Optimization of Sharpley Value Method of Cost Allocation in a Bimodal
Transport- Supply Chain Distribution Via Dynamic Programming
HARRISON OBIORA AMUJI*
Department of Statistics
Federal University of Technology, Owerri
PMB 1526, Owerri Imo State
NIGERIA
BRIDGET NWANYIBUIFE OKECHUKWU
Department of Statistics
Federal University of Technology, Owerri
PMB 1526, Owerri Imo State
NIGERIA
IHEANYI CHINEDU OBINWANNE
Department of Statistics
Federal University of Technology, Owerri
PMB 1526, Owerri Imo State
NIGERIA
VIVIAN NGOZI IKEOGU
Department of Logistics and Transport Technology
Federal University of Technology, Owerri
PMB 1526, Owerri Imo State
NIGERIA
SAMUEL INAKU EMRI
Department of Urban and Regional Planning
University of Cross River State
Calabar, Cross River State
NIGERIA
Abstract: - In this paper, we proposed a coalition between two modes of transportation, where one
provides cargo train and the other provides trucks of capacity 453 tons respectively for the coalition.
We have five grand coalitions and four coalitions. The coalitions were distributed along five paths
across Nigeria and specialized in the distribution of agricultural produce from the north to the south.
Each of the coalition was made up of four transport providers and composed of four legs. Since this is
a cooperative game scenario, Sharpley's value method of cost allocation was used to obtain the gains
accrued to the grand coalition. The coalition made a total savings of 1259.6 million naira within the
period under study. The researchers further developed and applied a Dynamic programming model to
the supply chain distribution and obtained an intelligent result. They discovered that if the five coalitions
were distributed among the four legs, in this other (1, 1, 2, 1), the grand coalition would make an
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
30
Volume 6, 2024
additional gain of 377.6 million naira. The distribution means that the allocation of the coalition to the
third leg should be doubled while the other legs remain as they were. The introduction of the
optimization method into the system brought additional revenue to the coalition and increased the total
gain to 1637.3 million naira.
Key-Words: - Bimodal mode of transportation, Sharpley value method, Transport and logistics,
Dynamic programming, Optimal decision policy, Gains to the grand coalition.
Received: June 13, 2023. Revised: February 16, 2024. Accepted: March 17, 2024. Published: May 20, 2024.
1 Introduction
Sharpley value method of cost allocation shares
some similarities with dynamic programming
[1]. It is a method based on some mathematical
axiom, which distributes average cost based on
the contribution of the participant to the grand
coalition; the method is based on marginal cost.
Also, dynamic programming is anchored on the
principle of marginal cost, in which the current
status is based on the immediate past event [2],
meaning that the principle of optimality is
dependent on marginal cost. Therefore, both
methods share a common feature of marginal
cost.
In the supply chain distribution, competition
among the transport providers is obvious and in
most cases counter-productive. This kind of
competition among players brings us to
competitive game theory. A competitive game
theory is an activity between two or more
rational players with strategies at the end each
player receives a reward or suffers a loss [3], this
is a competitive game with either pure or mixed
strategies. When a player is using a specific
course of action, it is said to be using a pure
strategy and we have this kind in a static game,
but when a player is using the strategy in a
certain proportion such that the opponent keeps
guessing on which the next move of the player
would be, the player is said to be using a mixed
strategy and we have such in a dynamic game.
A competitive game is also called a zero-sum
game because a loss to one player is a gain to
the other player and each player would like to
eliminate the other from existence, that is one of
the criteria for dominance and remaining
relevant in the market. In contrast, a cooperative
game theory proposes cooperation among
players, where each player will concentrate on
an area of specialization for the overall interest
of the players and the market and share
collectively the proceeds or loss from the
outcome of their cooperation. In this case, the
Sharpley value method offers a unique approach
to handling this kind of scenario in the
cooperative game theory [4]. In a supply chain
distribution powered by transportation,
cooperation among different modes of
transportation encourages specialization and
cost reduction. This is the principle behind
cooperative game theory.
In this paper, we propose a coalition in a
bimodal transport system. The cooperation is
between Rail transport via cargo trains and Road
transport via trucks [5] that ship cargoes from
origin to destinations. The train ships cargo
from the origin to the terminal where the truck
picks it up and supplies/distributes the cargo to
various terminals where they are either stored or
distributed immediately to the final consumers.
Our interest is in the collaboration between
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
31
Volume 6, 2024
these modes of transport and among the
coalitions. We encourage this kind of
cooperation not only for the economies of scale
but for healthy competition and the overall good
of the economy. We use the Sharpley value
method to obtain the savings among different
legs of the coalition and the sum of savings per
coalition. But instead of stopping there to
determine the percentage savings, our interest
was to discover how more gains could be made
on the savings using the optimization method.
Using the optimization method, we seek to
determine the optimal allocation of the
coalitions to different legs to maximize savings
using the Dynamic programming model. Our
interest is to determine how best the coalition
can be allocated to optimize savings for the
grand coalition. Therefore, we developed a
Dynamic programming model and the optimal
distribution policy [6] that will allocate
appropriate carriers to different legs (routes).
This will help the players gain more insight
about the coalition business thereby having
control of it.
For a better understanding of the work, we
organize the paper as follows: in section one, we
treated the introduction of the work, the
background of games theory and the types,
coalition among modes of transportation,
dynamic programming and its relationship with
Sharpley value method of cost allocation in a
cooperative game theory, the specific modes of
transportation that collaborate in this study. In
section two, we treated the literature review on
the related area, even though we acknowledge
that no direct work was done specifically on the
topic of this paper. In section three, we treat
materials and methods; here, we state the
method that would be applied to achieve the aim
of the paper. In section four, we presented data
and analysed it using both the Sharpley value
method in conjunction with the Dynamic
programming model stated in section three, and
finally, in section five, we treated the result and
general discussions of the work.
2 Literature Review
The literature in this area is rare because no
direct work was done on it, nevertheless, we
reviewed some related works. Competition
among players (transport providers) may be
destructive among rivals who seek to provide
the same service. However, the cooperation
among different modes of transportation where
each of them engages in different sections of the
distribution is called horizontal cooperation.
Each player performs a given assignment for the
overall interest of the coalition. A lot of benefits
are derived from horizontal cooperation [7]
because instead of unhealthy rivalry, the players
build an inclusive system that benefits the entire
players and stabilizes the market. Collaboration
among transport providers is cost-saving [8]
with a reduction in delivery time. The gains of
cooperative game theory, where players act in
unison is an aspect of game theory that should
be encouraged in real-life scenarios against a
zero-sum game where each player sees the other
as a rival. Cooperative game theory is a
characteristic function of an n-person
cooperative game with a set C of players
assigned to each subset S of C the maximum
value v(S) that coalition S can guarantee itself
by coordinating the strategies of its members
[9]. The gains from the coalition are shared
among the players.
Our focus in this paper is on the discrete serial
dynamic programming structure [10] where one
stage receives input from the previous stage and
sends its output to precisely one preceding
stage. This is a case where trains receive cargo
from the shipper, and convey it to a terminal
where trucks distribute them to various
destinations [11] demonstrates the various
applications of dynamic programming
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
32
Volume 6, 2024
modelling in solving real-life problems in which
supply chain distribution would not be an
exception. Dynamic programming breaks the
entire problem into different stages, with each
stage having independent decisions and policy;
the current stage is linked to the immediate past
stage through the principle of optimality [12]
and the recursive relationship is established
through the backward pass. Dynamic
programming is so flexible that it can be used to
model so many complex problems that other
optimization methods could not be used, it is a
branch of decision theory. For example, [13]
used Dynamic programming to model the
ASUU strike and from it offer a solution on how
to avert future ASUU strikes. We used dynamic
programming to model the gains from the
Sharpley value method from the bimodal
cooperation among the transport providers.
The applications of game theory - cooperative
and zero-sum game – are so wide, provided the
subject to be modelled has an element of
competition or decision making. Game theory
was used in the modelling of the cryptocurrency
[14] market and its players. The study was
specifically for Bitcoin and blockchains. The
author was motivated to model the Bitcoin
problem with a zero-sum game bearing in mind
that a loss to one player is a gain to its opponent.
Game theory is mostly concerned with decision-
making, in other words, game theory is within
the domain of decision theory and applied in
solving various economic problems [15]. Game
theory, be it zero-sum or cooperative, has one
common objective, that is, to apply the rational
decision to win over the opponent, depending on
what the opponent is. Apart from the empirical
application of game theory, other researchers
[16] did extensive work on the theoretical
application of game theory and viewed it as a
complex mathematical theory developed
specifically for decision-making. They
observed that game theory is associated with the
conflict situation that requires rationality to win
or minimize losses. They noted some areas of
application of games such as social life, political
and economic life; and that the players include
political parties, government authorities, firms
or businesses, prison inmates and professional
sports franchises, etc. To further demonstrate
the application of Game theory, [17] applied it
between doctors and patients to determine an
equilibrium point which is the value of the
game.
Some researchers [18] did review work, where
more insights about game theory were given.
They compared the popularity of game theory
and entrepreneurship using different journal
indexing databases such as EBSCO, Google
Scholar, etc., and the review showed that game
theory was more popular than entrepreneurship
based on the search made. Also, similar work
[19] was done on bibliometric analysis of game
theory to compare the popularity of game theory
with energy and natural resources. The research
was based on published works in Web of
Science (WOS) indexing. The result showed
that game theory was more popular. Hence,
game theory is gaining more popularity since
virtually all human endeavours involve
rationality and competition. In this study, we
seek to optimize the gains in cooperative game
theory offered by the Sharpley value method via
dynamic programming.
3 Materials and Methods
In this paper, we proposed a bimodal
transportation mode with a grand coalition of
five. The transport providers are generally
categorized into two groups with one providing
cargo trains and the other providing trucks of
capacities 453 tons respectively. The two
transportation modes are concerned with the
distribution and supply of agricultural produce
from northern to southern Nigeria. We have
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
33
Volume 6, 2024
observed that this collaboration between the
transportation modes and among players falls
within the purview of cooperative game theory.
Shapley value is the most fair and efficient cost
allocation method based on cooperative game
theory [20]. According to [5], cooperative game
theory is a pair (N, v) with a finite set of players
v: 2N R, and the function that assigns a real-
valued payoff v(S) to each coalition S N with
v () = 0. If we let |S| be the number of members
in the coalition S and N \ {k}, be the grand
coalition and xi denotes the share of the grand
coalition’s payoff that a player i N receives;
then, the Shapley value of player k is defined as:
(1)
Which can be written as;
(2)
where is the total cost of
transportation, the new cost obtained from
Sharpley value method and is the savings for
the ith partner of the coalition. However, our
focus is to optimize the allocation of the
coalitions to various legs of the coalition to
further maximize the savings through the
optimal allocation policy developed in this
paper.
To optimize the gains in equation (2), we
carefully observed that equation (2) is a special
case of the Dynamic programming model where
the total cost, , is composed of new cost
plus marginal cost (savings); hence, we can
write equation (2) as;
(3)
Equation (3) is a Dynamic programming model,
See [13], where is the optimization
function of two variables (stage and state
variables); is the function that assigns
values to gains due to the collaboration
(marginal cost);
is the new cost from
Sharpley value method.
The optimal allocation policy is determined as
follows; let Ni be the stage variables, i = 1, . . .,
n; vi be the state variables and di* be the optimal
decision variable at each stage, then the optimal
allocation policy is:
N1 = n1; v1* = d1
N2 = (n1 - d1) = dk2; v2* = d2,
………………. (4)
Nn-1 = (n-1 – dkn - 1); vn-1* = dn-1
Nn = (n – dn -1); vn* = dn
Therefore, the allocation of the coalition to
different legs in the following order (d1, d2, …,
dn), will yield additional gain to the grand
coalition. Hence, the total gain for the grand
coalition will be (K + m) = (Pk) million naira;
where K = gains from the Sharpley value
method, m = gains from the optimization
method, and Pk = the total gains in millions of
naira.
4 Data Presentation and Analysis
4.1 Data Presentation
We present the data from the origin to
destination, bimodal transportation route,
transport providers, unit cost, and total unit cost
in Appendix 4. Again, we presented the
computation of savings on each leg of the
coalitions in Appendix 5. We present Tables 1
and 2 as Appendix 1 and 2 and their
computation were based on Appendix 4 and 5.
In Appendix 6, we present the computational
tables for the problem from where we determine
the optimal decision policy.
4.2.1 Data Analysis 1
In this subsection, we present two types of
analysis, namely, the analysis based on the
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
34
Volume 6, 2024
Sharpley value method presented in Appendix 1
and the analysis based on the dynamic
programming model presented in Appendix 2.
In the analysis in Appendix 1, we saw that there
were five grand coalition (|N|) and four
operators per leg and each of the legs has
uniform weight (wi). We observed that the first
coalition (S1) made 21.86% savings from the
supply chain distribution through collaboration
thereby bringing down the actual cost from 1727
million to 1349.4 million naira. The second
coalition (S2) made a savings of 15% thereby
bringing down the cost from 1726 million to
1467.1 million, etc. We have noted that each
coalition from (S2) to (S5) made a gain of 15%
thereby reducing the cost by 15% respectively.
In general, due to the collaboration of these
transport providers, the cost of shipping for one
month was reduced from 7607 million to
63347.4 thereby saving 1259.6 million naira for
the grand coalition. This kind of collaboration is
encouraged and it is capable of stabilizing the
prices of staple food especially now Nigeria is
facing a food crisis and galloping inflation on
her staple food items.
4.2.2. Analysis 2:
In the second analysis, we apply a dynamic
programming model to further analyse the
supply chain distribution to determine how best
to allocate coalitions for optimal gains. The data
in Appendix 2 was used for the analysis. In
Appendix 3, we present the data in a Dynamic
programming format.
From Appendix 3, Xi represents the legs
(routes) and Si represents the coalitions. Our
interest is to find the best way to allocate the
coalitions to the routes to maximize savings for
the grand coalition. We use the Dynamic
programming model in equation (3) and
Optimal decision policy in equation (4) to arrive
at the optimal decision, see [13].
4.2.2.1. Optimal Decision Policy
For optimal solution, we apply equation (4) to
arrive at:
For a better understanding of the above optimal
decision policy, refer to Appendix 6. The
optimal decision variables were selected
from the Tables starting from stage 1 to stage 4.
Our interest in this section is to allocate the
coalition to different legs for optimal gains to
the grand coalition; we have four legs and five
coalitions. From our analysis, we observed that
if we allocate the five coalitions to the four legs
in this order (1, 1, 2, 1), we will maximize the
savings by 377.6 million naira. This is an
intelligent decision as we can see that the
appropriate allocation of these coalitions using
the optimization method will bring in more
savings in addition to the earlier savings through
the Sharpley value method. Therefore, the total
gain for the grand coalition (1259.6 +377.7) is
1637.3 million naira.
5. Discussion and Recommendation
5.1. Discussion
In this paper, we proposed a coalition between
two modes of transportation (Bimodal mode of
transportation). The coalition was between two
groups of transportation, where one provided
cargo train and the other provided trucks of
capacity 453 tons respectively. There were five
grand coalitions |N| and four coalitions |S|. The
coalitions were distributed along five paths
across Nigeria. The first coalition, S1 covers
Katsina-Enugu, the second coalition, S2 covers
Katsina-Jos, the third coalition, S3 covers
Kaduna-Calabar, the fourth coalition, S4 covers
Katsina-Benin and the fifth coalition, S5 covers
Agbor-Owerri. Each of the coalition is made up
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
35
Volume 6, 2024
of four transport providers (players). Each of the
coalition is composed of four legs. Since this is
a kind of cooperative game scenario where each
player cooperates to enhance the efficiency of
the distribution of the agricultural produce, the
Sharpley value method of allocation of cost was
appropriate for the study and was applied to
obtain the gains accrued to the grand coalition.
With the collaboration, the coalition was able to
make some gains from each leg and we
observed that the total savings for the grand
coalition was 1259.6 million naira within the
proposed period. But we did not stop there, we
developed and applied the Dynamic
programming model to the supply chain
distribution problem and obtained an intelligent
result. we discovered that if the five coalitions
were distributed among the four legs, in this
other (1, 1, 2, 1), the grand coalition would make
an additional gain of 377.6 million naira. The
distribution means that the allocation of the
coalition to the third leg should be doubled
while the other legs remain as they were. This
arrangement will bring additional revenue and
the total gain for the grand coalition was 1637.3
million naira. More optimization work should
be done in conjunction with the Sharpley value
method of cost or savings allocation using
different optimization methods. This will
enhance efficiency and encourage the
application of optimization methods in this rare
area of research.
5.2. Recommendations
From the findings in this paper, we recommend
as follows:
1. Efforts should be made to encourage coalition
among transport and logistics providers because
such cooperation eliminates destructive
competition inherent in the real-world system.
By the application of the Sharpley value
method, the players are encouraged to work as a
team.
2. Cooperation of this kind can stabilize the
prices of goods, especially agricultural produce.
Nigeria is facing challenges of the high cost of
transportation which resulted in the removal of
fuel subsidy and for this reason, there is
galloping inflation. This kind of cooperation can
cushion the effect of the high cost of
transportation thereby stabilizing the prices of
food items.
3. At each time, optimization methods should be
encouraged and brought into supply chain
distribution because we can see from the paper
that resources can be better allocated and
managed using optimization methods,
especially Dynamic programming.
4. Transport providers are encouraged to form
coalitions in different areas of supply chain
distribution because it is profitable and brings
gains to the grand coalition. The gains can be
shared using the Sharpley value method.
References:
[1] Aziz, H., Cahan, C., Gretton, C., Kilby, P.,
Mattei, N., Walsh, T., A Study of Proxies for
Shapley Allocations of Transport Costs,
(2014), pp.1-51.
[2] Amuji, H. O., Ugwuanyim, G. U., Ogbonna,
C. J., Iwu, H. C., Okechukwu, B. N., The
usefulness of dynamic programming in
course allocation in the Nigerian
Universities. Open Journal of Optimization,
Vol. 6(4), (2017), pp.176-186.
[3] Amuji1, H. O., Ugwuanyim, G. U., Anyiam,
K. E., Application of game theory in
maintaining the academic standard in the
Nigerian Universities, World Scientific
News: An International Scientific Journal,
Vol.125, (2019), pp.72-82.
[4] Malawski, M., Wieczorek, A., Sosnowska,
H., Konkurencja i kooperacja - teoria gier w
ekonomii i naukach społecznych
(Competition and Cooperation–Theory of
Games in Economics and Social Science),
Warszawa: Wydawnictwo Naukowe PWN,
(2006), pp.127-150.
[5] Kayikci, Y., Analysis of Cost Allocation
Methods in International Sea-Rail
Multimodal Freight Transportation, Journal
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
36
Volume 6, 2024
of Yasar University, Vol.15(57), (2020),
pp.129-142.
[6] Amuji, H. O., Onwuegbuchunam, D. E.,
Aponjolosun, M. O., Okeke, K. O., Mbachu,
J. C. Ojutalayo, J. F., The dynamic
programming model for optimal allocation
of laden containers to Nigerian seaports,
Journal of Sustainable Development of
Transport and Logistics, Vol. 7(2), (2022),
pp.69-79.
[7] Cruijissen, F., Cools, M., Dullaert, W.,
Horizontal cooperation in logistics:
opportunities and impediments, Transport
Research Part E: Logistics and Transport
Review, Vol.43(2), (2007), pp.129-142.
[8] Wang, Y., A collaborative approach based
on Shapley value for carriers in the supply
chain distribution, Heliyon, Vol.9, (2023),
e17967.
[9] Thomas, L., Games, Theory & Applications.
Chichester: Ellis Horwood (1986) p.86.
[10] Augustine, O. E., Barry, R. M., Non-Serial
Dynamic Programming: A Survey,
Palgrave Macmillan Journals, Vol. 25,
(1974), pp.253-265.
[11] Adelson, R. M., Norman, J. M., Laporte,
G., A dynamic programming formulation
with diverse applications, Operational
Research Quarterly, (1976), pp.119-121.
[12] Amuji, H. O., Onwuegbuchunam, D. E.,
Okoroji, L. I., Nwachi, C. C., Mbachu, J.
C., Development/Application of Dynamic
Programming Model for Students’
Academic Planning and Performance in
the Universities, Greener Journal of
Science, Engineering and Technological
Research, Vol.12(1), (2023), pp.20-25.
[13] Amuji, H. O., Olewuezi, N. P., Ohaeri, E.
O., Ikeogu, V. N., Okoh, J. O., On
Solution to ASUU Strike and
Consolidated University Academic Salary
Structure II (CONUASS II) in the
Nigerian Universities Using Optimization
Method, International Journal of Applied
Mathematics, Computational Science and
Systems Engineering, Vol.6, (2024), pp. 1-
13.
[14] Moustapha, B. A., With a transaction fee
market and without a block size limit in
Bitcoin network; there exists a Nash
equilibrium points of the mining game,
IJGTT., Vol. 6(1), (2020), pp.1-23.
[15] Allayarov, S., Kilicheva, F., Rakhimova,
K., Mamasadikov, A., Khamrayeva,S.,
Durmanov, A., Game Theory and Its
Optimum Application for Solving
Economic Problems, Turkish Journal of
Computer and Mathematics Education
Vol.12(11), (2021), 3432- 3441.
[16] RUFAI, A. A., MARTINS, B. A.,
ADEBANWA, A. A., Review of
Strategies in Games Theory and Its
Application in Decision Making, IOSR-
JEF., Vol. 12(1), (2021), pp.06-22.
[17] Song, L., Yu, Z., He, Q., Evolutionary
game theory and simulations based on
doctor and patient medical malpractice,
PLoS ONE, Vol.18(3), (2023), e0282434.
[18] Abdallah, M. H. A., Game theory in
entrepreneurship: a review of the
literature, Journal of Business and Socio-
economic Development, Vol. 4(1), (2024),
pp. 81-94.
[19] Dong, Y., Dong, Z., Bibliometric Analysis
of Game Theory on Energy and Natural
Resource, Sustainability, Vol. 15, (2023),
1-19.
[20] Shapley, L., A Value for n-person games,
Albert William Tucker and Harold W.
Kuhn (eds.), Contributions to the Theory of
Games, Princeton University Press,
(1953), pp.31–40.
rcentage
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
37
Volume 6, 2024
Appendix 1: Coalitions, total unit cost, grand coalition, Sharpley weight (wi), savings, percentage
savings, and the new cost.
Si
Unit Cost
Number of
operators |S| for
each route from
terminal to
terminal
Total number
of operators |N|
in grand
coalition
Savings
(%)
New
Cost
S1
1727
4
5
0.05
377.6
21.86
1349.4
S2
1726
4
5
0.05
258.9
15.00
1467.1
S3
1219
4
5
0.05
182.85
15.00
1036.15
S4
1770
4
5
0.05
265.5
15.00
1504.5
S5
1165
4
5
0.05
174.75
15.00
990.25
Total
7607
1259.6
81.86
6347.4
Appendix 2: Coalitions, unit cost savings, and total cost savings per leg
Leg1
Leg2
Leg3
Leg4
Total
S1
44.75
72.25
71.8
188.8
377.6
S2
44.6
72.15
71.8
70.35
258.9
S3
46.25
45.9
45.65
45.05
182.85
S4
48
72.8
72.5
72.2
265.5
S5
44
43.7
43.6
43.45
174.75
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
38
Volume 6, 2024
Si/Xi
1
2
3
4
0
0
0
0
0
1
44.75
72.25
71.8
188.8
2
44.6
72.15
71.8
70.35
3
46.25
45.9
45.65
45.05
4
48
72.8
72.5
72.2
5
44
43.7
43.6
43.45
Appendix 4: Route and combination of Trains and trucks from different transport providers, cost, and
frequency of supply per month.
Si
Origin-
Destination
(O-D)
Bimodal
freight
transport
route from
terminal to
terminal
Transport
providers
No. of operators
Unit Cost(m)
Frequency
Unit Cost(m)
Frequency
Unit Cost(m)
Frequency
Unit Cost(m)
Frequency
Total Unit Cost (b)
S1
Katsina -
Enugu
Katsina(+Tn) -
Kaduna(+Tk) -
Nsukka(+Tk) –
Enugu(+Tk)
Nigeria Train;
Admiral Trucker;
Country Service
Solution; Vennis
Truck
4
832
2
282
4
291
5
322
3
1727
S2
Katsina - Jos
Katsina(+Tn)-
Kano(+Tk)-
Kogi(+Tk)-
Jos(+Tk)
Nigeria Train;
Eccnosy Inte.
Solution; Admiral
Trucker; Maverick
Int. Soltn.
4
834
3
283
3
290
4
319
4
1726
S3
Kaduna -
Calabar
Kaduna(+Tk)-
Agbor(+Tk)-
Onitsha(+Tk) -
Calabar(+Tk) -
Admiral Trucker;
Country Service
Solution; Eccnosy
Int. Soln; Vennis
Trucker
4
294
3
301
4
306
3
318
4
1219
S4
Katsina -
Benin
Katsina(+Tn)-
Kano(+Tk) –
Lagos(+Tk)-
Benin(+Tk)
Nigeria Train;
Admiral Trucker;
Country Service
Solution; Maverick
Int. Soln.
4
810
2
314
3
320
4
326
3
1770
S5
Lagos - Sokoto
Lagos(+Tr) -
Kaduna(+Tr) -
Kastina(+Tr) -
Skoto(+Tk)
Admiral Trucker;
Country Service
Solution; Eccnosy
Int. Soln; Vennis
Trucker
3
285
5
291
5
293
3
296
3
1165
Appendix 3: Presentation of data in Table 2 in Dynamic programming format.
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
39
Volume 6, 2024
Appendix 5: Computation of savings on each leg of the coalition
Total Unit Cost (TL)
Unit Cost (U)
TL - U
Wi
Savings (yi)
1727
832
895
0.05
44.75
S1
1727
282
1445
0.05
72.25
1727
291
1436
0.05
71.8
1727
322
1405
0.05
188.8
377.6
1726
834
892
0.05
44.6
1726
283
1443
0.05
72.15
S2
1726
290
1436
0.05
71.8
1726
319
1407
0.05
70.35
258.9
1219
294
925
0.05
46.25
1219
301
918
0.05
45.9
S3
1219
306
913
0.05
45.65
1219
318
901
0.05
45.05
182.85
1770
810
960
0.05
48
1770
314
1456
0.05
72.8
S4
1770
320
1450
0.05
72.5
1770
326
1444
0.05
72.2
265.5
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
40
Volume 6, 2024
1165
285
880
0.05
44
S5
1165
291
874
0.05
43.7
1165
293
872
0.05
43.6
1165
296
869
0.05
43.45
174.75
Appendix 6.
For n = 4, we have the Table below:
for n = 4
S4
f*4
X*4
0
0
0
1
188.8
1
2
70.35
2
3
45.05
3
4
72.2
4
5
43.45
5
Total Unit Cost (TL)
Unit Cost (U)
TL - U
Wi
Savings (yi)
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
41
Volume 6, 2024
1
188.8
71.8
188.8
0
2
70.35
260.6
71.8
260.6
1
3
45.05
142.15
260.6
45.65
260.6
2
4
72.2
116.85
142.15
234.45
72.5
234.45
3
5
43.45
144
116.85
116
261.3
43.6
261.3
4
for n = 2
S2/X2
0
1
2
3
4
5
f*2
X*2
0
0
0
0
1
188.8
72.25
188.8
0
2
260.6
261.05
72.15
261.05
1
3
260.6
332.85
260.95
45.9
332.85
1
4
234.45
332.85
332.75
234.7
72.8
332.85
1
5
261.3
306.7
332.75
306.5
261.6
43.7
332.75
2
for n = 3
S3/X3
0
1
2
3
4
5
f*3
X*3
0
0
0
0
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
42
Volume 6, 2024
for n =1
S1/X1
0
1
2
3
4
5
f*1
X*1
0
0
0
0
1
188.8
44.75
188.8
0
2
261.05
233.55
44.6
261.05
0
3
332.85
305.8
233.4
46.25
332.85
0
4
332.85
377.6
305.65
235.05
48
377.6
1
5
332.75
377.6
377.45
307.3
236.8
44
377.6
1
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.4
Harrison Obiora Amuji, Bridget Nwanyibuife Okechukwu,
Iheanyi Chinedu Obinwanne, Vivian Ngozi Ikeogu,
Samuel Inaku Emri
E-ISSN: 2766-9823
43
Volume 6, 2024
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
