We study cost allocation problem arising from less-than-truckload collaboration among perishable product retailers. The relevant costs we consider include fixed transportation cost, variable transportation cost, and decay loss of perishable products. Cooperative game theory is applied to study this cost allocation problem. The corresponding cooperative game, called transportation facility choice game, is established. First, we show that the core of the transportation facility choice game is non-empty. Then, we identify some conditions for concavity and quasi-concavity of the transportation facility choice game with the linear decay and negative exponential decay functions, respectively. Finally, simulation is conducted to analyze how optimal solutions differ under the linear decay and exponential decay functions, and intuitive cost allocation schemes are proposed and compared with the\(\tau \)-value and the Shapley value of the corresponding game. Simulation results show that the optimal solution under linear decay function tends to choose facilities with higher fixed cost than that under exponential decay function. Additionally, among all the cost allocation schemes compared, the simple cost allocation scheme called A-IM, the\(\tau \)-value, and the Shapley value have better performance in terms of the percentage of allocations lying in the core.

The authors thank the associate editor and the referees for their insightful comments and suggestions. They helped the authors improve both the content and exposition of this work. The authors also express their gratitude to Mr. Zhuhui for his help on simulation. This work was supported by Major Program of the National Natural Science Foundation of China (71490725, 71490722) and Program of the National Natural Science Foundation of China (71271178).

Authors and Affiliations

1. School of Economics and Management, Southwest Jiaotong University, No. 111 Erhuan Road, Jinniu District, Chengdu, 610031, Sichuan, People’s Republic of China

   Jun Li

2. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

   Xiaoqiang Cai & Yinlian Zeng

Corresponding author

Correspondence toJun Li.

Appendix 1: Proof of Proposition3

Proof

For notational convenience, in this proof we usec(S) to denote\(c_{L}(S)\) (that is, the subscript are omitted). Other notions are simplified similarly. If\(c(S)=\underset{k\in F}{\min }c(S)_{k}\), then\(c(S)=\underset{k\in F}{\min }\left\{ f_{k}+\underset{j\in S}{\sum }a_{k} d_jq_{j}+\underset{j\in S}{\sum }t_{k}d_j\theta _{j}q_{j}\right\} \). Let\(f^{S}\),\(a^{S}\) and\(t^{S}\) be the fixed cost, unit variable cost and traveling time per unit traveling distance corresponding toc(S) , respectively. Then,

(1) For\(i,j\in N\), we have\(c(\{i\})=\underset{k\in F}{\min }\left\{ f_{k}+a_{k}d_iq_{i}+t_{k}d_i\theta _{i}q_{i}\right\} =f^{\{i\}}+a^{\{i\}}d_iq_{i} +t^{\{i\}}d_i\theta _{i}q_{i}\). Similarly\(c(\{j\})=f^{\{j\}}+a^{\{j\}}d_jq_{j}+t^{\{j\}}d_j\theta _{j}q_{j}\). Let\(c(S)_{W}\) be the total cost for coalitionS whenW is the optimal set of facilities used by coalitionS. Then

and

Thus,

If\(f_i\theta _{\min }\delta \ge a_i\), then\(\delta d_j\theta _{j}q_{j}-a^{\{j\}}d_jq_{j}/f^{\{j\}}\ge 0\), and\(\delta d_i\theta _{i}q_{i}-a^{\{i\}}d_iq_{i}/f^{\{i\}}\ge 0\).

If\(f^{\{i\}}\ge f^{\{j\}}\), we have\(c(\{i,j\})_{^{\{i\}}}-c(\{i\})-c(\{j\})\le -f^{\{j\}}<0\); otherwise\(c(\{i,j\})_{\{j\}}-c(\{i\})-c(\{j\})\le -f^{\{i\}}<0\).

If\(f_i\theta _{\max }\delta \le a_i\), then\(\delta d_j\theta _{j}q_{j} -a^{\{j\}}d_jq_{j}/f^{\{j\}}\le 0\), and\(\delta d_i\theta _{i}q_{i}-a^{\{i\}}d_iq_{i}/f^{\{i\}}\le 0\).

If\(f^{\{i\}}\le f^{\{j\}}\), then\(c(\{i,j\})_{\{i\}}-c(\{i\})-c(\{j\})\le -f^{\{j\}}<0\); otherwise\(c(\{i,j\})_{^{\{j\}}}-c(\{i\})-c(\{j\})\le -f^{\{i\}}<0\).

Given that at least one inequality is less than 0,\(\min \left\{ c(\{i,j\})_{\{i\}},c(\{i,j\})_{\{j\}}\right\} <c(\{i\})+c(\{j\})\), so\(c(\{i,j\})=\underset{_{k\in F}}{\min }c(\{i,j\})_{k}\).

(2) For\(S,T\in N\), and\(S\cap T=\emptyset \), let\(S\cup T=ST\), and suppose that\(c(S)=\underset{k\in F}{\min }c(S)_{k}\) on the setS andT. We then arrive at

Note that\(\underset{_{k\in F}}{\min }~c(S\cup T)_{k}\le f^{S}+t^{S}\underset{j\in S\cup T}{\sum }d_j\theta _{j}q_{j}+a^{S}\underset{j\in S\cup T}{\sum }d_jq_{j} \), and\(\underset{_{k\in F} }{\min }~c(S\cup T)_{k}\le f^{T}+t^{T}\underset{j\in S\cup T}{\sum }d_j\theta _{j}q_{j} +a^{T}\underset{j\in S\cup T}{\sum }d_jq_{j} \). Let\(c(S,T)=\underset{_{k\in F}}{\min }~c(S\cup T)_{k}-c(S)-c(T)\). Thus, we have

Combining ConditionA and ConditionB with (A.1) and (A.2), respectively, we can obtain

When\(f_i\theta _{\min }\delta \ge a_i\), we have\(\underset{j\in S}{\sum } \delta d_j\theta _{j}q_{j}-\underset{j\in S}{\sum }\frac{a^Sd_jq_{j}}{f^S}\ge 0\) and\(\underset{j\in T}{\sum }\delta d_j\theta _{j}q_{j}-\underset{j\in T}{\sum }\frac{a^Td_jq_{j}}{f^T}\ge 0\). Furthermore, if\(f^{S}\ge f^{T}\), then it follows from (A.4) that\(c(S,T)\le -f^{T}<0\); otherwise,\(c(S,T)\le -f^{S}<0\). Thus,\(c(S,T)<0\).

When\(f_i\theta _{\max }\delta \le a_i\), we have\(\underset{j\in S}{\sum }\delta d_j\theta _{j}q_{j}-\underset{j\in S}{\sum } \frac{a^Sd_jq_{j}}{f^S}\le 0\) and\(\underset{j\in S}{\sum }\delta d_j\theta _{j}q_{j}-\underset{j\in S}{\sum }\frac{a^Sd_jq_{j}}{f^S}\le 0\). Furthermore, if\(f^{S}\ge f^{T}\), then it follows from (A.3) that\(c(S,T)\le -f^{S}<0\); otherwise,\(c(S,T)\le -f^{T}<0\). Thus,\(c(S,T)<0\).

From the analysis above, we have\(\underset{_{k\in F}}{\min }c(S\cup T)_{k}<c(S)+c(T) \).

Given that

we have\(c(S\cup T)=\underset{k\in F}{\min }c(S\cup T)_{k}\).

(3) From (1), we know for\(S,T\in N\), and\(S\cap T=\emptyset \),\(\left| S\cup T\right| =2\), and so\(c(S\cup T)=\underset{k\in F}{\min }c(S\cup T)_{k}\). From (2), we know if\(\left| S\cup T\right| \le 3\), then\(c(S\cup T)=\underset{k\in F}{\min }c(S\cup T)_{k}\). It can be derived similarly that\(c(S\cup T)=\underset{k\in F}{\min }c(S\cup T)_{k}\), for all\(S\in N\).\(\square \)

Appendix 2: Proof of Proposition5

Proof

For\(l\notin S\subset T\subseteq N\), let\(h(S,\{l\})=c(S\cup \{l\})-c(S)\). Then,

Since

and

we have

Similarly, we can show that

Therefore,

Furthermore,

Given that

and

we have

Combining ConditionA and ConditionB with (A.5) and (A.6), respectively, we obtain

When\(f_i\theta _{\min }\delta \ge a_i\), we have\(\delta d_l\theta _{l}q_{l}-\frac{a^Td_lq_{l}}{f^T}\ge 0\) and\(\underset{j\in T-S}{\sum }\delta d_j\theta _{j}q_{j}-\underset{j\in T-S}{\sum }\frac{a^{S\cup \{l\}}d_jq_{j}}{f^{S\cup \{l\}}}\ge 0\). Furthermore if\(f^{S\cup \{l\}}\le f^{T}\), then from (A.7) we have\(h(S,l)-h(T,l)\ge 0\); otherwise from (A.8) we get\(h(S,l)-h(T,l)\ge 0\).

When\(f_i\theta _{\max }\delta \le a_i\), we have\(\delta d_l\theta _{l}q_{l} -\frac{a^Td_lq_{l}}{f^T}\le 0\) and\(\underset{j\in T-S}{\sum }\delta d_j\theta _{j} q_{j}-\underset{j\in T-S}{\sum }\frac{a^{S\cup \{l\}}d_jq_{j}}{f^{S\cup \{l\}}}\le 0\). Furthermore if\(f^{S\cup \{l\}}\le f^{T}\), from (A.8) we have\(h(S,l)-h(T,l)\ge 0\); otherwise from (A.7) we have\(h(S,l)-h(T,l)\ge 0\).

The analysis above shows that\(c(S\cup \{l\})-c(S)\ge c(T\cup \{l\})-c(T)\). This establishes the concavity of the game\((N,c_{L})\).\(\square \)

Appendix 3: Proof of Proposition7

Proof

Suppose that\(c_{E}(S)=\underset{k\in F}{\min }c_{E}(S)_{k}\) on the setS,T (\(S,T\subset N\)). For notational convenience, in this proof usec(S) to denote\(c_{E}(S)\) (other notations are simplified in a similar way). For\(S,T\subset N\), and\(S\cap T=\emptyset \), we have

and

Given that

and

then

and

If\(t^{T}\ge t^{S}\), then it follows from Lemma1 that

From (A.9), we have

Combining with ConditionB, we have

If\(t^{T}<t^{S}\), then it follows from Lemma1 that

From (A.10), we have

Combining with ConditionB, we have

Given that

then

and

Therefore\(c(S,T)<0\). When\(t^{T}\ge t^{S}\), from (A.11), we have\(c(S,T)<0\). Otherwise, (A.12) yields\(c(S,T)\le -f^{T}<0\). Similar to the proof of Proposition3, we can prove\(c(S)=\underset{k\in F}{\min }c(S)_{k}\), for all\(S\in N\).\(\square \)

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