Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 1283539 |

Ning Liu, Yaorong Cheng, "Allocating Cost to Shippers in Pickup and Delivery Service", Mathematical Problems in Engineering, vol. 2019, Article ID 1283539, 10 pages, 2019.

Allocating Cost to Shippers in Pickup and Delivery Service

Academic Editor: Vladimir Turetsky
Received16 Jul 2018
Revised18 Nov 2018
Accepted02 Jan 2019
Published29 Jan 2019


This paper is going to study the shipper collaboration cost allocation problem in pickup and delivery (SCAPPD), which includes two crucial aspects, namely, the optimal redistribution of transport requests for minimization of the total cost of the alliance and equitable allocation of the shipper’s cost for keeping the stability of the alliance. We focus on the second issue and use core concept from cooperative game theory to develop an allocation methods, called Location Cost Allocation Model (LCAM), which take geographical location factor into account. As the goal of this study is to design an allocation tool that will be accepted in practice, we use the criteria of stability, consistency, and robustness to evaluate the LCAM, Shapley, Nucleolus, and the equal profit method (EPM). All four allocation methods are applied to both a case study and randomly generated instances. Our computational results show that the stability of the LCAM is as good as the nucleolus and EPM. In terms of consistency the LCAM performs best. LCAM also performs well in terms of robustness.

1. Introduction

In order to lower the total cost and ensure the quality of logistics, horizontal partners or even competitors usually apply the collaborative transportation strategies [1]. Products that come from various corporations are transported on the same route by a logistics company for the purpose of shipping cost reduction. Since the article published by Dantzig and Ramser [2], literature on the vehicle routing problem (VRP) has enjoyed a significant growth. Most of the literature attempts to find the overall optimal solution for VRP [3]. However, current VRP models aim to incorporate real-life complexities and different kinds of models have different application scenarios and conditions. This means that different VRP models have different cost allocation rules that cannot be generalized. Pickup and delivery problem (PDP) can be recognized as important vehicle routing problem where objects or people must be transported between origin and destination. In this article, we use the PDP definition proposed by Dumas et al. [4]. Most of the literature attempts to find the overall optimal solution for PDP (e.g., [5, 6]). Although significant discounts are offered to the group as a whole through collaboration transportation, each entity engaged cares about its own interest, which leads to the problem how the combined transportation costs on the same route should be distributed among partners.

Mainstream approach that may be used to solve cost allocation problems for collaborative logistics is cooperative game approach. This problem has drawn widespread attention in recent years. Game theory provides several famous concepts of allocation, such as the core generated by Gilles [7], the Shapley value suggested by Shapley [8], and the nucleolus studied by Schmeidler [9]. The core theory requires individual, group, and coalition rationality, and these properties are the basis of cost allocation in the vast majority of the literature. But two drawbacks can be seen for core is not unique and may be empty. When it comes to certain calculation, Shapley values and nucleolus were compared with those suggested by the authors. Krajewska et al. [10] distributed the profit margins from the collaboration among partners by applying the Shapley value to cooperative game theory. There are many literatures using the Shapley value method for cost allocation or profit-sharing. (e.g., [1114]). In order to discuss the cost allocation of gas and oil companies, Engevall et al. [15] developed the concept of demand nucleolus in travel salesman games. Yin et al. [16] studied the nucleolus cost allocation method for the vehicle routing problem with split delivery. Some new methods are discussed by scholars based on classic solution concept. Frisk et al. [17] conducted a research on a cooperative forest transportation planning problem as well as some typical cost allocation methods (including Shapley values and nucleolus), which put forward an equal profit method (EPM). Dai and Chen [18] paid close attention to the problem of carrier collaboration in pickup and delivery services and suggested three profit distribution mechanisms based on the Shapley value, the concept of proportional allocation and the contribution of each bearer in providing and serving requests.

In terms of logistics cost analysis, the transportation cost allocation problem is basic for it is solely based on the route location. However, it causes some concerns about the method of distribution. A route cost allocation approach as well as a number of fairness standards was put forward by Fishburn and Pollak [19], where only the role of a single stop delivery cost was considered while other important factors were neglected, such as the relative position of clients. An approximate core was recognized and a geometric cost allocation approach with the idea of a moat was proposed by Faigle et al. [20]. According to the studies of Faigle et al. [20], Sun et al. [21] developed five fairness standards and contribution constrained packing model, which took the multiple fairness standards into consideration in cost allocation.

This paper has the following structure. In the next section we describe the problem. The cost allocation model will be discussed in Section 3, while, in Section 4, we conduct the comparison between our approach and others’ in terms of performance. And in Section 5, we will come to conclusion.

2. Problem Description

In terms of the shipper cooperation in the pickup and delivery services, a centralized planning framework is given to achieve the optimization of redistribution of transportation demands among the shippers to achieve the least amount of all-in costs after their alliance. In this case, it leads to a lower cost for the alliance as a whole comparing with the decentralized approach [22]. Previously, each shipper was only responsible for the fulfillment of his own requirement, which consists of the pickup node, delivery node, time window, and quantity. Before the alliance was set up, each shipper formulated his separate transportation planning to serve his own requirement with his own available transportation resources (vehicles). The calculation of the costs of shipper before the mutual partnership involved the individual cost of each shipper, respectively. Nonetheless, with the partnership relationship, the integration of the requests of all the shippers is made by a carrier, accompanied by the overall centralized transportation planning that is jointly worked out to reduce the total expenses to the smallest amount for all shippers. The total expenses of all the shippers after cooperating with one another can thus be derived from it. According to the collaborative transportation plan of the parties, the total costs shall be shared among the shippers with the aid of the cost allocation mechanism.

As was illustrated earlier, two major problems can be summarized to be solved in the SCAPPD. One problem is the collaborative planning. In order to achieve the lowest overall cost for the shippers, vehicle tours are supposed to meet the requirements within the capacity limit of each vehicle and the time window limit of each request and a series of demands and requirements (pickup/delivery) should be guaranteed. Each vehicle tour consisting of multiple arcs in a transportation network is required to set off and get back to the depot node of the corresponding carrier. To date, plenty of research has been carried out on the algorithm to solve PDPTW, for instance, Ropke and Pisinger in 2006 and Bent and Van in 2006. Another problem lies in the equitable allocation of the total cost amount to each shipper concerned, leading to a lower cost for each shipper as a result of the cooperation. The paper emphatically probes into the second problem.

To further explain the aforesaid SCAPPD, an instance analysis is conducted below. Given 8 nodes and one depot in the transportation network, the vehicle can hold 60 units. Assume that the point-to-point distance meets the Euclidean distance (see the transportation network as shown in Figures 1 and 2). A total of 4 transportation requests have been received for the shippers, which are signified with r1 to r4, respectively. Time windows, quantities, and node of the requests can be seen in Table 1. For example, 20 units of freight at node 7 is extracted by request r1 and the same quantity is delivered to node 8.

Request number1234

Pickup node7153
Delivery node8624
Pickup time window0,3090,100160,180260,290
Delivery time window40,80110,150170,250280,340
Pickup node coordinate(42,58)(67,85)(80,65)(85,35)
Delivery node coordinate(55,85)(58,70)(95,35)(64,46)

3. Allocation Models and Evaluation Criteria

3.1. Basic Definitions and Properties

It is said that each solution and concept for the total cost allocation can meet some fairness standards. However, none of them is able to meet all the standards according to the literature till now. Part of the properties that is most frequently used is enumerated as follows. A coalition is equal to a subset of participants, while the grand coalition refers to all players. We suppose that the opportunity to form and work together with others is provided for each participant in the coalitions. When the collaboration is established in the coalition , the total (normal) cost shall be produced. In cooperative game theory, the cost function is seen as the Characteristic Cost Function and each participant is identified as a participant. It is possible to make a comparison between the cost allocation issue and the cooperative game.

A cost allocation method is applied to the separation of total cost, i.e., . Among all the participants, is said to be efficient, which can be expressed as , where is the share of cost allocated to participant . The cost allocation can be recognized separate and reasonable, where no participant pays more than the independent cost, which is equal to the previous cost alone before the establishment of the alliance. Property can be presented as . The core of the game is identified as those who meet the requirements of the cost allocation.

3.2. The Shapley Value

The Shapley value [8] concept gives us a unique solution to deal with the cost allocation problem. As can be seen from the computation formula below, it is assumed that only one participant is accessible to the grand coalition each time. The participant in the coalition is responsible for the marginal cost which means his accession will increase the total cost of the entire coalition. The amount received by participants in this program depends on the order in which participants joined. The cost assigned to participant is equal shows the number of participants in the coalition considered. The sum of this formula is equal to that over all coalitions that include the participant . The value of refers to the increase of the cost of the alliance with the increased the participant , which is expressed by the marginal cost of participant regarding the coalition .

3.3. The Nucleolus

The cost allocation is identified in the calculation of nucleolus of a game to reduce the worst inequity to the lowest degree, so that the individual rationality can be fulfilled. That is to say, aiming at reducing the maximum dissatisfaction of any coalition as much as possible, we investigated how dissatisfied it is with the proposed allocation among the coalition . We delivered the dissatisfaction of cost allocation about the coalition , as a measurement of the amount by which coalition falls short of its potential in the allocation . The nucleolus refers to the cost allocation with the largest correlated excess vector in dictionary order. The concept was further defined by Schmeidler [9]; i.e., the nucleolus indicates the unique vector which has the nonincreasing sorted vector of excess lexicographically minimized. The vector of excesses of an allocation is the vector defined as

3.4. Equal Profit Method

In 2010, for the purpose of minimization of maximum difference between pairs of related savings, Frisk et. al 2010 came up with a method inspired by finding a stable allocation, which was identifies as equal profit method (EPM). The relative saving of participants is expressed as follows: . Assuming the cost allocation is stable, we have that . Thus, refers to the difference in relative savings between two participants of and. LP problem is supposed to be addressed so as to find the allocation.

The first constraint set acts as a measurement of the pairwise difference between the revenues of participants. Variable in the goal minimize the biggest differences. All the stable allocations are defined with the other two constraint sets.

3.5. Location Cost Allocation Model (LCAM)

The formulation of LCAM is in fact a two-stage model. The first stage, we build the following linear programming model to achieve the minimum excess of every coalition under the condition of efficiency. The model (M1) is as follows:

Objective function (5) requires minimum . Constraint (6) corresponds to the efficiency condition, which states total route cost should be completely allocated. Constraint (7) corresponds to the coalition rationality conditions. By introducing , we add minimal penalties to the constraints that define the core to maintain the grand coalition “stability” and to obtain suboptimal solutions. Constraint (8) demonstrates the value scope of and . By solving this model, we can get the value of , denoted as .

The second stage, as the geometric characteristics of the route, is the main basis for cost allocation; the objective of the model (M2) is to achieve a stable allocation set, considering the marginal cost and the dispersion degree of the nodes. The model (M2) is as follows:

Objective function (9) requires minimum . Constraint (10) shows that the cost to the shipper allocated by alliance equals the cost combination of relative pickup and delivery nodes. Constraint (11) shows shipper single stop delivery cost from depot. Constraint (12) corresponds to the efficiency condition, which states total route cost should be completely allocated. Constraint (13) shows that, when , the core exists; we select the solution meeting constraints (10) through (21) in the core. When , the core is empty or the solution meeting constraints (10) through (21) is not in the core. Constraint (14) corresponds to the individual condition that each shipper’s cost should not exceed its single stop delivery cost. Constraint (15) shows that the cost allocation should be positively correlated with the cost contribution factors and the correlation should be as high as possible. Constraint (16) represents the cost contribution factors, including and . In Constraint (17), demonstrates the marginal cost contribution rate of node . In Constraint (18), represents the dispersion degree of node with other nodes. In Constraint (19), the marginal cost is the reduction of the total route length after the node is removed, while preserving the original access sequence. Constraint (20) reflects the consistent relation between cost allocation and cost contribution; namely, the larger the cost contribution, the higher the cost allocation to the shipper. Constraint (21) demonstrates the value scope of . Relative parameter and variable are shown in Table 2.


Total cost is calculated by PDP algorithm (e.g. [5, 6]).
Cost of shipper allocated by alliance.
Shipper single stop delivery cost from depot.
Cost between nodes and , without loss of generality, we set . is Euclidean distance between nodes and .
Cost contribution factors, including and .
Marginal cost contribution rate of node .
Dispersion degree of node with other nodes.
Marginal cost, which is the decreased value of the total route length after removing node while keeping others with the original visiting sequence.
Let be the index of the th shipper in the sequence of ordered cost contribution factors.
Set of node delivered by same vehicle.
The set of nodes.
The set of pickup nodes.


Cost of node allocated by alliance.
A minimum penalized slack in the constraints defining the core to keep the grand coalition “stable”.
Minimum difference between the cost of shipper and its cost based on geographical location.

In this allocation model, the two factors of and are considered. The reason for considering both factors is because they represent different aspects. is calculated from , and is marginal cost of the node, which considers decreased value of the total route length after removing the node while keeping others with the original visiting sequence. is calculated from and can be understood as a geographical isolation value, which is a distance between node and every other node. Just considering only one of these factors has drawbacks. We will use two examples to illustrate that considering two factors at the same time will make the distribution result more fair and reasonable.

In the first example, we analyze the situation mentioned in Figure 3. The red line represents request 1 of Shipper A (r1, from node 1 to node 4) and the blue line represents request 2 of Shipper B (r2, from node 2 to node 3). Request 2 is just right on the straight path of request 1, and its marginal cost is 0. If we only consider the factor (marginal cost contribution rate of node ), the result of cost allocation for request 2 is 0. Normally, this kind of allocation would, in practice, be unacceptable to the affected shippers. In this example, Shipper A will not be willing to bear all the costs.

In the second example, we analyze the situation mentioned in Figure 4. The relevant data comes from Table 1. The only difference is that the position of the nodes has changed.

The calculation results show that the dispersion degree of request 2 (from node 1 to node 6) is approximately equal to 0.256, which is equal to the dispersion degree of the request 4 (from node 3 to node 4). If we only consider the factor (dispersion degree of node with other nodes), request 2 and request 4 should be allocated the same cost. But the marginal cost contribution rate of request 2 is 0.41, while the marginal cost contribution rate of request 4 is 0.15, indicating that the impact of request 4 to total cost is much smaller, and it should be allocated with smaller costs.

Model LCAM is a linear programming model, which is implemented in LINGO 11 using linear solver to solve the linear programming problems. There are many classical algorithms for linear programming problems, including simplex algorithm [23], ellipsoid method [24], and Karmarkar’s algorithm [25].

3.6. Evaluation Criteria

To evaluate the allocation methods we use several evaluation criteria. As the goal of this study is to design an allocation tool that will be accepted in practice, we have selected evaluation criteria to assess which tool is most likely to be accepted in practice. We focus mainly on important criteria from the perspective of the shippers. First of all, the method is preferred to generate stable allocations. Stable allocations are in the core of the game, a well-known concept from cooperative game theory. For such allocations, no subset of shippers has an incentive to withdraw from the collaborative planning on the basis of the allocation. Secondly, the allocation should be consistent with the geographical location of the shippers. That is, if geographical location varies, the allocation should change accordingly in a consistent way. We evaluate the consistency by using an ordinary least squares regression. Thirdly, an allocation method is preferably robust. If a shipper propose a similar transport request periodically, they will resent being allocated significantly different allocation at each shipment. To quantify robustness we use the coefficient of variation of the allocation.

4. Numerical Results

4.1. Case Study

We study the methods applied to a case study to gain insight into the performance in practice. Going back to Figures 1 and 2 and Table 1, the specific description of the case has been described in Section 2. This instance allows us to evaluate the feasibility and effectiveness of LCAM method.

In the first stage of LCAM, we can get ; the core consists of all the stable profit imputations; that is, . However, cost is negative profit. So we use the cost saving replacing cost obtained from the cooperation among the shippers to further study the nature of the game. The cost saving of the coalition is as follows: As Table 3 indicates, each possible coalition the minimum total cost and related cost savings can be calculated.



Provided the cost savings for the individual shippers in the separate operation is zero, the cost savings generally increase with the expansion of the scale of the coalition. Ideally, largest cost savings would correspond with the grand coalition. For this instance, the function is monotonic, super additive, and balanced. As can be revealed in Figure 5, the core of this four-shipper game can be figured out by TUGlab [26]. The corresponding core vertices are shown in Table 4.



What is noteworthy is, as can be observed from this example, that the core-vertex solutions and core solutions are unfair when allocating small savings to one or more shippers. For example, Shipper A is allocated with nothing among five of the core vertices. In fact, this kind of distribution is unacceptable to related shippers as a rule. It justifies why it is far from enough to simply require the solution to be stable. It is more convenient to impose the fairness with the geographical location criteria as in the case of LCAM dose.

As Table 5 indicates, in this instance, the cost allocation shall meet the constraints of Shipper C > Shipper D > Shipper A > Shipper B according to Constraint (20). However, as shown in Table 5, the postcollaboration cost of each shipper given by Shapley or EPM is Shipper C > Shipper D > Shipper B > Shipper A, and they all do not satisfy condition (20). Nucleolus satisfy the condition (20) luckily. Because Shapley, nucleolus, and EPM do not consider the impact of geographical location of nodes on cost allocation by definition. So in terms of geographical location criterion the LCAM would be preferred over the other methods discussed in paper. It should be noted that, in some cases, there may be coincidences to meeting constraints (20) via Shapley value, nucleolus, or EPM.



4.2. Randomly Generated Instances

In this section, we use three criteria presented in Section 3.6 to evaluate the Shapley, nucleolus, EPM, and LCAM, through two randomly generated examples. All results are obtained by using an Intel® Core™ i5-5200 CPU @ 2.2GHz with 8 GB of RAM.

Stability. First, the core turned out to be nonempty for each instance. Because the nucleolus, EPM, and LCAM guarantee a solution in the core whenever it is nonempty, their allocations are stable by definition. But the Shapley value is not constructed by implicitly using any of the core criteria. Therefore, from a stability point of view, the Shapley value performs worst. However, for each instance in Tables 6 and 8, all four mechanisms develop a cost allocation that stays at the core.


Pre-collaboration costs76.3143.9355.2282.3224.8168.986.4593.5632.5638.6
Post-collaboration costs171.0306.0439.3524.7347.8502.7176.01050.81097.31038.3
Cost collaboration of Shapley32.962.9298.1155.681.650.326.0366.9363.7450.1
Cost collaboration of Nucleolus35.958.2301.9169.092.068.440.0273.1505.8580.0
Cost collaboration of EPM33.171.8217.5132.681.377.431.5338.1348.3381.6
Cost collaboration of LCAM36.764.3261.4157.181.760.323.8342.3338.5415.4


Pre-collaboration costs533.2241.5675.2202.7109.1134.6288.8619.1291.8467.7
Post-collaboration costs820.3549.7838.5444.6608.0244.9422.11002.1338.0758.2
Cost collaboration of Shapley252.0107.5371.780.152.147.6106.9177.2126.1333.7
Cost collaboration of Nucleolus279.8105.1392.3178.373.955.2163.8192.5207.9367.7
Cost collaboration of EPM226.9122.2328.5108.665.852.9112.7222.8116.3256.8
Cost collaboration of LCAM258.4109.9348.

Consistency. In order to assess the consistency of the different allocation methods, we need to design random instances. We evaluate consistency based on marginal distance and average distance. Marginal distance is the decreased value of the total route length after removing node while keeping others with the original visiting sequence. Average distance is the average distance between node and every other node. We evaluate the consistency by using an ordinary least squares regression.

So far, without benchmark instance for SCAPPD, 20 randomly generated instances are applied to assess the performance of this mechanism. Each instance contains a vector. Set the number of nodes to 9 (1 node as depot node and 8 others as shipper node). The coordinates of each node are freely developed by the computer. The capacity of each vehicle is 60 units. Each request is related to 2 nodes, pickup node as well as delivery node. These requests are developed through random selection of one pickup node ( and a delivery node , each related to at most one request. Each request is related to a randomly introduced quantity of freight units no bigger than a predefined number that is defined to 60. The time window for all request service is set to (480 minutes = 8 hours).

According to the experimental results, the cost reduction of each shipper resulting from the four mechanisms can be different, which is due to the fact that the core of each instance includes multiple possible allocations and that different definitions of fairness in each mechanism can result in different core allocations. The results of the LCAM pay more attention to the impact of geographical location on the allocation. We evaluate it based on the marginal distance and the average distance to other shippers. As these two factors simultaneously influence the allocate result, we have performed an ordinary least squares regression to evaluate each allocation method, where the allocate result is the dependent variable. The explanatory variables are a constant, the marginal distance, and the average distance to other shippers.

A summary of the regression results is shown in Table 7. For each allocation method we have reported the estimated coefficient and its significance level. We use as a measure to compare the allocation methods. is the proportion of variation in the dependent variable that is explained by the explanatory variables. We see that the LCAM has the highest and hence performs best on the geographical location criterion, while Shapley and EPM perform relatively poor. Nucleolus performs worst.

Explanatory variableShapleyNucleolusEPMLCAM

Marg. dist0.1070.0000.1040.0000.0650.0000.0860.000
Avg. dist. to other cust.0.4460.0000.4330.0000.5140.0000.5110.000



Pre-collaboration costs76.372.279.073.573.582.475.479.172.886.4
Post-collaboration costs171.0157.2163.4166.5175.2168.6173.1175.3177.0181.6
Cost collaboration of Shapley32.825.031.426.233.937.335.135.932.644.0
Cost collaboration of Nucleolus35.829.132.228.337.439.741.938.535.046.1
Cost collaboration of EPM33.129.032.531.332.936.532.334.333.040.0
Cost collaboration of LCAM36.728.


Pre-collaboration costs73.685.975.474.879.881.979.865.883.169.0
Post-collaboration costs167.8173.9166.4170.7174.7174.0173.3169.1190.1182.1
Cost collaboration of Shapley29.639.731.431.330.734.933.831.340.736.0
Cost collaboration of Nucleolus34.339.833.835.232.537.138.134.945.942.6
Cost collaboration of EPM30.537.831.732.633.635.433.330.737.432.8
Cost collaboration of LCAM33.441.834.735.434.

Robustness. In order to assess the robustness of the different allocation methods, we need to design another random example. We are interested in the variation of the target shipper’s allocation in a similar task. So we randomly generated 20 instances similar to the instance in Table 1, in which there is no change in request 3, and observe the variation of the allocation of Shipper C to different methods.

We measure the robustness as the standard deviation relative to the average allocated emission, which is the coefficient of variation (CoV). To see which allocation method performs best in terms of robustness, we compare the CoV for the different methods. The coefficient of variation for the instances is shown in Table 9. We see that EPM and LCAM are the best method in which CoV does not exceed 5%. LCAM performs worse than EPM in terms of robustness because LCAM considers more constraints; it is closer to the actual situation and seeks solutions in a smaller feasible domain.

Target shipperAllocation methodStd. dev. of allocationAverage allocationCov

Shipper CShapley3.461.60.0546

5. Conclusion

In this paper we study SCAPPD, in which we need to allocate the shipper’s cost fairly. We analyze three other allocation methods based on well-known solution concepts in cooperative game theory: the Shapley value, nucleolus, and EPM. Furthermore, LCAM are proposed which consider marginal cost and the dispersion degree of nodes in cost allocation on the basis of the core allocation method. Each allocation method is evaluated in terms of stability, consistency, and robustness. In our numerical experiments, the stability of the LCAM is as good as the nucleolus and EPM. In terms of consistency the LCAM performs best. LCAM also performs well in terms of robustness. LCAM offers more choices for the shippers to choose the cost allocation mechanism they want.

In this study, each shipper is assumed to have only one pickup and delivery request in model LCAM. The model considering one-to-one (1-1) problems is the basis of many-to-many (M-M) problems. Hence, it is interesting to extend this research to consider a more general case where each shipper has multiple pickup and delivery requests.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


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Copyright © 2019 Ning Liu and Yaorong Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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