In this paper we consider the problem of shipping several products from an origin to a destination, when a discrete set of shipping frequencies is available only, in such a way that the sum of the transportation and inventory cost is minimized This problem, which is known to be NP-hard, has applications in transportation planning and in location analysis. We derive some dominance rules for the problem solutions which allow to tighten the bounds on the problem variables. Moreover, we present a branch-and-bound algorithm and we evaluate its performance on randomly generated problem instances.

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Bertazzi, L.

Speranza, M.

Ukovich, W.

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65 An algorithm for the transportation problem with given frequencies L. Bertazzi, M. G. Speranza Dept. of Quantitative Methods - University of Brescia Corso G. Mameli, 27 - 1-25122 Brescia, Italy. Tel: 39-30-2988.1. Fax: 39-30-2400925. e-mail: speranza@master. cci. unibs. it W. Ukovich D.E.E.I.- University of Trieste Via A. Valerio, 10 - I - 34127 Trieste, Italy. Tel: 39-40-6767135. Fax: 39-40-6763460. e-mail: ukovich@univ.trieste.it Abstract In this paper we consider the problem of shipping several products from an origin to a destination, when a discrete set of shipping frequencies is available only, in such a way that the sum of the transportation and inventory cost is minimized. This problem, which is known to be NP-hard, has applications in transportation planning and in location analysis. We derive some dominance rules for the problem solutions which allow to tighten the bounds on the problem variables. Moreover, we present a branch-and-bound algorithm and we evaluate its performance on randomly generated problem instances. Keywords Transportation, inventory, dominance rules, branch-and-bound. 1 INTRODUCTION We consider the situation in which each product of a given set is made available at an origin and needed at a destination at a constant rate. A discrete set of available shipping frequencies is given and the shipment is carried out by trucks with given capacity. The problem is to decide how much of each product to ship at each frequency in such a way that the sum of the transportation and inventory cost is minimized. This problem was introduced in Speranza and Ukovich (1994) together with other models for shipping products from an origin to a destination. Similar models, where the shipping frequencies are allowed to take any continuous value, are considered in Burns et al (1984), Blumenfeld et al (1985), Anily and Federgruen (1990). The problem was shown to be NP-hard in Speranza and Ukovich (1991), where an optimal branch-and-J. Doležal et al. (eds.), System Modelling and Optimization© Springer Science+Business Media Dordrecht 1996536 Contributed Papers bound algorithm was presented. The computational results showed that the algorithm was efficient on problem instances of small/medium size. The more complex problem of shipping products from several origins to a destination, when a given set of shipping frequencies is available, was considered in Bertazzi, Speranza and Ukovich (1994), where different heuristics were presented, based on solving in a first phase a one origin - one destination problem for each of the given destinations, and then on improving the solution through local search techniques. In this paper, we derive some dominance rules for the solution of the one-origin-one-destination problem. Such rules allow to tighten the bounds on the problem variables. Then we present a new branch-and-bound algorithm in which we apply the dominance rules in order to improve its efficiency and we compare it with the branch-and-bound proposed in Speranza and Ukovich (1991). The paper is organized as follows. In Section 2 the problem is described and its math-ematical programming formulation is given. The dominance rules and their implications are presented in Section 3. In Section 4 the new branch-and-bound is described and the computational results are given in Section 5. 2 PROBLEM DESCRIPTION AND FORMULATION A set of products is made available at an origin and needed at a destination at a constant rate. Trucks, with given capacity, are available for shipping products, and each product can be partly shipped by trucks travelling at different frequencies. Each product can be continuously divided. Two cost factors are considered, namely the transportation and the inventory cost. Shipping products at a high frequency generates low inventory costs but high transportation costs, while shipping at a low frequency generates high inventory costs and low transportation costs. The problem is to decide the fraction of each product which is shipped at each frequency in such a way that the sum of the transportation and the inventory costs is minimized. The following notation is used. I: set of products; J: set of frequency indices; n: number offrequencies (=IJJ); i: products; j: index of frequency; Ii: jth frequency, with j E J; tj: period offrequency h (= l/h); H: time horizon (equal to the minimum common multiple of the periods); qi: rate at which product i is produced at the origin and needed at the destination; Vi: unit volume of product i; hi: inventory cost of one unit of product i per unit time; 1"j: capacity of each truck travelling at frequency fj; Cj: cost of a single trip of a truck travelling at frequency h; An algorithm for the transportation problem with given frequencies 537 Xij: fraction of product i shipped at the frequency h; Yj: number of trucks which travel at the frequency h· The problem can be formulated as follows. min L L hitjqiHxij + L cj(H/tj)yj iEI jEJ JEJ tj L viqiXij ::; 7'jYj iEI LXij = 1 JEJ iE/ jEJ 0< x < 1 _ tJ_ iE/,jEJ Yj ~ 0 integer j E J. (1) (2) (3) (4) (5) The ob jecti ve function (1) represents the sum of the inventory cost and the transportation cost on the time horizon H. Constraints (2) are capacity constraints, which state that the number of trucks Yj must be sufficient to load all the products assigned to frequency h. Constraints (3) impose that all the quantity of product i produced at the origin must be shipped to the destination at some of the given frequencies. The mixed integer programming problem defined through (1 )-(5) will be referred to as Problem P. In the following, we assume that the unit time is chosen in such a way that all tj are integer and that the frequencies are ordered in the decreasing order, that is i1 > h > .. , > in. Moreover, we assume that all trucks have the same capacity, independently of the frequency, that is that 7'j = 7', Vj, and, without loss of generality, we normalize the capacity to 7' = 1. As all trucks have the same capacity, we assume Cj = c, V j. 3 DOMINANCE CONDITIONS In this section we present the most important theoretical results we have obtained for the Problem P. For their formal proofs, we refer to Bertazzi, Speranza and Ukovich (1995). Dominance is a concept, common in multiobjective optimization, that turns out to be convenient for our problem. In fact, Problem P can be considered as stemming from an optimization problem with two conflicting objectives: minimize the inventory cost and minimize the transportation cost. In general terms, a feasible solution for a multiobjective optimization problems is dom-inated if another feasible solution exists with not worse performance with respect to each objective, and a strictly better performance with respect to at least one objective (d. for instance Keeney and Raiffa, 1976). The following result states a general condition guaranteeing that a solution is dominat-ed. 538 Contributed Papers Lemma 1 A solution (x, y) for Problem P is dominated if 3 0: E ~+ (the set of positive real numbers), j, k E J such that: (6) (7) (8) It is worth to point out that the above result generalizes the saturation property for Problem P presented in Speranza and Ukovich (1994): Corollary 1 Consider an optimal solution (x, y) for Problem P. Suppose 3j, k E J such that j > k, "Vi > O. Then Yk = Vk. As a consequence of the last result, in the sequel we always consider solutions satis-fying the saturation condition. Then Lemma 1 yields a more convenient condition for domination: Theorem 1 A feasible solution (x,y) for Problem P is dominated if 30: E ~+, j,k E J such that conditions (6) and (7) of Lemma 1 are satisfied, and (9) Corollary 2 A solution (x, y) for Problem P is dominated if yjtk/tj :c:: 1 for some j > k with tj /tk integer. Corollary 2 allows to tighten the bounds for the variables Yi> j > 1, whenever tj admits a proper submultiple in the given set of periods {tj}. Let us denote by J' <; J the set of indices of the periods which admit a proper submultiple. Then Yj ::; UEl j = m!n{tj/tk - 1: tj/tk is integer} j E J'. (10) In the case J' = J, then Yj ::; UElj, Vj, and the range for YI can be made tighter. In fact, n n YI :c:: ttC~= viqi - LYj/tj) :c:: tI(L viqi - L min(U EOj , U Blj)/tj) = LEI (11) j=2 j=2 where U EOj = r tj I:iEI viqi 1· Observe that, in this case (J' = J), the width of the range of the variables does not depend on the volume to be shipped, but on the periods only. Let us emphasize this feature. Since U Elj ::; tj -1, then YI :c:: rII(I:i Viqi -n+ 1 + I:j=21i)1 :c:: U EOI -tl (n-l). An algorithm for the transportation problem with given frequencies 539 A complete enumeration of the solutions generates a number of solutions not larger than (tt (n - 1) + 1 )t2 ... tn. This quantity depends on the number and on the values of the periods only, which in practice are quite moderate. Thus, for fixed n, such a "brute force" enumeration turns out to be just pseudo-polynomial. A different kind of bounds for the variables Yj, j > 1 can be inferred from a new domination result, which in turn is another consequence of Theorem 1: Corollary 3 A solution (x, y) for Problem P is dominated if V; ~ tj for some j > 1. As a consequence of the last result, then V; < tj in any optimal solution, hence rv; 1 = Yj :::; tj, Vj > 1. Furthermore, note that if Yj = tj, then the above Corollary still applies (i.e. the corresponding solution is still dominated), thus yielding a tighter upper bound for Yj, j > 1, equal to tj - 1. Another different dominance rule is given by the following theorem. which states that a solution is dominated if all products travelling at a certain frequency can be shipped by full load trucks at a higher frequency. Corollary 4 A solution (x, y) for Problem P is dominated if V;tk/tj is an integer positive number for some j > k. Corollary 4 can be further generalized as follows. Theorem 2 Given a solution (x,y), if a subset J of the frequencies and a frequency fk exist such that tj > tk, Vj E J, and l: "EJ V;tk/tj is an integer positive number, then the " J solution is dominated. 4 A BRANCH-AND-BOUND ALGORITHM In this section we present a branch-and-bound algorithm for the solution of the Problem P. In this algorithm we apply the dominance rules we have derived in Section 3; this algorithm will be compared in Section 5 with the branch-and-bound proposed in Speranza and Ukovich (1991) in order to evaluate its performance. The branch-and-bound works on a search tree where each level corresponds to a fre-quency fj and each node v represents the vehicles Yj used at this frequency. This number of vehicles belongs to a set of values defined in the following way. We recall that, by the Corollary 3, Yj < tj Vj > 1 and that, in the particular case in which the period tj admits a submultiple in the given set of the periods {tj}, by the Corollary 2, Yj:::; UBlj = mink{tj/tk -1: tj/tk is integer}. Furthermore, given a partial solution (Yt, ... , Yj-d, Yj :::; U B2j with U B2j = f(l:i Viqi -l:t;,~ Yk/tk)tl Therefore, the upper bound for Yj, j > 1, is, in general, Yj :::; min(tj - 1, U B2j) and, in the particular case in which tj has a submultiple in {tj}, Yj :s: min(tj-l,UBl j ,UB2j). The lower bound for Yj is LBI for j = I, as defined in (11) and Yj = 0 for all j > 1. The tree is explored in the depth first way. The fathoming criteria are the following. An upper bound U is initially set to the minimum value obtained by solving the heuristics presented in Bertazzi, Speranza and 540 Contributed Papers Ukovich (1995) on the instance of the problem; this upper bound mantains the best current upper bound during the algorithm. In each node v, a lower bound Lv is calculated by solving the relaxed problem by the Procedure Ext-Greedy presented in Speranza and Ukovich (1991). If the solution of the relaxed problem is integer, this node is fathomed. If Lv > U, the node v is fathomed. Furthermore, in each node v the dominance rules presented in Section 3 are applied; if the obtained solution is dominated, the node v is fathomed. When the number of vehicles Yj are fixed in all the level of the tree, if the solution is feasible, the optimal value is calculated and the node v is fathomed. If the optimal value is lower than the current upper bound U, then U is modified. 5 COMPUTATIONAL RESULTS The branch-and-bound proposed in Speranza and Ukovich (1991) and the new branch-and-bound have been implemented in FORTRAN on a Personal computer with an Intel 486dx/66 processor. The computational results have been obtained by generating random instances with the following characteristics: • set of frequencies: 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10; • number of products: 1000, 5000, 10 000; • quantity per product per unit time (qi): randomly generated between 0.1 and 5, between 5 and 100 and between 0.1 and 100; • unit volume per product (Vi): randomly generated between 0.001 and 0.01; • unit inventory cost per product (hi): randomly generated between 0.001 and 0.05, between 0.05 and 1 and between 0.001 and 1; • transportation cost per trip (c): 300. The results are shown in Table 1. Each row corresponds to 5 problem instances with the characteristics described in columns 1-3. Column 1 gives the number of products; Column 2 gives the range of the quantity per product and Column 3 the range of the inventory cost hi. Columns 4-7 show the computational results obtained by the branch-and-bound pro-posed in Speranza and Ukovich (1991), referred as Old branch-and-bound; Column 4 gives the average number of visited nodes (the maximum number of possible nodes has been fixed in 30 000); Column 5 gives the number of times we have obtained the optimal solution on 5 instances; Column 6 the node in which the optimal solution has been found (average on the 5 instances); finally, Column 7 gives the average CPU time (minutes and second-s). In columns 8-11 the computational results obtained by the new branch-and-bound, referred as New branch-and-bound, are shown as in columns 4-7. The computational results show that the new branch-and-bound is more efficient than the one proposed in Speranza and Ukovich (1991), both in terms of number of visited nodes and CPU time. An algorithm for the transportation problem with given frequencies 541 Table I Comparison of the branch-and-bound algorithms Parameters Old branch-and-bound New branch-and-bound Prod. Quant. Inventory Nodes N Opt. Time Nodes N Opt. Time 1000 0.1-5 0.001-0.05 13 784 3 1064 05.09 3431 5 1031 00.53 1000 0.1-5 0.05-1 148 5 125 00.02 51 5 37 00.00 1000 0.1-5 0.001-1 192 5 160 00.03 136 5 114 00.02 1000 5-100 0.001-0.05 22290 3 4676 06.44 8411 5 2021 01.41 1000 5-100 0.05-1 1363 5 431 00.26 410 5 185 00.06 1000 5-100 0.001-1 3996 5 1899 01.23 1100 5 728 00.18 1000 0.1-100 0.001-0.5 9274 4 1387 02.47 1697 5 589 00.22 1000 0.1-100 0.05-1 949 5 221 00.18 227 5 47 00.03 1000 0.1-100 0.001-1 1970 5 1123 00.39 644 5 392 00.10 5000 0.1-5 0.001-0.05 15255 3 1175 16.37 6251 5 1344 03.55 5000 0.1-5 0.05-1 3341 5 944 03.36 681 5 300 00.31 5000 0.1-5 0.001-1 1321 5 1090 01.26 610 5 507 00.30 5000 5-100 0.001-0.05 30000 0 0 34.28 19070 5 690 12.15 5000 5-100 0.05-1 2070 5 434 01.58 249 5 117 00.14 5000 5-100 0.001-1 24608 1223 26.56 4776 5 666 03.06 5000 0.1-100 0.001-0.05 30000 0 0 34.41 13097 5 645 08.33 5000 0.1-100 0.05-1 2212 5 305 02.08 364 5 122 00.18 5000 0.1-100 0.001-1 24 798 1 92 27.08 6588 5 368 04.68 10000 0.1-5 0.001-0.05 30000 0 0 59.02 10454 5 1730 12.01 10 000 0.1-5 0.05-1 3097 5 683 06.17 854 5 211 01.13 10 000 0.1-5 0.001-1 2562 5 689 05.13 850 5 345 01.14 10 000 5-100 0.001-0.05 4079 5 1306 07.18 584 5 515 00.52 10000 5-100 0.05-1 3388 5 217 05.52 244 5 75 00.24 10000 5-100 0.001-1 3615 5 794 06.21 418 5 348 00.39 10 000 0.1-100 0.001-0.05 4722 5 1954 08.42 803 5 577 01.10 10000 0.1-100 0.05-1 3917 5 377 07.00 494 5 157 00.45 10000 0.1-100 0.001-1 4725 5 1995 08.43 804 5 649 01.10 REFERENCES Anily, S. and Federgruen, A. (1990) One Warehouse Multiple Retailer Systems with Ve-hicle Routing Costs. Management Science, 36, 92-114. Bertazzi, 1., Speranza, M.G. and Ukovich, W. (1994) Minimization of logistic costs with given frequencies. Technical Report 79, Department of Quantitative Methods, Univer-sity of Brescia (submitted). Bertazzi, 1., Speranza, M.G. and Ukovich, W. (1995) Dominance Rules and Approxi-mate Solutions for the Transportation Problem with Discrete Available Frequencies, (in preparation). Blumenfeld, D.E., Burns, L.D., Diltz, J.D. and Daganzo, C.F. (1985) Analyzing trade-offs between transportation, inventory and production costs on freight networks. Trans-portation Research, 19B 361-80. 542 Contributed Papers Burns, L.D., Hall, R.W., Blumenfeld, D.E. and Daganzo, C.F. (1984) Distribution S-trategies that Minimize Transportation and Inventory Cost. Operations Research, 33, 469-90. Keeney, R. 1., and Raiffa, H. (1976) Decisions with multiple objectives. Wiley, New York. Speranza, M.G. and Ukovich, W. (1991) A capacitated transportation problem with fac-toring costs. Technical Report 26, Department of Quantitative Methods, University of Brescia (submitted). Speranza, M.G. and Ukovich, W. (1994) Minimizing transportation and inventory costs for several products on a single link. Operations Research, 42, 879-94. 

An algorithm for the transportation problem with given frequencies

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