The traveling purchaser problem with budget constraint

Abstract

Let us consider a set of markets plus a depot and a set of products for each of which a positive demand is specified. Each product is made available in a subset of markets in each of which only a given quantity, less than or equal to the required one, can be purchased at a given unit price. The distance between each couple of markets and between each market and the depot is known. The Traveling Purchaser Problem with Budget constraint (TPP-B) looks for a simple cycle starting at and ending to the depot and visiting a subset of markets at a minimum traveling cost and such that the demand for each product is satisfied and the cost globally spent for purchasing the products does not exceed a defined budget threshold. As the TPP also this problem arises in several application domains, but while the former has been largely studied, very few contributions exist in the literature for the TPP-B. We propose and compare two solution algorithms, an enhanced local-search heuristic and a variable neighborhood search (VNS) approach also tested in a multi-start variant. The proposed algorithms have been used to solve both the capacitated and the uncapacitated version of the problem. Test problems have been obtained by adding a budget constraint to known benchmark instances for the TPP.

Introduction

The capacitated Traveling Purchaser Problem with Budget constraint (capacitated TPP-B) can be described as follows. Consider a set of markets $M≔\{1,\dots ,m\}$ plus a depot (indexed 0) and a set of products $K≔\{1,\dots ,n\}$. Each product $k,k\in K$, can be purchased at a subset $M_k\subseteq M$ of markets at a non-negative unit price $f_{\mathit{ki}}$ depending on the market $i,i\in M_k$. A demand $d_k$ of units is specified for each product $k\in K$ and only $q_{\mathit{ki}}$ units of product k are available at market i, $i\in M_k$. We assume that $0<q_{\mathit{ki}}⩽d_k$ and ${\sum }_{i\in M_k}{q}_{\mathit{ki}}⩾d_k$ for all $k\in K$ and $i\in M_k$. A traveling cost between each couple of markets and between each market and the depot is known and given by $c_{\mathit{ij}}$, $\forall i,j\in M\cup \{0\}$. A budget B is also specified and represents a threshold on the total amount which can be spent on purchasing products. The problem consists in selecting a subset of markets and visiting them with a tour starting at and ending to the depot such that the demand $d_k$ for each product $k\in K$ is satisfied at a global purchasing cost which does not exceed the given budget threshold B while minimizing the total traveling cost. The special case where $d_k=1\forall k\in K$ and $q_{\mathit{ki}}=1$ for each $i\in M_k,$ defines the uncapacitated version of the problem (uncapacitated TPP-B).

The TPP-B is a problem involving two types of costs. The first one is the traveling cost which is minimized in the objective function. The second one is the purchasing cost, the value of which is bounded in the constraints. Most of the previous studies on similar problems have restricted the analysis to the case where the two cost measures were both included in the objective function. This single-objective combinatorial optimization problem is known as TPP and has found several applications in scheduling and routing contexts (see Ramesh [1] and Singh and van Oudheusden [2]) as well as in the telecommunication network design (see Ravi and Salman [3]). The TPP is known to be NP-hard in the strong sense, reducing to the traveling salesman problem (TSP) when each market offers a product which is not provided by the remaining ones. It also contains the uncapacitated facility location problem and the set covering problem as special cases. Due to its computational hardness several heuristic procedures have been proposed to solve it (see, for instance, Golden et al. [4], Ong [5], Pearn and Chien [6], Voß [7], Renaud et al. [8] and Mansini et al. [9]) while recently a branch-and-cut procedure has allowed the optimal solution of problems with up to 200 markets and 150 products (see Laporte et al. [10] and references therein for other exact methods previously proposed in literature for the same problem). An evident drawback of the TPP is the possible different nature of the two costs merged in the objective function: the traveling cost is usually represented as distance or traveling time whereas the purchasing cost is typically measured as currency. Moreover, a trade-off frequently characterizes the two objectives since reducing the purchasing cost may imply an increase in the distance traveled. To avoid such disadvantage Riera-Ledesma and Salazar-González (see [11]) introduce the bi-objective TPP: this is a bi-criteria version of the TPP where minimizing the purchasing cost and the traveling distances are two separate objectives. The bi-objective is a generalization of the TPP whose solution provides insight into the trade-off between the two costs. The authors tackle the problem by generating the set of all supported and non-supported efficient points in the objective space. For each efficient point a Pareto optimal solution in the decision space is computed by solving a single-objective problem while bounding the remaining cost function. In the case the single objective is represented by traveling costs while purchasing costs are bounded in the constraints, the problem solved is the TPP-B. The authors provide exact solutions by means of a branch-and-cut approach for instances with up to 100 markets and 200 different products but only for the uncapacitated version of the problem.

The uncapacitated TPP-B is also analyzed in Ravi and Salman [3] as an application of telecommunication network design. In that paper, the authors introduce an algorithm for the bi-criteria version of the UTPP with metric distances based on the rounding of an LP relaxation solution whose worst-case ratio is poly-logarithmic. They also show that, for a special case of the UTPP which models the ring-star network design problem with proportional costs, a constant factor approximation algorithm exists.

In the present paper, we propose two different algorithms to solve the TPP-B. To the best of our knowledge these procedures represent the first heuristics proposed in the literature to solve this bounded version of the TPP. The first procedure is based on an enhanced local-search scheme where a neighbor solution is obtained by the current one by removing l-consecutive markets and inserting as many markets as required to restore solution feasibility. Since the value of the parameter l is changed during the search, the procedure is more than a simple local search. The idea of removing l-consecutive vertices is not new in the literature. One of the most effective heuristic algorithm for the TPP (see Riera-Ledesma and Salazar-González [12]) makes use of such idea obtained as generalization of the procedures proposed by Voss [7] and Keller [13]. With respect to this heuristic, we introduce a tabu structure characterized by a variable length tabu list to help the search to escape from local optimal solutions and a problem-specific procedure to improve the objective function value without losing feasibility. Moreover, in our algorithm the maximum number of consecutive markets which can be removed is not predefined as in [12] but changes during the search. Extensive computational results have shown that our heuristic is very efficient and, in those instances where budget constraint is not too tight (i.e. not too close to the minimum purchasing cost), it is also able to provide very effective solutions. It represents the immediate choice for a decision maker interested in a practical tool to obtain good solutions in few seconds.

The second procedure is based on a variable neighborhood search (VNS) scheme (see Mladenović and Hansen [14] and references therein). We generate a sequence of neighborhoods $N_q(·),$ $q=1,\dots ,q_{\max }$. For a given q and a given solution $s,$ a solution $s^{\prime }$ belonging to the neighborhood $N_q(s)$ is generated from s by inserting q random markets. Then a local search procedure is applied to $s^{\prime }$. At this aim we use a variant of the proposed enhanced local search algorithm. If no solution improvement is obtained, the parameter q is increased allowing the search to explore farthest and possibly more promising parts of the solution space. We also implement a variant of this VNS scheme where, if no solution improvement is obtained, a maximum number $t_{\max }$ of random solutions is generated in the same neighborhood $N_q(s)$ before the value of q can be changed (Multi-start VNS). The proposed VNS procedure and its variant are straightforward and produce very effective solutions. To test all the proposed algorithms we have introduced a broad class of benchmark problems obtained by inserting budget thresholds to the TPP instances proposed in [10]. We have solved instances with both tight and untight budget constraints and tested both the capacitated and the uncapacitated version of the problem. Our experimental analysis provides the first computational results available for the TPP with budget constraint. Both optimal (when available) and heuristic solutions of the solved benchmark problems can be downloaded at the web site http://www.ing.unibs.it/or-group/testproblems.htm.

The paper is organized as follows. In Section 1, we present the mathematical formulation of the TPP with budget constraint. The proposed solution methods are introduced in Section 2 where we also provide the description of a lower bound for the problem used to evaluate our algorithms performance. Section 3 is devoted to computational results. For the uncapacitated case we have used the instances solved in [11], where the authors deal with the UTPP with budget constraint to compute Pareto optimal solutions of a bi-objective TPP. For the capacitated version of the problem, since no optimal solutions are available, we compare the algorithms each other and with respect to the problem lower bound in terms of efficiency and effectiveness. Finally conclusions and future developments are drawn in Section 4.