Assessing the Reliability of a Multistate Logistics Network under the Transportation Cost Constraint

Reliability and transportation cost are two important indicators to measure the performance of logistics network. As a combination of reliability and transportation cost, the performance index R(d,c) of logistics network is defined as the probability that at least d units of flow demand can be successfully transmitted from the source to the destination with the total transportation cost less than or equal to c. In this paper, an algorithm is developed to calculate R(d,c) in terms of (d, c)-minimal paths ((d, c)-MPs for short). The proposed algorithm employs a decomposition technique to divide the search space of (d, c)-MPs such that the search space of (d, c)-MPs can be dramatically reduced, and thereby (d, c)-MPs can be efficiently obtained. An example is provided to illustrate the proposed algorithm. Finally, computational experiments conducted on one benchmark network indicate that the proposedmethod has an advantage over the existing methods.


Introduction
With the swift development of E-commerce, express logistics in China is presenting a vigorous development trend. Meanwhile, most of enterprises begin to pay more attention to logistics performance for improving business efficiency. Logistics networks which provide the infrastructure for the storage and distribution of products undertake the mission of promoting the efficient and safe movement of goods over time and space and thus play an increasingly important part in sustaining the national economic and social development. It is widely accepted that an efficient, reliable, and cost-effective logistics network not only affects a firm's short-term benefits but also influences a firm's long-term development.
The performance assessment of logistics networks is a popular issue in the field of logistics and supply chain management. In a real-world environment, the performance of logistics networks is always affected by various unexpected events, such that it is subject to degradation. Hence, it is of importance from the perspective of logistics management to assess the capability of logistics networks to ensure the delivery of required quantity of goods to the right demand point. In addition, the transportation cost is also a major concern for logistics providers. Therefore, combining capacity reliability and transportation cost for performance measure of logistics networks, in a sense, is of significance to possible improvement of logistics performance.
Theoretically, logistics network can be represented as sets of both nodes and arcs, where each node stands for a supplier, a transfer center, or a market, and each arc connecting a pair of nodes stands for transportation medium (traffic tools, traffic routes, or both) [1]. In fact, the transportation medium, which is multistate due to the nature of the traffic tools (truck, railway, cargo ship, etc.), may be in a failure state, partial failure state, or maintenance state [1,2]. That is, the number of available traffic tools is not fixed in some sense, and thus each arc has several possible capacities. Therefore, logistics network can be modeled as a multistate flow network in which arcs are associated with multiple integer capacities, operational reliability, and unit transportation cost. And, the goods transported through such logistics network are looked upon as a flow. The reliability index ( , ) of logistics network is defined as the probability that at least units of flow demand can be successfully transmitted from the source to the destination with the total transportation cost less than or equal to c [3][4][5][6][7][8][9]. One of the general algorithms for computing the above-mentioned methods [5][6][7][8][9] in solving the ( , )-MP problem.
The rest of this paper is organized as follows. Section 2 introduces the network model and basic results. In Section 3, the proposed algorithm is described in detail, and its time complexity is also analyzed. In Section 4, an illustrative example is provided to demonstrate the proposed algorithm. Computational experiments are conducted in Section 5 to compare the proposed algorithm with the existing methods. The final section presents the concluding remarks. . Specifically, if is a feasible state vector (i.e., it satisfies the flow conservation law), then ( ) = ∑ =1 . The network model satisfies the following assumptions [7][8][9]: (1) the state of each arc is a random variable which takes integer values from 0 to according to a given distribution; (2) the states of different arcs are statistically independent; (3) all flows in the network obey the conservation law: that is, total flows into and from a node (other than the source and destination nodes) are all equal.

Lemma 1. A state vector is a ( , )-MP if and only if satisfies the following conditions:
where (V, ⋅) is the set of arcs emanating from node V and (⋅, V) is the set of arcs pointing to node V. It should be pointed out that (2)-(4) are established on the basis of the well-known flow conservation law [9]. By lemma, it is easy to obtain the following conclusion. (6), then is a ( , )-MP.

Corollary 2. For a state vector with ( ) = , if satisfies conditions (3), (4), (5), and
By Lemma 1, the enumeration method can be used to search for ( , )-MPs. Given a multistate network ( , , , ), (4) shows that the total number of state vectors contained in search space of ( , )-MPs is ∏ =1 (min{ , }+1) which is a pretty huge number. Thus, it is inefficient to search for ( , )-MPs by Lemma 1. Below, we will introduce how to divide the search space into disjoint subspaces from which all ( , )-MPs can be efficiently obtained.

The Proposed Algorithm
Given a set of state vectors where is the set of states of with respect to , that is, = { | ≤ ≤ for 1 ≤ ≤ }, if we limit the search space of ( , )-MP in set , it is apparent that Lemma 1 can be transformed into the following conclusion.

Corollary 3. A state vector is a ( , )-MP in set if and only if satisfies the following conditions:
It should be noted that the only difference between Lemma 1 and Corollary 3 is the state range of . Next, we will consider a special state vector in set X: the smallest state vector . It is clear that there are three cases with respect to  Proof. Since is the smallest state vector in , we have ≥ for any vector ∈ . If ( ) > , then ( ) ≥ ( ) > for any ∈ . That is, ( ) > for any ∈ ; then, by the definition of ( , )-MP, there is no ( , )-MP in .
According to Theorem 4, once we have verified that ( ) > holds, set will be discarded (no ( , )-MP exists in ). Thus, the search space of ( , )-MPs will be reduced, which is extremely beneficial in solving ( , )-MPs.
Case 2 ( ( ) = ). In such case, we can claim that the following theorem holds. Case 3 ( ( ) < ). In such case, we need to analyze the largest state vector in set . If ( ) < , we have the following statement.
Proof. Since is the largest state vector in set X, we have ≤ for any ∈ . If ( ) < , it is clear to derive ( ) ≤ ( ) < for any ∈ . That is, ( ) < for any ∈ ; then, by the definition of ( , )-MP, there is no d-MP in set X.
It is noteworthy that all of the three theorems contribute to reducing the search space of ( , )-MPs. If ( ) ≥ , then we have ( ) < ≤ ( ). In such case, the enumeration algorithm can be used to search for ( , )-MPs in set by Proof. Since ( ) ≥ , at least units of flow can be transmitted from the source to the destination when the network is under . Also, such a -flow ( 1 , 2 , . . . , ) can be obtained by the following steps [10,12,13]: (a) Add a fictitious node * , and add an arc with a fixed state from to * .
(b) The state of each arc ∈ E is defined by .
(c) Determine the max-flow from to * . Since ( ) ≥ , and the state of the arc from to * is d, the maxflow from to * is d.
Since the flows through are upper bounded by , we have ≤ for all ∈ . Meanwhile, it is apparent that ≥ 0 for all ∈ E. Thus, we have 0 ≤ ≤ for all ∈ ; that is, The definition of -flow indicates that a d-flow satisfies the flow conservation law; thus, it satisfies (7) (1) Pivot on arc 1 : (2) Pivot on arc 2 : (3) . . . Readers may refer to the work of Jane and Laih [13] for the proofs. Now, we first consider subset 0 . Since the d-flow ( 1 , 2 , . . . , ) derived from Theorem 7 is the smallest state vector in set 0 , if it is a ( , )-MP, it is the unique ( , )-MP in set 0 ; otherwise, there is no ( , )-MP in set 0 . As a result, there is no need to check other state vectors in 0 ; that is, 0 can be discarded as soon as d-flow ( 1 , 2 , . . . , ) has been checked, which will reduce the number of enumerated state vectors to a large extent.
Step 4. For 0 , if there is no directed cycle in it and ∑ =1 ≤ , it is a ( , )-MP; otherwise, it is not a ( , )-MP.
The time complexity of the proposed method is discussed below. Either Step 0 or Step 6 takes ( ) time.
Step 3 takes ( ) time to decompose . Step 4 requires ( + ) time. Therefore, the algorithm totally takes ( 3 ) time from Steps 0 to 4. It is apparent that the time complexity of Step 5 is determined by the number of enumerated state vectors. The number of state vectors in Step 5 is less than ∏ =1 (min{ , } + 1); thus, the time complexity of Step 5 is less than (( + )∏ =1 (min{ , } + 1)). Therefore, the time complexity of the proposed algorithm is less than (( + )∏ =1 (min{ , } + 1)).

An Illustrative Example
In this section, we use a simple logistics network to demonstrate the proposed algorithm step by step. The network, as shown in Figure 1, is cited from the literature in [6][7][8][9]. The data of each arc is given in Table 1. If the manager is assigned a budget of 14 (c = 14) to transport 3 units of commodity (demand level d = 3) from to t, the reliability index R (3,14) can be obtained as follows (for convenience, the set of state vectors is denoted by its smallest and largest state vectors as [ , ]).
It is observed from Table 2 that the proposed algorithm spends less time than Yeh's [8] in searching for different ( , )-MPs. In particular, the proposed algorithm has a better searching efficiency as the demand level grows. This is as expected, because Yeh's algorithm is an exhaustive enumeration method in nature, whereas the proposed algorithm utilizes a decomposition technique to divide the search space, such that the search space of ( , )-MPs can be dramatically reduced, and thereby ( , )-MPs can be efficiently obtained. Furthermore, Table 2 shows that the proposed algorithm displays an advantage over the algorithm of Niu and Xu [9]. Both of them are partial-enumeration methods, but the proposed technique is more effective in shortening the search space of ( , )-MPs. Therefore, the proposed algorithm outperforms the two methods in solving ( , )-MPs.

Conclusions
Reliability analysis is a fundamental tool for understanding the operational level of logistics networks. Logistics network can be modeled as a typical multistate flow network in which arcs are associated with multiple integer capacities, operational reliability, and unit transportation cost. The reliability index ( , ) which takes into account both reliability and transportation cost can serve as a comprehensive performance assessment of logistics networks. This paper is devoted to proposing a new efficient algorithm for solving all ( , )-MPs. A major contribution of this paper is that we develop a new approach to shorten the search space of ( , )-MP. Specifically, the proposed algorithm employs an effective decomposition technique to divide the search space of ( , )-MPs such that the search space of ( , )-MPs can be dramatically reduced. A real example is used to illustrate the applicability of the proposed algorithm. Also, to check the performance of the proposed algorithm, computational experiments on a medium-sized logistics network are conducted, and the results show that the proposed algorithm has an advantage over the existing methods [5][6][7][8][9].
For future research, there is still a room for developing efficient methods for ( , ) . For example, as with the existing algorithms [4][5][6][7][8][9], the proposed algorithm is also an enumeration method in nature, and thus it still needs to perform the enumeration of state vectors. Therefore, it is more desirable to develop new methods which are not grounded on the enumeration of state vectors. Furthermore, we only consider the transmission of single type of commodity in the network. Thus, it is more practical to extend the proposed algorithm to the reliability evaluation of multicommodity multistate networks under cost constraint in which multiple types of commodity are transmitting from the source to the destination.