Exact and Heuristic Solutions for a Shipment Problem with Given

Exact and Heuristic Solutions for a Shipment Problem with Given Frequencies Luca Bertazzi • Maria Grazia Speranza • Walter Ukovich Dipartimento di Metodi Quantitativi, University of Brescia, Italy Dipartimento di Metodi Quantitativi, University of Brescia, Italy Dipartimento di Elettrotecnica, Elettronica ed Informatica, University of Trieste, Italy [email protected] • [email protected] • [email protected]

W

e consider the problem of shipping several products from an origin to a destination when a discrete set of shipping frequencies is available, in such a way that the sum of the transportation and inventory costs is minimized. This problem, which is known to be NP-hard, has applications in transportation planning and in location analysis. In this paper we derive dominance rules for the problem solutions that allow a tightening of the bounds on the problem variables and improve the efficiency of a known branch-and-bound algorithm. Moreover, we present some heuristics and compare them with two different modifications of an EOQ-type algorithm for the solution of the problem with continuous frequencies. (Transportation; Inventory; Dominance Rules; Branch-and-Bound; Heuristics)

Introduction A set of products is made available at an origin and demanded at a destination at a constant rate. A discrete set of available shipping frequencies is given and the shipments are 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 has applications in transportation planning and in location analysis. For instance, the origin may represent a consolidation center and the destination a warehouse. A group of suppliers regularly sends products to the consolidation center, and from there they are shipped to the warehouse. In the case where several alternative locations exist for the consolidation center, the results of this paper could be used for assessing the operational costs for each possible location. This problem was introduced in Speranza and Ukovich (1994b) together with other models for shipping products from an origin to a destination. The practical interest for the case in which the set of possible 0025-1909/00/4607/0973$05.00 1526-5501 electronic ISSN

shipping frequencies is given has been acknowledged, for instance, by Hall (1985), Maxwell and Muckstadt (1985), Jackson et al. (1988), and Muckstadt and Roundy (1993). As they point out, shipping anything every 公2 days is often impractical to implement. At least, it is reasonable to assume that a minimum interval between shipments exists, and usable shipping frequencies correspond to integer multiples of such a base period (cf. again Muckstadt and Roundy 1993). In this case, the solution provided by models where shipping frequencies can be any real number, such as those proposed by Burns et al. (1985) and by Blumenfeld et al. (1985), may turn out to be infeasible. Imposing feasibility by rounding off a continuous solution, as suggested by Hall (1985), often fails to produce the true optimum and, in some cases, produces costs that are significantly larger than the minimum. A simple example has been thoroughly discussed in Speranza and Ukovich (1994b), where the cost of the rounded solution turns out to be more than 10% larger than the optimum. A systematic analysis of the cost increase produced in this way has been Management Science © 2000 INFORMS Vol. 46, No. 7, July 2000 pp. 973–988

BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

carried out in Speranza and Ukovich (1994a) on an experimental basis. The problem with discrete frequencies was shown to be NP-hard in Speranza and Ukovich (1996), where a polynomial transformation to it from the Integer Knapsack problem has been produced. In the same paper, an optimal branch-and-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, or, symmetrically, from a single origin to several destinations, has been studied by several authors, such as Federgruen and Zipkin (1984), Burns et al. (1985), Daganzo (1985, 1988), Daganzo and Newell (1985), Federgruen et al. (1986), and Anily and Federgruen (1990a, b, 1993), among others. However, in all these papers continuous frequencies are allowed. The results available for this problem have been discussed in more detail in Anily and Federgruen (1990a) and in Bertazzi et al. (1997). The case with a given discrete set of frequencies has been considered in Bertazzi et al. (1997), 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. The branch-and-bound algorithm of Speranza and Ukovich (1996) was used for the solution of the one-originone-destination problems. The computational results of the heuristics were satisfactory for problems with up to 20 destinations and 5 products, both in terms of quality of the solution and in terms of solution time. However, the use of the branch-and-bound algorithm made the computational time for problems of larger size too large to allow an extensive computational experience. This motivated studying effective and efficient heuristic procedures for the problem with one origin and one destination. In this paper, we derive some dominance rules for the problem solutions. We show that a solution which satisfies certain conditions is dominated by a solution which has strictly lower inventory cost and not greater transportation cost. The following example may be of

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help in explaining the concept of dominance and the way we use it. Example 1. Suppose shipments can be performed either every day or every 2 days, and a total volume equal to 1.5 is produced at the origin every day and needs to be shipped. If all products are shipped every day, then 2 trucks are needed at frequency 1, that is every day, and 0 at frequency 21 . If all products are shipped at frequency 21 , then 3 full-load trucks are needed every two days. In the latter case the transportation cost is lower but the inventory cost is certainly higher than in the former case. Then it is not possible to identify on the basis of the given data only— frequencies and volume—which solution is better. However, if 1 truck travels every day, only 1 truck is needed every two days. This solution dominates the solution with 3 trucks every two days, because it has the same transportation cost, but lower inventory cost, and such a conclusion does not depend on all the data of the problem, but only on volumes and available frequencies. On the contrary, the solution with 2 trucks every day neither dominates nor is dominated by the solution with 1 truck every day and 1 truck every two days, because the latter solution has lower transportation cost but higher inventory cost. Dominance rules allow a tightening of the bounds on the problem variables and can be used to improve the efficiency of the branch-and-bound algorithm. Moreover, we propose some heuristics and consider two different modifications of the EOQ-type algorithm given in Blumenfeld et al. (1985) for the solution of the problem where any continuous shipping frequency is allowed. The solutions obtained by the heuristics are compared with the optimal solution for problems with up to 10,000 products and 15 frequencies. The computational results show that one of the proposed heuristics generates smaller errors than the others with respect to the optimum. The paper is organized as follows. In §1 the problem is described and its mathematical programming formulation is given. The dominance rules and their implications are presented in §2. The new branch-andbound algorithm is described in §3, while the computational experience is presented in §4. Then §5 discusses the practical impact of relaxing some

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

assumptions of the model considered in this paper. It is numerically shown that the restrictions induced by the assumptions we use are not too severe. Finally, in §6 the heuristics are described and the computational results are given in §7.

Variables x ij fraction of product i which is shipped at frequency f j y j number of trucks which travel at frequency f j . The problem can be formulated as follows.

冘 冘 h t q x ⫹ 冘 共c /t 兲y , t 冘 vqx ⱕry j 僆 J, 冘 x ⫽ 1 i 僆 I,

min

1. Problem Description and Formulation A set of products is made available at an origin and demanded 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 traveling at different frequencies. Each product can be continuously divided. It is assumed that, for each product, the production rate is equal to the demand rate. Two cost factors are considered, namely the transportation and the inventory cost (which is charged in the same way both at the origin and at the destination). All shipments having the same frequency are supposed to be simultaneous, i.e., performed at the same time. For each frequency, an unlimited number of trucks is available. The problem is to decide the fraction of each product which has to be 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. Data I set of products J set of frequency indices n number of frequencies (n ⫽ 兩J兩) i index of product (i 僆 I) j index of frequency ( j 僆 J) f j jth frequency, with j 僆 J t j headway, or period, corresponding to frequency f j (t j ⫽ 1/f j ) q i rate at which product i is produced at the origin and demanded at the destination v i unit volume of product i h i inventory cost of one unit of product i per unit time r j capacity of each truck traveling at frequency f j c j cost of a single trip of a truck traveling at frequency f j .

Management Science/Vol. 46, No. 7, July 2000

i j i ij

j

i僆I j僆J

j

j

j

(1)

j僆J

i i ij

j j

(2)

i僆I

ij

(3)

j僆J

0 ⱕ x ij ⱕ 1 yj ⱖ 0

i 僆 I, j 僆 J,

integer

j 僆 J.

(4) (5)

The objective function (1) represents the sum of the inventory cost and the transportation cost per unit time. Constraints (2) are capacity constraints, which state that the number of trucks y j must be sufficient to load all the products assigned to frequency f j . Constraints (3) impose that the whole 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 ᏼ. It is worth pointing out that, as it has already been observed in Speranza and Ukovich (1996), if the variables y j are restricted to be either 0 or 1, then a special case of the Capacitated Plant Location Problem (CPLP), or of the Simple Plant Location Problem (SPLP) is obtained. (It was interesting to discover that while the latter is an almost trivial problem, the former one is NP-hard; cf. Speranza and Ukovich 1996.) However, Problem ᏼ is not a special case of a CPLP, since in it the variables y j can take any integer value, while in the location problems y j can be only 0 or 1. Of course, Problem ᏼ can be reduced to a special case of an ordinary CPLP by multiplying the warehouse locations, but this is a pseudo-polynomial transformation. So Problem ᏼ is a variant of CPLP, with costs c ij for variables x ij given by c ij ⫽ h i q i t j , and variables y j taking any integer value. These two particular features make it meaningfully different from both CPLP and SPLP, as it is also proved by the just-mentioned complexity results.

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

In the following subsection, we assume that the unit time is chosen in such a way that all t j are integers and that the frequencies are ordered in nonincreasing order, that is f 1 ⱖ f 2 ⱖ . . . ⱖ f n . Moreover, we assume that all trucks have the same capacity, independently of the frequency, that is r j ⫽ r, @j, and, without loss of generality, we normalize the capacity to r ⫽ 1. As all trucks have the same capacity, we assume c j ⫽ c, @j. 1.1. Model Features It seems appropriate to discuss to some extent some relevant features of the model expressed by Problem ᏼ. It complies with the following assumptions. Constant and Known Demand. This is the most classic and common assumption since 1913 (cf. Harris 1913). A great deal of literature exists that discusses such an assumption (see relevant references, among the others, in Chika´ n 1990, Lee and Nahmias 1993, and Baita et al. 1998), so it seems un-necessary to go further through this topic here. Periodicity. Shipments are supposed to be performed according to a periodic pattern. There is a given set of available periods that can be used for shipments. It is the duty of the decision maker to decide which shipment frequencies to use (possibly several), among those available, for each product type. Moreover, the decision maker must also decide, once and for all, the quantity of each product to be shipped at each frequency. This implies that each time a shipment with a certain frequency is made, the same constant quantity is shipped. This feature is the same as the one used in order-quantity models and strategies (cf., for instance, Silver and Peterson 1985 or Lee and Nahmias 1993). More generally, this is a classic assumption for the so-called “frequency domain” approach to inventory problems (cf. again Baita et al. 1998). It is used in several models, e.g., among others, in Anily and Federgruen (1990a), where an interesting discussion on this kind of assumptions and different motivations for them are presented. Also in this case, the reader may be referred to these discussions for a thorough analysis of these issues. Relaxing the constant quantity assumption would

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shift the model towards a “time domain” approach (cf. again Baita et al. 1998), which has several interesting properties, but is outside the scope of the present paper. Nevertheless, in §5 an experimental assessment of how such kind of assumptions may affect the performance of the shipping strategies is presented. Simultaneous Shipments. As pointed out above, all shipments having the same frequency are supposed to be performed at the same time. Moreover, shipments taking place at different frequencies are coordinated in such a way that there is a time instant at which all shipments relative to each used frequency are performed. In other words, “staggering” is not contemplated. The simultaneous shipments assumption has several justifications. It may constitute an operative constraint when prescheduled trips of some carrier are used for shipments. For instance, a ship may leave a port each Friday. So the period is 7 (days). Then the decision is just how many containers to load on it. If one decides to ship 2 containers each week, for example, they will necessarily leave simultaneously, i.e., on Friday, since there are no ships calling on the port any other day of the week. Clearly, relaxing the simultaneous shipments assumption may allow reduction of costs, in particular inventory costs, without raising transportation costs. However, the solution provided by the model of Problem ᏼ is a feasible one also for the case in which staggering is allowed. Moreover, any solution for Problem ᏼ may be easily improved by introducing staggering: It suffices to “shift” shipments that use periods having a largest common divider larger than unity (thus including isofrequency shipments, when frequency is not equal to unity). It is worth pointing out that an optimal solution for Problem ᏼ with no more than one shipment per frequency remains optimal also when the simultaneous shipments assumption is relaxed, if the feasible shipment periods have a largest common divider equal to unity. The issues involved by the simultaneous shipments assumption have been addressed by Hall (1991) in his comments on the paper by Anily and Federgruen (1990a). Hall produces a simple example showing that if deliveries are coordinated, cost can be reduced. In their rejoinder, Anily and Federgruen (1991) confirm

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

that in their model deliveries are never coordinated, as it is in our case, and justify this feature of their model. Other models that implicitly assume simultaneous shipments can be found in Nahmias (1992).

2. Dominance Results In Speranza and Ukovich (1996) it is shown that Problem ᏼ is NP-hard. Furthermore, in Speranza and Ukovich (1994b) it is shown that in any optimal solution all the trucks carry a full load, with the only possible exception of one nonfull truck traveling at the smallest used frequency (the saturation property). If the number of trucks y៮ j at each frequency f j is given, then, by a suitable variable substitution, Problem ᏼ can be transformed into a transportation problem in which transportation costs of the variables x ij have a product form; that is they are the product of a quantity (h i q i ), depending on the index i only, and of another quantity (t j ), depending on the index j only. In this case, the problem can be solved by a greedy algorithm, which first orders the products in nonincreasing order of the inventory cost per unit volume h i /v i , and the frequencies in decreasing order, and then assigns products to the available trucks, according to the given orderings (Speranza and Ukovich 1996). Furthermore, a partially relaxed version of Problem ᏼ in which some variables y j , j 僆 J 1 傺 J are fixed to given values, the other variables y j , j 僆 J 2 ⫽ J⶿J 1 are relaxed and may take any real value, and t j ⱕ t k @j 僆 J 1 , k 僆 J 2 can be solved using the Procedure Ext-Greedy proposed in Speranza and Ukovich (1996). The Procedure Ext-Greedy first orders products and frequencies as before, then assigns products to the frequencies in J 1 , according to the given orderings. At this point, if all the frequencies of J 1 are saturated, and some products still have to be assigned, the remaining products i are assigned in turn to the most profitable frequency f j , with j 僆 J 2 , i.e., to the frequency f j such that g ij ⫽ min k g ik , where hi cj g ij ⫽ t j ⫹ . vi rj In this way, the set I of the products turns out to be split into two parts: I ⫽ I 1 艛 I 2 . The products of I 1

Management Science/Vol. 46, No. 7, July 2000

have the highest h i /v i ratios, and are assigned to frequencies in J 1 . The other products are in I 2 , and are assigned to frequencies in J 2 (there may be an intermediate product, shared between J 1 and J 2 ). For a thorough discussion on the Procedure Ext-Greedy, the interested reader is referred again to Speranza and Ukovich (1996). Conversely, if the fraction of each product traveling at each frequency is given, that is if the values x៮ ij are given, then the values of the variable y៮ j are immediately determined as y៮ j ⫽ t j

冘 v q x៮

.

i i ij

i僆I

In general, the number of trucks needed to ship all the products at the frequency f j represents a trivial upper bound on the variables y j : y j ⱕ UB0 j ⫽ t j

冘 vq

j 僆 J.

i i

(6)

i僆I

Obviously, given a partial solution y៮ 1 , . . . , y៮ j⫺1 , y j ⱕ UB1 j ⫽

冉冘

冘 yt៮

j⫺1

v iq i ⫺

i

k⫽1

k

k

冊

tj

j 僆 J.

(7)

The above bounds can be made tighter by means of the dominance results derived in the following. Let us define ៮ s ⫽ ts V

冘 vqx , i i is

i

the total volume shipped at frequency f s . The following result states a general condition guaranteeing that a solution is dominated. Lemma 1. A solution ( x៮ , y៮ ) for Problem ᏼ is dominated if ?j, k 僆 J, ␣ 僆 ᑬ ⫹ (the set of positive real numbers), such that: k ⬍ j,

␣ⱕ

៮j V , tj

៮ j ⫺ ␣ t j  V ៮ k ⫹ ␣ t k  y៮ j y៮ k V ⫹ ⱕ ⫹ . tj tk tj tk

(8) (9)

(10)

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

Proof. Take yˆ 僆 ᑬ n as follows: yˆ s ⫽

再

៮ j ⫺ ␣ t j  if s ⫽ j, V ៮ k ⫹ ␣ t k  if s ⫽ k, V otherwise. y៮ j

erty. Lemma 1 can be rewritten to take this property into account. (11)

In this way some products, whose global daily volume is ␣, are moved from frequency j (in y៮ ) to frequency k (in yˆ ). This reduces the inventory cost resulting from (8). The new solution yˆ is a feasible solution, since in (11) yˆ j ⱖ 0 due to (9). Then (10) says that the transportation cost per unit time in yˆ is not larger than the transportation cost per unit time in y៮ . 䊐 It is worth pointing out that the above result generalizes the saturation property for Problem ᏼ (which was first proven in Speranza and Ukovich 1994b): Corollary 1. Consider an optimal solution ( x៮ , y៮ ) for ៮ j ⬎ 0. Problem ᏼ. Suppose ?j, k 僆 J such that j ⬎ k, V ៮ Then y៮ k ⫽ V k . ៮ k ⬍ y៮ k . Then take ␣ ⫽ min{V ៮ j /t j , Proof. Suppose V ៮ ( y៮ k ⫺ V k )/t k }. Condition (9) is satisfied. Let us separately consider two cases: ៮ j /t j ⱕ ( y៮ k ⫺ V ៮ k )/t k . In this case 1. ␣ ⫽ V ៮ j ⫺ ␣ t j  V ៮ k  y៮ j y៮ k ៮ k ⫹ ␣ t k  V ៮ k ⫹ y៮ k ⫺ V V ⫹ ⱕ ⱕ ⫹ , tj tk tk tj tk and condition (10) is satisfied. ៮ k)/t k ⱕ V ៮ j/t j. Then 2. ␣ ⫽ (y៮ k ⫺ V t ៮ j ⫺ 共y៮ k ⫺ V ៮ k兲 j V ៮ ៮ V j ⫺ ␣ t j  V k ⫹ ␣ t k  tk y៮ k ⫹ ⫽ ⫹ tj tk tj tk ⱕ

៮ j  y៮ k V ⫹ tj tk

ⱕ

y៮ j y៮ k ⫹ , tj tk

and Condition (10) holds. Since (8) holds in both cases, Lemma 1 can be applied. It follows that ( x៮ , y៮ ) is dominated and hence cannot be optimal. This shows the contradiction. 䊐 As a consequence of the last result, in the sequel we only consider solutions satisfying the saturation prop-

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Theorem 1. A feasible solution ( x៮ , y៮ ) for Problem ᏼ is dominated if ?j, k 僆 J, ␣ 僆 ᑬ ⫹ , such that Conditions (8) and (9) of Lemma 1 are satisfied, and t ៮ j ⫺ ␣ t j  ⫹  ␣ t k  j ⱕ y៮ j . cˆ T ⫽ V (12) tk Proof. Condition (12) is a consequence of Condition (10) of Lemma 1 since, according to the saturation ៮ k ⫹ ␣ t k  ⫽ y៮ k ⫹ ␣ t k  ⫽ y៮ k ⫹  ␣ t k  (since property, V y៮ k is an integer). 䊐 The above result allows us to derive some dominance rules by selecting particular values for the parameter ␣. Corollary 2. A solution ( x៮ , y៮ ) is dominated if y៮ j t k /t j ⱖ 1 for some j ⬎ k, with t j /t k integer. ៮ j /t j , 1/t k }, Condition (9) Proof. Taking ␣ ⫽ min{V is satisfied. Let us consider separately two cases. ៮ j /t j ⱕ 1/t k . This implies V ៮ j t k /t j ⱕ 1 and 1. ␣ ⫽ V ៮ j t k /t j  ⫽ 1, since y៮ j t k /t j ⱖ 1. In this case thus V ៮j cˆ T ⫽ V

tk tj tj ⫽ ⱕ y៮ j , tj tk tk

and Condition (12) is satisfied. ៮ j/t j. In this case, 2. ␣ ⫽ 1/t k ⱕ V ៮ j ⫺ t j /t k  ⫹ t j /t k cˆ T ⫽ V ៮ j  ⫺ t j /t k ⫹ t j /t k ⫽ V ៮ j  ⫽ y៮ j , ⫽ V since t j /t k is assumed to be an integer and Condition (12) is satisfied. Since (8) holds in both cases, then Theorem 1 applies. 䊐 In plain words, the above result says that a solution is dominated if part of the products shipped at frequency f j can be shipped by full load trucks at a higher frequency f k such that t j is a multiple of t k . It follows that if frequency f j satisfies these conditions in any nondominated solution ( x, y), then y j ⬍ t j /t k , i.e. y j ⱕ t j /t k ⫺ 1. Corollary 2 allows us to tighten the bounds on the

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

variables y j , j ⬎ 1 whenever t j admits a proper submultiple in the given set of periods {t j }. Let us denote by J⬘ 債 J the set of indices of the periods which admit a proper submultiple. Then y j ⱕ UB2 j ⫽ min k

再

tj tj ⫺ 1: is integer tk tk

冎

y1 ⱖ t1 j 僆 J⬘.

(13)

A different kind of bound for the variables y j , j ⬎ 1 can be derived from a new dominance rule, which follows from Theorem 1. Corollary 3. A solution ( x៮ , y៮ ) for Problem ᏼ is dominated if y៮ j ⱖ t j for some j ⬎ 1. Proof. Condition (8) obviously holds with any k ៮ j /t j , 1}, and Condition (9) ⬍ j. Then take ␣ ⫽ min{V also holds. Now consider two cases: ៮ j ⬍ t j : then ␣ ⫽ V ៮ j /t j and y៮ j ⫽ t j since y៮ j 1. V ៮ j  ⱕ t j ; in this case, ⫽ V ៮ j /t j t j /t k ⱕ t k t j /t k ⫽ t j ⫽ y៮ j , cˆ T ⫽ t k V and Theorem 1 applies. ៮ j ⱖ t j: then ␣ ⫽ 1; in this case, 2. V ៮ j ⫺ t j  ⫹ t k t j /t k ⫽ V ៮ j  ⫺ t j ⫹ t j ⱕ y៮ j , cˆ T ⫽ V and Theorem 1 applies again. 䊐 As a consequence of the last result, in any optimal solution y j ⱕ t j ⫺ 1, for all j ⬎ 1. Then we have a new upper bound UB2 j ⫽ t j ⫺ 1,

j 僆 J⶿J⬘.

(14)

For the variables y j , with j 僆 J⬘, the bound given in (13) is tighter. By using the upper bounds obtained for the variables, the range of values for y 1 can be made tighter. In fact, y1 ⱖ t1

冉冘 冉冘 i

ⱖ t1

i

冘 yt n

v iq i ⫺

j⫽2

j

j

冊

冘 min共UB0t , UB2 兲 j

j⫽2

冉

冘 v q ⫺ n ⫹ 1 ⫹ 冘 1/t n

i i

i

j⫽2

j

j

⫽ LB1 .

Management Science/Vol. 46, No. 7, July 2000

冊 (15)

j

冊

ⱖ UB0 1 ⫺ t 1 共n ⫺ 1兲. A complete enumeration of the solutions generates a number of solutions not larger than 共t 1 共n ⫺ 1兲 ⫹ 1兲t 2 · · · t n . This quantity depends on the number and on the values of the feasible headways only, which in practice are quite moderate. The following result holds. Theorem 2. For any fixed n, Problem ᏼ can be solved in pseudo-polynomial time. The range for the y j can be made tighter if each period is a multiple of the previous one, i.e., if t j /t j⫺1 is an integer, j ⬎ 1. As an example, consider the case in which replenishment order intervals must be power-of-two multiples of some basic interval (cf. for instance Maxwell and Muckstadt 1985, and Roundy 1989). In this case, from Corollary 2 it must be y j ⱕ (t j /t j⫺1 ) ⫺ 1, for j ⬎ 1, and it follows that y1 ⱖ t1 ⱖ t1

冉

冘 v q ⫺ 冘 共t /t t 兲 ⫺ 1 n

j

j⫺1

i i

i

j

j⫽2

冊

冘 v q ⫺ 1 ⫹ tt . 1

i i

n

i

In this case (t j /t j⫺1 is an integer, j ⬎ 1), a complete enumeration would generate a number of solutions not larger than 2

n

v iq i ⫺

Observe that 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 UB2 j ⱕ t j ⫺ 1, @j, then

tn t2 ··· . t1 t n⫺1

A different dominance rule is given by the following result which states that a solution is dominated if all products traveling at a certain frequency can be shipped by full-load trucks at a higher frequency.

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

Corollary 4. A solution ( x៮ , y៮ ) for Problem ᏼ is ៮ j t k /t j is a positive integer for some j ⬎ k. dominated if V ៮ j /t j , Condition (9) holds. Proof. Taking ␣ ⫽ V ៮ ៮ j t k /t j )t j /t k ⫽ V ៮ j ⱕ V ៮ j Then cˆ T ⫽ V j t k /t j t j /t k ⫽ (V ⫽ y៮ j , and Condition (12) holds. Since Condition (8) also holds, Theorem 1 applies. 䊐 The hypotheses of Corollary 4 may be satisfied in cases in which Corollary 2 does not apply and one truck at frequency f j carries a partial load (for exam៮ j ⫽ 3.5, t j ⫽ 7, and t k ⫽ 2), and in cases in ple, V which Corollary 2 does not apply and the trucks at ៮ j ⫽ 3, t j frequency f j carry a full load (for instance, V ⫽ 3, and t k ⫽ 2). This proves that the conditions of the two results are independent, in the sense that neither is a consequence of the other one. Given any partial solution, both dominance rules given by Corollaries 2 and 4 can be checked in O(n 2 ). Corollary 3 can be checked in O(n). It is worth pointing out that in both the situations considered by Corollaries 2 and 4, it happens that a solution is dominated when part of the products shipped at a certain frequency f j can be shipped by full-load trucks at a higher frequency f k , thus reducing the inventory cost without increasing the transportation cost. However, this condition is not sufficient in general to guarantee dominance, as the following example shows. Example 2. Two frequencies are available: f 1 ⫽ 1/19 and f 2 ⫽ 1/ 20. The volume produced in unit time is equal to 1/10. Assume c ⫽ 1. A solution of the problem is y៮ 1 ⫽ 0, y៮ 2 ⫽ 2. The transportation cost per unit time is equal to 2/20. If part of the products are shipped with a full-load truck at frequency f 1 , then the solution yˆ 1 ⫽ 1, yˆ 2 ⫽ 1 is obtained, with transportation cost equal to 1/19 ⫹ 1/20 ⬎ 2/20. In this case the transportation cost increases, although part of the products are shipped by full-load trucks at a higher frequency. Corollary 4 can be generalized as follows. Theorem 3. Given a solution ( x៮ , y៮ ), if a subset J៮ of the frequencies and a frequency f k exist such that t j ⬎ t k , @j ៮ j t k /t j is a positive integer, then the solution 僆 ៮J , and ¥ j僆J៮ V is dominated. Proof. A new solution xˆ ,yˆ with lower or equal

980

transportation cost and lower inventory cost can be obtained by modifying the number of trucks at frequencies f j , j 僆 J៮ , and f k as follows: ៮ j yˆ j ⫽ y៮ j ⫺ V yˆ k ⫽ y៮ k ⫹

j 僆 J៮ ,

冘 V៮t t . j k

j僆J៮

j

In the new solution the inventory cost is lower as some products are shipped at a higher frequency than in ( x៮ , y៮ ). The difference ⌬ between the transportation cost in the new solution and the transportation cost in ( x៮ , y៮ ) is ⌬⫽c

冉冘 冊 冉冘 冊 j僆J៮

៮ jt k 1 V ⫺c tj tk

j僆J៮

៮ j V ⱕ0 tj

and then the new solution dominates the solution ( x៮ , y៮ ). 䊐

3. An Improved Branch-and-Bound Algorithm In this section we present a new branch-and-bound algorithm for the solution of the Problem ᏼ. In this algorithm we apply the dominance rules we derived in §2. It will be compared in §4 with the branch-andbound proposed in Speranza and Ukovich (1996) to evaluate its performance. The branch-and-bound algorithm works on a search tree where each level corresponds to a frequency f j and each node at level j represents the vehicles y j used at frequency f j . With reference to the results of §2, we recall that, given a partial solution y៮ 1 , . . . , y៮ j⫺1 , it is y j ⱕ min(UB1 j , UB2 j ), where UB1 j is defined in (7) and UB2 j is defined in (13) and (14). The lower bound for y j is LB 1 for j ⫽ 1, as defined in (15), and 0 for all j ⬎ 1. The tree is explored in the depth first way. In each node at level j, the value for y j is determined as follows. At the first level, we first choose the maximum value in the range for y 1 defined above and then the other values in decreasing order. At the other levels, with the exception of the leaves of the tree, i.e., the nth level, the value for y j is chosen in increasing order, starting from the minimum value in the abovedefined range. Finally, in the leaves of the tree only the

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

value for y n corresponding to the number of vehicles needed to ship all the products not yet shipped is possible. If such value is greater than the upper bound for y n , the node can be fathomed. The following fathoming criteria are applied. An upper bound U on the optimum is initially set to the minimum value obtained by solving the set of heuristics described in §6. This upper bound is updated during the algorithm execution and maintains the best current value of the objective function. In each node s, a lower bound L s is calculated by solving the relaxed problem by the Procedure Ext-Greedy presented in Speranza and Ukovich (1996). If L s ⬎ U, the node s is fathomed. Furthermore, in each node s the dominance rules stated in Corollaries 2, 3, and 4 are tested; if the partial solution associated with the node is dominated, the node s is fathomed. When the value of y j is fixed, for all j, the optimal values of the x variables are obtained through the algorithm described at the beginning of §2.

4. Computational Results for the Branch-and-Bound Algorithm The branch-and-bound algorithm described in the previous section has been compared with the branchand-bound presented and tested in Speranza and Ukovich (1996). In principle, the latter algorithm operates like the new one, as specified in §3, except that no bounds nor dominance rules like the ones presented in §2 are used. For a thorough discussion on the old branch-and-bound algorithm, the interested reader is referred to Speranza and Ukovich (1996). Table 1

The two algorithms have been implemented in Fortran and the experiments have been carried out on a personal computer with an Intel Pentium II processor. The computational results have been obtained for randomly generated instances with the following characteristics. • Set of frequencies: four sets of frequencies have been tested. The first three sets of frequencies contain frequencies of the form 1/k, with 1 ⱕ k ⱕ 5, 1 ⱕ k ⱕ 10, 1 ⱕ k ⱕ 15 in the first, second, and third set, respectively. The fourth set contains frequencies of the form 1/ 2 k , with 0 ⱕ k ⱕ 10 (as considered in Roundy 1989); • Number of products: 10, 100, 1000, 10000; • Quantity per product per unit time (q i ): randomly generated in two different intervals (0.1–5) and (5– 100); • Unit volume per product (v i ): randomly generated between 0.001 and 0.01; • Unit inventory cost per product (h i ): randomly generated between 0.001 and 1; • Transportation cost per trip (c): 300. Let us briefly comment on the ranges chosen for the experiments. Since trucks have unit capacity, products are such that a truck can carry from 100 to 1,000 units of a product. If the day is chosen as the time unit, the production capacity ranges from 1 unit of product every 10 days to 100 units per day. Finally, taking as the monetary unit $1 and assuming that the yearly inventory cost of a product is 10% of its value, the value of a product ranges from about $3.65 to $3,650. The computational results are shown in Tables 1, 2,

Computational Results with Frequencies 1/k, k ⫽ 1, . . . , 5

Parameters

Old Branch-and-Bound

New Branch-and-Bound

Prod.

Quant.

Nodes

N

Opt.

Time

Nodes

N

Opt.

Time

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

14 59 40 348 173 2,963 1,427 28,181

5 5 5 5 5 5 5 5

10 22 19 33 23 57 34 18

0.00 0.00 0.00 0.00 0.00 0.05 0.25 8.07

10 30 26 38 24 69 41 72

5 5 5 5 5 5 5 5

6 16 12 25 15 41 25 14

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

Table 2

Computational Results with Frequencies 1/k, k ⫽ 1, . . . , 10

Parameters

Old Branch-and-Bound

New Branch-and-Bound

Prod.

Quant.

Nodes

N

Opt.

Time

Nodes

N

Opt.

Time

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

47 414 280 1,552 764 13,313 6,357 100,000

5 5 5 5 5 5 5 0

43 268 201 228 105 761 391 0

0.00 0.00 0.00 0.00 0.01 0.23 1.54 1.37.25

27 93 61 114 82 504 354 210

5 5 5 5 5 5 5 5

23 60 27 96 46 451 261 177

0.00 0.00 0.00 0.00 0.00 0.01 0.06 0.04

Table 3

Computational Results with Frequencies 1/k, k ⫽ 1, . . . , 15

Parameters

Old Branch-and-Bound

New Branch-and-Bound

Prod.

Quant.

Nodes

N

Opt.

Time

Nodes

N

Opt.

Time

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

141 1,542 1,190 3,640 2,556 37,941 21,034 100,000

5 5 5 5 5 5 5 0

137 1,214 1,018 572 1,092 7,787 6,927 0

0.00 0.00 0.00 0.00 0.05 1.06 6.23 2.13.50

44 166 153 93 524 4,696 3,337 2,186

5 5 5 5 5 5 5 5

40 102 88 64 490 4,525 3,286 2,152

0.00 0.00 0.00 0.00 0.01 0.09 1.07 0.42

Table 4

Computational Results with Frequencies 2 k , k ⫽ 0, . . . , 10

Parameters

Old Branch-and-Bound

New Branch-and-Bound

Prod.

Quant.

Nodes

N

Opt.

Time

Nodes

N

Opt.

Time

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

337 3,159 2,037 30,266 17,207 100,000 100,000 100,000

5 5 5 5 5 0 0 0

335 173 592 790 3,405 0 0 0

0.00 0.00 0.00 0.05 0.30 2.46 59.54 1.25.16

31 27 43 27 29 46 42 128

5 5 5 5 5 5 5 5

29 25 20 22 24 39 38 122

0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.02

and 3 for the first three sets of frequencies and in Table 4 for the set {1/ 2 k , k ⫽ 0, . . . , 10}. Each of the four tables is organized as follows. The first two columns give the parameters of the experiments. More exactly, the first column gives the number of products and the

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second the range of the quantity of product q i . For each combination of the ranges of the parameters, 5 experiments were made. A maximum limit of 100,000 nodes was set on each experiment for both the old and the new branch-and-bound algorithms. The four col-

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

umns on the old branch-and-bound algorithm and the four columns of the new branch-and-bound algorithm give the average number of nodes generated by the algorithm (Nodes), the number of experiments completed within 100,000 nodes (N), the average number of nodes after which the optimum was found (Opt.), and the average computational time, in minutes and seconds (Time), respectively. In the cases in which the algorithm does not find the optimum within 100,000 nodes, the number of nodes explored—that is 100,000 —is used for the calculation of Nodes. Similarly, in the calculation of the average solution time, the time to explore 100,000 nodes was taken when an instance was unsolved within 100,000 nodes. From each of the tables it can be observed that the new branch-and-bound substantially improves the old one, both in terms of number of nodes and of computational time. In particular, the performance of the new approach in Table 1 clearly performs well enough to make the problem seem almost too easy. However, it should be first pointed out that the problem is NP-hard. Moreover, the old algorithm still shows high computation times and many scanned nodes in the lower part of that table. For instance, it has an average computation time of about 4 minutes with 10,000 products. Thus the fact that the new algorithm takes on the average only less than 1 second for the same cases seems more a merit of the new algorithm than a consequence of dealing with too easy a problem. By evaluating the average solution time, on all the experiments on the first three sets of frequencies, the number of nodes explored by the new branch-andbound algorithm is on the average 0.93% of the number of nodes generated by the old one on the instances with 5 frequencies, the 1.18% on the instances with 10 frequencies, and the 6.67% with 15 frequencies. The computational time of the new branch-and-bound algorithm is on the average only 0.19% of the computational time of the old one on the instances with 5 frequencies, 0.18% on the instances with 10 frequencies, and 1.4% on the instances with 15 frequencies. By the way, note that the above statistics take the time to explore 100,000 nodes whenever the problem has not been solved to optimality, thus un-

Management Science/Vol. 46, No. 7, July 2000

derestimating the performance of the new algorithm with respect to the old one. Moreover, while the old branch-and-bound did not solve within 100,000 nodes all instances, the new one solved all instances. On the instances with set of frequencies {1/ 2 k , k ⫽ 0, . . . , 10}, while the new branch-and-bound solved all the instances within 100,000 nodes, the old branch-and-bound solved within 100,000 nodes 62.5% of the instances only.

5. Assessment of Optimal Solutions vs. Strategies with Relaxed Assumptions As it was pointed out in §1.1, the model we use is based on a certain set of well-defined assumptions, which were discussed at length there, and the solutions we can produce with our algorithm are certainly optimal only within the class of strategies complying with these assumptions. Now it may be interesting to investigate what happens when some of our assumptions are dropped. In other words, how good may the optimal solutions of our model turn out to be when they are compared with other strategies, not feasible for Problem ᏼ? This is an especially interesting question, since, as it is shown in §§6 and 7, we can provide, in an extremely efficient way, solutions for Problem ᏼ which are in practice very good, if not optimal, most of the time. Therefore it would be very convenient to use our solutions for problems other than Problem ᏼ, if they would turn out to be good enough. In particular, it would be interesting to relax two assumptions: the constant quantity assumption and the simultaneous shipments assumption. In fact, although they are perfectly justified in some situations, as it has been discussed in §1.1, it is easy to produce other situations in which they are not. Unfortunately, the modeling of situations allowing for variable quantities and staggering is not immediate, especially within the framework of the “frequency domain,” which is the one we adopt in this paper. Some work toward this direction is presently reported in Bertazzi and Speranza (1997) and Favaretto et al. (2000). To evaluate how good our solutions are when

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

applied to these other situations, we take another, easier way. We compare the cost of the solutions we produce with a lower bound on the cost that could be charged if our two assumptions are relaxed; and this lower bound can be computed very easily. To show how this can be done, consider the optimal cost for the EOQ model z EOQ, the optimal cost for Problem ᏼ, z P , and the optimal obtainable cost for Problem ᏼ without the constant quantity and the simultaneous shipment assumptions z R . Clearly (see also Bertazzi and Speranza 1997), z EOQ ⱕ z R ⱕ z P . Then define

冉

z LB ⫽ max z EOQ,

冘 h q ⫹ Hc 冘 v q H 冊 . i i

i

i i

(16)

i

Clearly, z LB is a lower bound for z R since the second term in (16) is nothing else than the sum of the least achievable inventory cost and of the least achievable transportation cost. Then ␳, defined as

␳⫽

z P ⫺ z LB ⴱ 100 z LB

gives an upper bound for the relative value of the gap between z P and z R . Thus small values for ␳ prove that the solutions we can produce for Problem ᏼ are also satisfactory when variable quantities and staggering are allowed. The values experienced for ␳ in a series of randomly generated instances for Problem ᏼ are shown in Table 5. Experiments have been carried out according to the same procedures described in §4. Table 5 is structured accordingly. All the average percent values for ␳ reported in Table 5 are very low. Only in three cases are values higher than 1% shown. In particular, in the lower part of the table, where more complex problems are considered because of the quantity of different item types involved, all reported values are below 0.01%. Note that these (and all) values may be reduced by increasing the set of available frequencies. So we may conclude, from this empirical evidence, that, unless

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Table 5

Average Values for ␳ (Percent) with Frequencies 1/k, k ⫽ 1, . . . , 5

Prod.

Quant.

␳

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

1.1503 2.0875 5.0749 0.1452 0.2578 0.0038 0.0087 0.0001

shipped volumes are very low, allowing variable quantities or staggering should not be expected to significantly drop costs. As a consequence, the solutions we may provide for Problem ᏼ are still quite good for these relaxed cases also.

6. Heuristic Procedures The computational results of §4 show the improved efficiency of the new branch-and-bound algorithm. However, with larger instances, even this new algorithm is not always able to prove optimality with the available computational resources. This motivates the need for efficient and effective heuristic procedures for the solution of Problem ᏼ. Some such procedures are presented in this section. A natural heuristic is based on the idea of filling trucks as soon as products are made available at the origin and shipping a truck as soon as it is filled up. This heuristic, referred to as Full, can be described as follows. Full 1. Initialize the volume per unit time which requires shipment to W :⫽ ¥ i v i q i . 2. For each frequency f j , 1 ⱕ j ⱕ n, compute the number y j of full-load trucks at frequency f j and update the value of W as follows: yj y j :⫽ t j W W :⫽ W ⫺ . tj 3. If W ⬎ 0, then ship W at the frequency, among all, which gives rise to the minimum total (transportation and inventory) cost.

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In the heuristic Full and in the following heuristics, only the values of the variables y j are determined. Given the number of trucks y 1 , . . . , y n at the different frequencies, the trucks are loaded with the products in nonincreasing order of h i /v i . A somewhat more complex heuristic is based on shipping at the highest frequency f 1 a number of trucks equal to LB 1 as given in (15), and in shipping at the following frequencies full-load trucks with the only possible exception of one truck traveling at the smallest frequency f n. Then a local search is performed with the objective of consolidating loads of trucks traveling at different frequencies and evaluating higher frequencies for the consolidated loads. The procedure, referred to as Lower, is described as follows. Lower 1. Set the number of trucks at frequency f 1 to y 1 :⫽ LB 1 and the volume per unit time which still requires shipment to W :⫽ ¥ i v i q i ⫺ y 1 /t 1 . 2. For each frequency f j , 1 ⬍ j ⱕ n, compute the number y j of full-load trucks at frequency f j , yj y j :⫽ t j W W :⫽ W ⫺ . tj 3. If W ⬎ 0 then y n :⫽ y n ⫹ 1. 4. Compute, for each frequency f j , the total cost z j of the products shipped by the y j trucks. 5. For each frequency f j , 1 ⬍ j ⱕ n: • Select, among the frequencies f 1 , . . . , f j , the frequency f * which gives rise to the minimum total cost to ship the products of y j . • If f * ⬎ f j , then ship all the volume of y j , . . . , y n at frequency f *; stop. • If f * ⫽ f j , then, for each pair of frequencies f k and f s , with f k ⬍ f j ⬍ f s , compute the total cost c s to ship the volume of y j , . . . , y k at frequency k f s ; if c s ⬍ ¥ l⫽j z l , then ship the volume of y j , . . . , y n at frequency f s ; stop. Inspection of the optimal solutions of Problem ᏼ shows that they very often satisfy two characteristics. The first characteristic is that t 1 ¥ i v i q i  trucks travel at the highest frequency f 1 . The second one is that only two frequencies are actually used. Heuristic Best is based on this idea. A number of trucks equal to t 1 ¥ i v i q i  is used at frequency f 1 . The remaining

Management Science/Vol. 46, No. 7, July 2000

volume is shipped at the best among all the frequencies. Best 1. Compute the number of trucks at frequency f 1 : y 1 ⫽ t 1 ¥ i v i q i . Compute the volume per unit time that still requires shipment W :⫽ ¥ i v i q i ⫺ y 1 /t 1 . 2. For each frequency f j , 1 ⱕ j ⱕ n, compute the total cost of shipping the volume of W at frequency f j . Ship W at the frequency that gives rise to the minimum total cost. Finally, we consider two discretized versions of the continuous solution obtained in Blumenfeld et al. (1985). More precisely, the relaxation of Problem ᏼ where any continuous value is allowed for the shipping frequencies has a very simple solution. In particular, there is a single optimal period t* for all products, with t* ⫽ min

冉冑

冊

c 1 , . ¥ i h iq i ¥ i v iq i

(17)

The first of the two quantities over which the minimum is taken is the classic “Wilson’s formula” for the economic order quantity (see for instance Erlenkotter 1989), while the second one accounts for the finite capacity of the trucks. In order to obtain a feasible solution for Problem ᏼ, the period t* can be modified in different ways. We consider the two following simple adjustments which give rise to two heuristics for Problem ᏼ. EOQ-l: All products are shipped at the frequency with period obtained by rounding off the solution (17) to the nearest available period larger than t*, say t k . Then the maximum number of full-load trucks t k ¥ i v i q i  is shipped at frequency f k . The remaining volume is shipped at the lowest available frequency f n . If t* ⬎ t n , then t n is selected and all products are shipped at the minimum available frequency. EOQ-s: All products are shipped at the frequency with period obtained by rounding off the solution (17) to the nearest available period smaller than t*, say t k . All products are shipped at frequency f k . If t* ⬍ t 1 , then t 1 is selected and all products are shipped at the maximum available frequency. Some computational results on the performance of

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

Table 6

Comparison Between Optimal Solution and Heuristics with Frequencies 1/k, k ⫽ 1, 2, . . . , 5

Prod.

Quant.

EOQ-l

EOQ-s

Full

Best

Lower

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

0.4760 (4) 4.1904 (0) 9.0629 (0) 0.2513 (0) 0.5393 (1) 0.0256 (1) 0.0458 (0) 0.0016 (1)

0.2364 (4) 5.8092 (1) 28.2710 (0) 0.9963 (1) 3.9566 (0) 0.1836 (0) 0.2772 (0) 0.0117 (1)

0.0000 (5) 2.6448 (0) 4.9074 (0) 0.1963 (0) 0.5288 (1) 0.0234 (0) 0.0426 (0) 0.0023 (0)

0.0000 (5) 0.9001 (2) 0.0000 (5) 0.0812 (1) 0.2154 (2) 0.0137 (1) 0.0130 (1) 0.0025 (0)

0.0000 (5) 8.8529 (0) 22.3727 (0) 0.4862 (0) 1.2997 (0) 0.0287 (0) 0.0598 (0) 0.0024 (0)

1.8241 (7)

4.9677 (7)

1.0432 (6)

0.1532 (17)

4.1378 (5)

Table 7

Comparison Between Optimal Solution and Heuristics with Frequencies 1/k, k ⫽ 1, 2, . . . , 10

Prod.

Quant.

EOQ-l

EOQ-s

Full

Best

Lower

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

44.6778 (0) 10.0578 (0) 11.1817 (0) 0.4178 (0) 0.4409 (1) 0.0145 (0) 0.0354 (0) 0.0008 (0)

0.3688 (3) 6.2533 (1) 29.2215 (0) 1.0510 (0) 4.0960 (0) 0.1886 (0) 0.2845 (0) 0.0119 (1)

36.5585 (0) 2.0770 (1) 2.0374 (2) 0.1783 (0) 0.2338 (2) 0.0200 (0) 0.0375 (0) 0.0020 (0)

0.0000 (5) 1.3132 (1) 0.7783 (2) 0.0619 (2) 0.1527 (2) 0.0082 (1) 0.0063 (1) 0.0016 (0)

21.4738 (1) 9.5605 (0) 19.0692 (0) 0.4947 (0) 1.8312 (0) 0.0318 (0) 0.0806 (0) 0.0022 (0)

8.3533 (1)

5.1844 (5)

5.1431 (5)

0.2903 (14)

6.5680 (1)

the two latter heuristics for Problem ᏼ have been reported in Speranza and Ukovich (1994a).

7. Computational Results for the Heuristics The computational results for the heuristics of §6 have been obtained on the instances tested in §4 on which the optimum was found. The availability of the optimal solutions for the tested instances allows a careful evaluation of the performance of the heuristics. The cost values obtained by the heuristics have been compared with the optimal costs. The results are shown in Tables 6 –9 for the same sets of frequencies and parameter ranges as in Tables 1– 4. Each row corresponds to the problem instances on which the branch-and-bound algorithms have been tested with the characteristics described in Columns 1–2. In Columns 3–7 the average percent relative

986

errors with respect to the optimum are shown for the different heuristics, within parentheses the number of instances solved to optimality. The last row shows the average error values and the cumulative number of instances solved optimally. Computation times are always below 1 second. For each heuristic, the error tends to decrease when the number of products increases from 100 to larger values (however, no clear pattern appears for 10 items). The large errors experienced with few products, despite the fact that the optimal solution is often found, could be explained by observing that in these cases a low number of trucks is involved; therefore, sending one of them in a wrong day can have a large relative impact on the overall costs. Conversely, with many products, many trucks are considered, thus producing many nearly optimal solutions. However, it also must be pointed out that cases with few products are easier, thus they do not

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

Table 8

Comparison Between Optimal Solution and Heuristics with Frequencies 1/k, k ⫽ 1, 2, . . . , 15

Prod.

Quant.

EOQ-l

EOQ-s

Full

Best

Lower

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

63.1760 (0) 15.9334 (0) 17.2455 (0) 0.5997 (1) 0.4200 (1) 0.0138 (0) 0.0277 (1) 0.0010 (0)

0.3688 (3) 6.2533 (1) 29.6308 (0) 1.0551 (0) 4.1544 (0) 0.1886 (0) 0.2866 (0) 0.0119 (1)

28.1569 (0) 1.6967 (0) 2.4187 (0) 0.0961 (1) 0.2035 (1) 0.0154 (0) 0.0280 (0) 0.0011 (0)

0.0000 (5) 1.3132 (1) 1.1046 (2) 0.0631 (2) 0.1718 (2) 0.0063 (1) 0.0068 (1) 0.0013 (0)

12.0876 (2) 9.3702 (0) 19.8047 (0) 0.4983 (0) 2.9026 (0) 0.0384 (0) 0.1073 (0) 0.0014 (0)

12.1771 (3)

5.2437 (5)

4.0770 (2)

0.3334 (14)

5.6013 (2)

Table 9

Comparison Between Optimal Solution and Heuristics with Frequencies 1/ 2 k , k ⫽ 0, . . . , 10

Prod.

Quant.

EOQ-l

EOQ-s

Full

Best

Lower

10 10 100 100 1,000 1,000 10,000 10,000

0.1–5 5–100 0.1–5 5–100 0.1–5 5–100 0.1–5 5–100

2,018.9100 (0) 1,244.3600 (0) 1,356.9810 (0) 44.6936 (0) 24.0034 (0) 0.2159 (0) 1.2263 (0) 0.0367 (0)

0.0536 (4) 6.1865 (1) 27.9003 (0) 1.0308 (1) 4.1627 (0) 0.1883 (0) 0.2867 (0) 0.0119 (1)

47.7216 (0) 2.7860 (0) 2.1958 (1) 0.0643 (0) 0.0198 (1) 0.0185 (1) 0.0033 (0) 0.0009 (0)

0.0000 (5) 1.8823 (2) 2.4741 (3) 0.0568 (2) 0.2981 (0) 0.0052 (1) 0.0328 (0) 0.0019 (0)

0.0000 (5) 4.3947 (1) 19.9290 (0) 0.1046 (1) 0.2365 (0) 0.0024 (0) 0.0054 (0) 0.0002 (1)

586.3034 (0)

4.9779 (7)

6.6013 (3)

0.5939 (13)

3.0841 (8)

require heuristics: All cases with 10 and 100 products have been optimally solved by our new branch-andbound algorithm in less than one second. The EOQ-type heuristics produce relatively larger, and, in some cases, very large errors. In particular, the huge errors for the EOQ-l heuristic in Table 9 are due to the fact that it uses the lowest available frequency, which turns out to be unrealistically large, especially with few products in low quantities. Among the other heuristics, the Best heuristic, with few exceptions, gives the smallest average error. It produces an average error less than 0.34% over all the instances of Tables 6 –9. Moreover, the Best heuristic found the optimum on about 36% of the instances. Finally, the errors generated by this heuristic do not seem to strongly depend on the number of frequencies.

8. Conclusions In this paper we presented dominance rules for the problem of minimizing the sum of the transportation

Management Science/Vol. 46, No. 7, July 2000

and inventory costs in the shipping of several products between two nodes, given a set of possible shipping frequencies. The dominance rules allow one to substantially improve the performance of a known branch-andbound algorithm. It was possible on a personal computer to optimally solve problems with up to 10,000 products and up to 15 frequencies. Furthermore, the enhanced branch-and-bound algorithm has been used for computing benchmark solutions to assess a set of heuristics on the instances on which the optimum was available. The experimental results show that the heuristic which ships at the maximum frequency the largest possible number of full load trucks and ships the remaining products at the best of all frequencies outperforms the other heuristics. These results allow one to solve very large instances with very good values for the objective function, using limited computational resources. This is of particular interest since it allows one to use the single-origin, single-destination problem as a subroutine for more complex problems.

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BERTAZZI, SPERANZA, AND UKOVICH Solutions for a Shipment Problem with Given Frequencies

In the authors’ opinion, the conclusions of this paper may suggest new interesting research directions. One among others deals with studying the application of the results of this paper to practical situations. In this case, modeling issues are relevant, and the efficient and effective solution tools provided by this paper should turn out to be of value in attacking practical cases, possibly as subroutines in more complex methodologies. Furthermore, other topologies could be explored for the same shipment problem with discrete frequencies. As an example, the sequence of simple links and the multiple-origin, multiple-destination cases are certainly of interest both from the theoretical and from the practical point of view. 1 1 The research of the authors was partially supported by Progetto Finalizzato Trasporti 2 of the CNR under Contracts 93.01898.PF74 and 94.01472.PF74. The authors gratefully thank two anonymous referees for their constructive comments.

References Anily, S., A. Federgruen. 1990a. One warehouse multiple retailer systems with vehicle routing costs. Management Sci. 36 92–114. , . 1990b. A class of Euclidean routing problems with general route cost functions. Math. Oper. Res. 15 268 –285. , . 1991. Rejoinder to “Comments on one warehouse multiple retailer systems with vehicle routing costs.” Management Sci. 37 1497–1499. , . 1993. Two-echelon distribution systems with vehicle routing costs and central inventories. Oper. Res. 41 37– 47. Baita, F., R. Pesenti, W. Ukovich, D. Favaretto. 1998. Dynamic routing-and-inventory problems: A review. Transportation Res. 32A 585–598. Bertazzi, L., M. G. Speranza. 1997. Optimal periodic shipping strategies for the single link problem. Technical report No. 132, Department of Quantitative Methods, University of Brescia, Italy. , , W. Ukovich. 1997. Minimization of logistic costs with given frequencies. Transportation Res. 31B 327–340. Blumenfeld, D. E., L. D. Burns, J. D. Diltz, C. F. Daganzo. 1985. Analyzing trade-offs between transportation, inventory and production costs on freight networks. Transportation Res. 19B 361–380. Burns, L. D., R. W. Hall, D. E. Blumenfeld, C. F. Daganzo. 1985. Distribution strategies that minimize transportation and inventory cost. Oper. Res. 33 469 – 490. Chika´ n, A. 1990. Inventory Models. Kluwer.

Daganzo, C. F. 1985. Supplying a single location from heterogeneous sources. Transportation Res. 19B 409 – 419. . 1988. A comparison of in-vehicle and out-of-vehicle freight consolidation strategies. Transportation Res. 22B 173–180. , G. F. Newell. 1985. Physical distribution from a warehouse: Vehicle coverage and inventory levels. Transportation Res. 19B 397–407. Erlenkotter, D. 1989. An early classic misplaced: Ford W. Harris’s Economic Order Quantity model of 1915. Management Sci. 35 898–900. Favaretto, D., R. Pesenti, W. Ukovich. 2000. Discrete frequency models for inventory management. Internat. J. Production Econom. To appear. Federgruen, A., G. Prastacos, P. Zipkin. 1986. An allocation and distribution model for perishable products. Oper. Res. 34 75– 82. , P. Zipkin. 1984. A combined vehicle routing and inventory allocation problem. Oper. Res. 32 1019 –1037. Hall, R. W. 1985. Determining vehicle dispatch frequency when shipping frequency differs among suppliers. Transportation Res. 19B 421– 431. . 1991. Comment on “One warehouse multiple retailer systems with vehicle routing costs.” Management Sci. 37 1496 –1497. Harris, F. W. 1913. What quantity to make at once. Factory, The Magazine of Management 10 135–136. Jackson, P. L., W. L. Maxwell, J. A. Muckstadt. 1988. Determining optimal reorder intervals in capacitated production-distribution systems. Management Sci. 34 938 –958. Lee, H. L., S. Nahmias. 1993. Single-product, single-location models. S. C. Graves, A. H. G. Rinnooy Kan, P. H. Zipkin, eds., Handbooks in Operations Research and Management Science, Vol. 4: Logistics of Production and Inventory. North-Holland, Amsterdam 3–55. Maxwell, W. L., J. A. Muckstadt. 1985. Establishing consistent and realistic reorder intervals in production-distribution systems. Oper. Res. 33 1316 –1341. Muckstadt, J. A., R. O. Roundy. 1993. Analysis of multistage production systems. S. C. Graves, A. H. G. Rinnooy Kan, P. H. Zipkin, eds., Handbooks in Operations Research and Management Science, Vol. 4: Logistics of Production and Inventory. NorthHolland, Amsterdam 59 –131. Nahmias, S. 1992. Production and Operations Analysis. Irwin, Roundy, R. 1989. Rounding off to powers of two in continuous relaxations of capacitated lot sizing problems. Management Sci. 35 1433–1442. Silver, E. A., R. Peterson. 1985. Decision Systems for Inventory Management and Production Planning. Wiley, New York. Speranza, M. G., W. Ukovich. 1994a. Analysis and integration of optimization models for logistic systems. Internat. J. Production Econom. 35 183–190. , . 1994b. Minimizing transportation and inventory costs for several products on a single link. Oper. Res. 42 879 – 894. , . 1996. An algorithm for optimal shipments with given frequencies. Naval Res. Logist. 43 655– 671.

Accepted by Awi Federgruen; received December 12, 1995. This paper was with the authors 16 months for 3 revisions.

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