truncation leaves one digit, we may conclude that any smallest positive difference between x and x’ cannot exceed 10-1, etc. Consequently, to the preceding description of the smallest positive differences, the minimal difference (called ‘mindiff’) of the inventory and the routing objective must be selected and is defined as:
{
min diff = min wk Δmin k k
}.
(5)
Let LBk be a lower bound on gk and UBk an upper bound on gk. It follows, then, from our assumptions that ε must be strictly smaller than the relation:
ε
