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A Game-Theoretic Approach to the Freight Transportation Pricing Problem in the Presence of Intermodal Service Providers in a Competitive Market

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Abstract

This article studies a competitive freight transportation pricing problem in the presence of two Intermodal Service Providers (ISPs) and a Direct Transportation System (DTS). The ISPs apply both rail and road transportation modes to carry the demands of a network of customers. The DTS uses only roads to carry the demands, without any transhipment at a distribution center. Each customer chooses its best transportation service based on the prices offered by the ISPs and the expenses of using the DTS. The ISPs determine their prices to maximize their profits, considering the customers’ choice behaviour. In order to determine the equilibrium decisions, a non-cooperative game-theoretic approach based on Stackelberg leader-follower competition is applied. Mixed-integer linear programming models are proposed to formulate this competition. A real-life case study is also conducted to demonstrate the validity of the models. We find that a barrier pricing strategy from the leader to deter the entrance of the follower ISP is not recommended for both of them because it may even lead to a negative value of profits for the leader. Finally, some sustainability objectives of the government, as the strategic decision maker, are examined. The results could help the government assess the effects of its policies on the transportation market, the environment, and the society.

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Abbreviations

ISP:

Intermodal Service Provider

DTS:

Direct Transportation System

a, a ′ :

The index of the Intermodal Service Provider (ISP) companies, a, a′ ∈ {1, 2}.

i, j, m :

The indices of customers, i, j, m ∈ {1, 2, …, n}.

M :

A very big positive number.

ε :

A very small positive number (the smallest monetary unit, e.g., one cent).

n :

Total number of customers.

D a :

The distance between the manufacturing company and the distribution center of ISPa (km).

d a, i :

The distance between the distribution center of ISPa and customer i (km).

\( {d}_{a,i}^T \) :

Total distance between the manufacturing company and customer i for ISPa, \( {d}_{a,i}^T={d}_{a,i}+{D}_a \).

e a :

Transportation cost of ISPa for carrying the demands of customers from the manufacturing company to the distribution center of ISPa (per ton − km).

\( {e}_a^{\prime } \) :

Transportation cost of ISPa for carrying the demands of customers from the distribution center of ISPa to a customer (per ton − km).

h a :

Operational cost of ISPa including loading, unloading, batching costs (per ton).

c a, i :

Total transportation cost for ISPa for carrying the demand of customer i (per ton), \( {c}_{a,i}={h}_a+\left({D}_a\times {e}_a\right)+\left({d}_{a,i}\times {e}_a^{\prime}\right) \).

k i :

The ratio of total distance of ISP1 with respect to total distance of ISP2 for customer i, \( {k}_i=\frac{d_{1,i}^T}{d_{2,i}^T} \).

F i :

Transportation expense of customer i when using the DTS (per ton).

w a :

The threshold price in which ISPa exits the competitive market, \( {w}_a=\mathit{\operatorname{Max}}\left\{\frac{F_i}{d_{a,i}^T}|i=1,2,\dots, n\right\}+\varepsilon \) (per ton − km).

Q i :

The demand of customer i (ton).

β a, i :

The probability of choosing ISPa by customer i, if the customer is indifferent between the transportation expenses of ISPa and those of the other ISP, ∀a, i : βa, i ∈ [0, 1], ∀ i : β1, i + β2, i = 1.

θ a, i :

The probability of choosing ISPa by customer i, if the customer is indifferent between the transportation expenses of ISPa and those of the DTS, ∀a, i : θa, i ∈ [0, 1].

I :

Investment costs of establishing an intermodal infrastructure.

p a :

The transportation price of ISPa (per ton − km).

L a, i :

The willingness-to-pay of customer i for ISPa.

\( {L}_{a,i}^{\prime } \) :

The indifference price of customer i for ISPa.

x a, i :

(Binary) 1, if ISPa chooses La, i as its price pa; 0, otherwise.

r a, i :

(Binary) 1, if ISPa chooses \( {L}_{a,i}^{\prime } \) as its price pa; 0, otherwise.

R i :

(Auxiliary variable) equals to \( {r}_{2,i}{L}_{2,i}^{\prime } \).

\( {r}_{a,i}^{\prime } \) :

(Binary) 1, if customer i partially chooses ISPa and is different between the expenses of ISPa and those of the DTS, i.e. \( {p}_a=\frac{F_i}{d_{a,i}^T}\&{p}_{a^{\prime }}>\frac{F_i}{d_{a^{\prime },i}^T}\ \left({a}^{\prime}\ne a\right) \); 0, otherwise.

ω a, i :

(Binary) 1, if \( {p}_a<\frac{F_i}{d_{a,i}^T} \); 0, otherwise.

τ a, i :

(Binary) 1, if \( {p}_a>\frac{F_i}{d_{a,i}^T} \); 0, otherwise.

\( {r}_{a,i}^{\prime \prime } \) :

(Binary) 1, if customer i partially chooses ISPa and is different between the expenses of ISPa and those of the other ISP, i.e. \( {p}_1=\frac{p_2}{k_i}\le \frac{F_i}{d_{1,i}^T} \); 0, otherwise.

\( {\tau}_i^{\prime } \) :

(Binary) 1, if \( {p}_1>\frac{p_2}{k_i} \); 0, otherwise.

\( {\omega}_i^{\prime } \) :

(Binary) 1, if \( {p}_1<\frac{p_2}{k_i} \); 0, otherwise.

z a :

(Binary) 1, if ISPa decides to exit the market; 0, otherwise.

y a, i :

(Binary) 1, if customer i (partially or completely) chooses ISPa; 0, otherwise.

b i, j :

(Binary) 1, if L2, i ≤ L2, j, or equivalently, \( {L}_{2,i}^{\prime}\le {L}_{2,j}^{\prime } \); 0, otherwise.

\( {b}_{i,j}^{\prime } \) :

(Binary) 1, if L2, i = L2, j, or equivalently, \( {L}_{2,i}^{\prime }={L}_{2,j}^{\prime } \); 0, otherwise.

λ i. j :

(Binary) 1, if L2, i > L2, j, or equivalently, \( {L}_{2,i}^{\prime }>{L}_{2,j}^{\prime } \); 0, otherwise.

μ i. j :

(Binary) 1, if L2, i < L2, j, or equivalently, \( {L}_{2,i}^{\prime }<{L}_{2,j}^{\prime } \); 0, otherwise.

s a, i :

(Binary) 1, if customer i prefers the expenses of ISPa over the expenses of the DTS, i.e. \( {p}_a\le \frac{F_i}{d_{a,i}^T} \); 0, otherwise.

g a, i :

(Auxiliary variable) equals to pasa, i.

U a, i :

(Auxiliary variable) equals to paya, i.

\( {U}_{a,i}^{\prime } \) :

(Auxiliary variable) equals to \( {p}_a{r}_{a,i}^{\prime } \).

\( {U}_{a,i}^{\prime \prime } \) :

(Auxiliary variable) equals to \( {p}_a{r}_{a,i}^{\prime \prime } \).

X i :

(Auxiliary variable) equals to L2, ix2, i.

B i, j :

(Auxiliary variable) equals to L2, ibi, j.

\( {B}_{i,j}^{\prime } \) :

(Auxiliary variable) equals to \( {L}_{2,i}^{\prime }{b}_{i,j} \).

\( {B}_{i,j}^{\prime \prime } \) :

(Auxiliary variable) equals to \( {L}_{2,i}^{\prime }{b}_{i,j}^{\prime } \).

V i :

(Binary) 1, if \( {p}_1\le \frac{F_i}{d_{a,i}^T} \); 0, otherwise.

f a andπ a :

The profit function of ISPa.

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Acknowledgement

The authors acknowledge the editors and the two anonymous reviewers for their helpful comments that improved the article. The authors are also grateful to Mr. A. Taghizadegan for providing GIS maps.

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  1. Department of Transportation Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Mohammad Tamannaei

  2. Department of Industrial and Systems Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Hamid Zarei & Sajede Aminzadegan

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  1. Mohammad Tamannaei
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Tamannaei, M., Zarei, H. & Aminzadegan, S. A Game-Theoretic Approach to the Freight Transportation Pricing Problem in the Presence of Intermodal Service Providers in a Competitive Market. Netw Spat Econ 21, 123–173 (2021). https://doi.org/10.1007/s11067-020-09511-8

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