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Research Article
A comparative study of artificial neural
network (ANN) and adaptive neuro-fuzzy
inference system (ANFIS) models in
distribution system with nondeterministic
inputs
Modestus O Okwu
1
and Olufemi Adetunji
2
Abstract
Most deterministic optimization models use average values of nondeterministic variables as their inputs. It is, therefore,
expected that a model that can accept the distribution of a random variable, while this may involve some more com-
putational complexity, would likely produce better results than the model using the average value. Artificial neural
network (ANN) is a standard technique for solving complex stochastic problems. In this research, ANN and adaptive
neuro-fuzzy inference system (ANFIS) have been implemented for modeling and optimizing product distribution in a multi-
echelon transshipment system. Two inputs parameters, product demand and unit cost of shipment, are considered
nondeterministic in this problem. The solutions of ANFIS and ANN were compared to that of the classical transshipment
model. The optimal total cost of distribution using the classical model within the period of investigation was 6,332,304.00.
In the search for a better solution, an ANN model was trained, tested, and validated. This approach reduced the cost to
4,170,500.00. ANFIS approach reduced the cost to 4,053,661. This implies that 34% of the current operational cost was
saved using the ANN model, while 36% was saved using the ANFIS model. This suggests that the result obtained from the
ANFIS model also seems marginally better than that of the ANN. Also, the ANFIS model is capable of adjusting the values
of input and output variables and parameters to obtain a more robust solution.
Keywords
Meta-heuristics, nondeterministic input, artificial neural network, ANFIS, transshipment, fuzzy
Date received: 22 December 2017; accepted: 1 March 2018
Background
Transshipment problem allows the movement of goods
from some source points to some destination points, with
the possibility of transshipping through some other source
or destination points if it would provide cost benefit. This
model is a further generalization of the transportation
model because it relaxes the assumption that movement
can only be made to a destination point from one or more
supply point/s without passing through an intermediate
point. This model should be capable of producing a better
solution than that of typical transportation problem, if such
exists, but it is also known to be more difficult to solve.
The transshipment problem itself is deterministic, and
being non-deterministic polynomial-time (NP) hard,
1
Department of Mechanical Engineering, Federal University of Petroleum
Resources, Warri, Delta State, Nigeria
2
Department of Industrial and Systems Engineering, University of
Pretoria, Pretoria, South Africa
Corresponding author:
Modestus O Okwu, Department of Mechanical Engineering, College of
Technology, Federal University of Petroleum Resources, Effurun (FUPRE),
P.M.B. 1221, Effurun, Delta State, Nigeria.
Email: mechanicalmodestus@yahoo.com
International Journal of Engineering
Business Management
Volume 10: 1–17
ªThe Author(s) 2018
DOI: 10.1177/1847979018768421
journals.sagepub.com/home/enb
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
(http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
solving it is a challenge on its own. Making some of the
input parameters nondeterministic makes its solution even
more arduous. However, the deterministic transshipment
model only uses approximate values of the model para-
meters, and the extension of the problem to that with some
nondeterministic variables is expected to produce a better
solution, if tractable. This is what has been done in many
stochastic models, and the marginal difference is usually
referred to as the value of the stochastic solution.
1
Representing nondeterminism in input parameters is
another interesting issue. While traditionally the nondeter-
ministic variables of most models have been represented in
stochastic forms, recent advances in artificial intelligence
have brought the use of fuzzy models to prominence. Solu-
tions to stochastic models of linear programming problems
(LPPs), for instance, use various techniques that exploit the
block separability of the variables, while fuzzy representa-
tion makes use of fuzzy operations and fuzzy arithmetic
theories. A popular approach of fuzzy logic is based on the
fuzzy decomposition theorem and utilizes the properties of
alpha cuts along various sections of the fuzzy set. Birge and
Louveaux
1
and Klir and Yuan
2
are good readings in order
to compare the stochastic LPP and fuzzy modeling and
solution approaches to problems.
Finding solution to NP hard problems in itself is diffi-
cult; hence, finding solutions to transshipment problems
with nondeterministic input parameters could be even more
challenging. A number of solutions have been proposed in
each case. The main difference in their representation is
that while a stochastic model uses an indicator function
to allocate a variable to the most likely class in its set based
on the estimate of the likelihood of its membership of each
set, fuzzy logic assumes that the variable is a member of all
of the sets, but by different likelihood of membership. It
uses a function ranging from zero to unity to represent the
membership across its different sets and uses a different
type of set operations from those of probability models to
find the variables’ joint membership values. Gaines
3
tried
to explain the difference between fuzzy logic and stochastic
process in relationship with probability logic, where sto-
chastic processes assume statistical independence and
fuzzy logic is purely logical inference.
This article presents solutions to a transshipment prob-
lem with nondeterministic input variables in order to esti-
mate the value of the nondeterministic solutions. The
demand and unit cost of product distribution were consid-
ered to be nondeterministic and solved through adaptive
models. Artificial neural network (ANN) model was used
to solve the model with stochastic demand and cost vari-
ables, while adaptive neuro-fuzzy inference system
(ANFIS) was used to solve the same problem with both
variables considered fuzzy. The solutions obtained in both
cases were then compared. The main difference in these
model solutions is in the training paradigm. The data in
both cases were separated into training and test samples,
and the performance of the test samples in both cases was
compared to that of the traditional solution approach.
While the manufacturing area seems replete with compar-
ison of these different paradigms of solution in diverse
contexts, no author seems to be known in logistics func-
tions, and in transshipment of goods in particular, that has
compared the quality of solutions obtained from these two
paradigms.
This article is further sectioned as follows: the next
section is a presentation of the articles comparing the per-
formance of ANN models to those of ANFIS models in
various contexts. The third section is a presentation of the
ANN and ANFIS models of the transshipment problem
with the two nondeterministic input variables (demand and
cost). In the fourth section, the ANN and the ANFIS models
were trained, and the levels of fitness achieved were pre-
sented. The final section discusses the result of the test
sample on the two trained models.
Comparative performance of ANN and
ANFIS
ANN and ANFIS performances have been compared in
many instances, but mostly in mechanical systems. He
et al.
4
used both ANN and ANFIS to predict the flow of
river in a small river basin in a semiarid mountain region,
where the actual mathematical relationships were not fully
understood, but the results were observable under diverse
conditions. In the study, support vector machine (SVM),
ANFIS, and ANN methods were compared. The values of
coefficient of correlation (R), root mean squared error,
Nash–Sutcliffe efficiency coefficient, and mean absolute
relative error were used in assessing the strengths of the
three modeling paradigms. The outcome suggests that the
three techniques were good enough, while SVM outper-
forms both ANN and ANFIS. Anari et al.
5
did a study on
infiltration rate using ANN, ANFIS, local linear regression
(LLR), and dynamic LLR and observed that ANFIS
appeared to have performed better than all other three
techniques.
Do and Chen
6
used ANN and ANFIS to forecast post-
admission evaluation of students using some preadmission
parameters as inputs. They claimed that ANFIS produced
better results than ANN. Kamali and Binesh
7
used ANN
and ANFIS to study the diffusion of water through nano-
tubes using the molecular dynamics data. They also experi-
mented with data set of different sizes and concluded that
ANFIS outperformed ANN. Sredanovic and Dica
8
used
both ANN and ANFIS to predict the strength of minimal
quantity lubrication and high-pressure cooling in the turn-
ing task. Both were said to have performed well. Azeez
et al.
9
compared the performance of ANN and ANFIS in
the triage of emergency patients using various vital signs of
patients as input parameters. Data were collected from a
Malaysian University Hospital. They used ANN backpro-
pagation technique for training, testing, and validation of
data set. Fuzzy rules were developed with alpha cuts for
2International Journal of Engineering Business Management
ANFIS model, and fuzzy subtractive clustering was used to
find the fuzzy rules for the ANFIS model. ANN model gave
a better result.
Esen
10
conducted experiment on heat pump system by
comparing the performance of ANFIS and ANN in the
upright ground structure. The author stated preference for
ANFIS over ANN. Vieira
11
modeled the temperature evo-
lution of a kiln using both the feed-forward ANN and
ANFIS system and concluded that ANFIS produced results
with less error than the ANN model. Kisi and Ay
12
com-
pared the performance of radial basis ANN with ANFIS in
the study of dissolved oxygen in evaluating water quality. It
was established that the ANN model outpaced the ANFIS
model. Farahmand et al.
13
analyzed the temporal distribu-
tion of groundwater contaminants in Iran. They found that
ANFIS model outperformed the ANN models, having
higher efficiency of estimation. Research areas in other
fields of comparative study of ANN and ANFIS are avail-
able in the literature but it cannot be said emphatically that
either technique always performs better than the other. The
performance depends on the context.
There have also been cases where the performance of
ANN or ANFIS (or both) has been compared to those of
other techniques (like meta-heuristics); or other situations
where they have been complementarily applied with other
techniques. Li and Chen
14
used both the least square
method (LSM) and the ANFIS to train the artificial systems
used in modeling traffic flow in Beijing ring roads and
connected lines in order to develop and evaluate the intel-
ligent transport systems. The results obtained from the two
approaches were validated against real-life data, and they
concluded that model trained with ANFIS produced better
results than that trained with LSM. Azadeh and Zarrin
15
used data envelopment analysis, and meta-heuristics
trained ANN and ANFIS logics to evaluate the efficiency
of staff along three parameters. It was concluded that
ANFIS models trained with genetic algorithm (GA) and
particle swarm optimization fared best. Barak and Sadegh
16
used a combination of autoregressive integrated moving
average (ARIMA) and ANFIS to model energy demand.
The linear time series parameters were modeled with the
ARIMA logic, and the nonlinear residual was later captured
using the ANFIS logic in building an expanded model.
They concluded that this model performed better than some
others. Mansouri et al.
17
compared the performance of
M5Tree, ANN, ANFIS, and multivariate adaptive regres-
sion splines (MARS) in analyzing the axial compression of
fiber-reinforced polymer and found that ANN predicted an
accurate result. Mostafa and El-Masry
18
used gene expres-
sion programming (GEP), ANN, and ARIMA to forecast
the evolution of oil prices between January 1986 and June
2016. They reported that both ANN and GEP outperformed
ARIMA, and that GEP produced the best result of the three
using R
2
as the measure of performance. Oduro et al.
19
did
a comparative analysis of the prediction of vehicular emis-
sion using boosting MARS (BMARS), classification and
regression trees (CART), ANN, and a hybrid of CART–
BMARS. They reported that the combined system proved
superior to the models of ANN and those of each of the
techniques when not hybridized.
While the transshipment problem has been solved using
a number of heuristics and meta-heuristics, it is hard to
come about those solved using artificial intelligence. Most
transshipment models presented recently have been done
under the theme of cross-docking. Augustina et al.
20
pre-
sented a summary of recent works done on transshipment.
They classified the literature into three main categories
based on strategic intent and planning horizon: operational,
tactical, and strategic. They reviewed some 50 papers. At
operational level, the two main issues addressed are about
scheduling the cross-docking of trucks and dock-door
assignment problems. The most common solution
approaches are the different variants of modified Johnson,
modified linear programming solution techniques, and
meta-heuristics. Tactical-level decision focuses mainly on
facility layout decisions, and a number of these problems
are solved using meta-heuristics. The long-term decision
relates to network design: cross-dock area location and
sizes and choice of vehicle sizes and routes. Some other
recent papers in the area include a multi-objective GA
solution for transportation, assignment and transshipment
problems,
21
and a Just-in-time (JIT) multi-objective truck
forecast problem with door assignment using differential
evolution in the cross-dock system.
22,23
Model notations and formulation
The following notations are adopted for the modeling
purpose:
Iis the set of all source points, indexed by iand includ-
ing transshipment points (TPs) as necessary
Jis the set of all demand points, indexed by jand includ-
ing TPs as necessary
Kdenotes set of TPs, indexed by k
Pdenotes set of product types, indexed by p
X
ijp
is the amount of product ptransported from produc-
tion point ito depots
X
ikp
is the amount of product pshipped from production
point ito TP k
X
kjp
is the amount of product pshipped from TP kto
depot j
Z
pi
is the capacity of source point i, including the TP, to
produce or store product p, whichever is smaller
d
pj
is the demand for product pat the destination point j,
including the TP
c
pij
is the unit amount of transporting product pfrom
production point ito depot j
c
pik
is the unit amount of transporting product pfrom
production point ito TP k
c
pkj
is the unit amount of transporting product pfrom TP
kto depot j
Okwu and Adetunji 3
is a time indexed random vector on which the outcome
of the demand and cost depends, which helps to
partition decisions into periods before and after the
evolution of !, a random event that affects demand
and shipment cost
!is a random event, !2, that influences demand and
cost of product, p
qis the second-stage objective vector of a stochastic
programming model, q2Q
xis the first-stage feasible decision vector, x2X
yis the second-stage feasible decision vector, y2Y
E() is the expectation operator
M() is the membership function operator
Ais the first-stage constraint matrix in a two-stage
stochastic model with recourse
his the RHS vector of the second stage (recourse action)
of the stochastic program;
Wis the Recourse matrix, based on the values from the
evolution of
Tis called the Technology matrix such that Wy ¼hTx
ais the lower bound value of a triangular distribution
bis the upper bound value of a triangular distribution
mis the modal value of a triangular distribution
The deterministic model of the transshipment problem
can be represented as:
Minimize
X
iX
jX
p
cpijXpij þX
iX
kX
p
cpik Xpik
þX
jX
kX
p
cpjk Xpjk
ð1Þ
subject to
X
j
Xpij þX
k
Xpik zp;i8p;ið2Þ
X
i
Xpij þX
k
Xpkj dpj 8j;pð3Þ
X
i
Xpij X
j
Xpjk dpk 8k;pð4Þ
Xpij;Xpik ;Xpkj 0ð5Þ
Equation (1) is the objective function and is the total
cost of shipment of product p, consisting of three cost
terms: the first term being the cost of shipment from source
idirectly to destination j, the second term being that from
source ito TP, k, and the third term being that from TP, k,to
destination j. Equation (2) is the set of supply constraints,
ensuring that the quantity of items delivered to TPs, k, and
directly to final destination points, j, do not exceed the
capacity of the supply point, i. Equation (3) is the set of
demand constraints, ensuring that the quantity of items
delivered from TPs, k, and directly from supply points, i,
do not fall below the demand at the destination point, j.
Equation (4) is the set of balancing equation for TPs to
ensure that the difference between the sum of item p,
shipped to the TP k, and the sum of items shipped away
from the point is the demand at the TP if the TP also serves
as a destination point (or zero otherwise). The last set of
equations is the non-negativity constraints of the variables.
Allowing for nondeterministic variables (demand and
cost) in the model, the two-stage stochastic model with
recourse is presented next. The general equivalent of the
deterministic objective function is minimize c
T
X, with c
and Xbeing the cost and demand vectors, respectively.
To modify equations (1) to (5) as a stochastic programming
model, according to Birge and Louveaux,
1
the model equa-
tions (6) to (9) become:
Minimize
cTXþE½min qð!ÞTyð!Þ ð6Þ
subject to
AX ¼bð7Þ
Tð!ÞxþWyð!Þ¼hð!Þð8Þ
x;yð!Þ0ð9Þ
c
T
Xis the objective function presented in equation (1),
which is the deterministic component of equation (6).
E½min qð!ÞTyð!Þ is the expected value of the stochastic
component of the objective function and is added to the
deterministic function to be minimized together for the
optimal values of xand y, where the Xvector is the set of
the foremost decision variables, and the Yvector is the set
of the subsequent decision variables, which are made after
the evolution of the random event, , which affects the
result observed after the first decisions, and thus, guides
the optimal decision taken in the second stage. Equations
(2) to (4) from the first stage (before evolves) are rolled
into the first-stage constraint matrix, A, as shown in equa-
tion (7) in the stochastic model. Equation (8) is the set of
matrices and vectors defining the set of constraints that the
decision vector, y(!), must satisfy at the optimal point,
when choosing recourse values based on the time-
dependent random variable, , which have been realized
based on the evolution of the random event, !. Equation
(9) is the extended non-negativity constraint for the deci-
sion variables, xand y(!).
The model in equations (6) to (9) representing the two-
stage stochastic programming model with recourse has
been solved using various solution approaches to large
matrix linear programs (LP) such as the column generation
technique, the Lagrange decomposition technique, and var-
ious meta-heuristics. In this article, two artificial intelli-
gence approaches, ANN and ANFIS, are utilized, and the
results of the two are compared to that of the traditional
4International Journal of Engineering Business Management
transportation tableau. The modeling approach is presented
next.
The second term in equation (6) could be expanded in
different ways depending on whether it is being considered
to be stochastic or fuzzy, and on whether it is discrete or
continuous. For the purpose of this work, a triangular dis-
tribution is assumed for both the cost and demand vari-
ables, howbeit with different parameter values for the
demand and cost variables involved. The function can be
written out as either a stochastic component or a fuzzy
component for the second term of equation (6), respec-
tively, in the following equations
1,2
fðxÞ¼
0forx<a
2ðxaÞ
ðbaÞðmaÞfor axm
2ðbxÞ
ðbaÞðbmÞfor mxb
0ifx>b
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
ð10Þ
MðxÞ¼
0forxa
xa
mafor a<xm
bx
bmfor mx<b
0forxb
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
ð11Þ
The deterministic component (i.e. the first term of equa-
tion (6)) still remains as defined in equation (1).
Company of case study
The company used as a case study is a multinational bev-
erage bottling company with a single production plant, n¼
1, a range of products represented by the single product
family, i¼1, servicing the south western part of Nigeria
and some neighboring cities via 10 depots, j¼10, and three
TPs, k¼3, which also serve as depots. Data collected from
the bottling plant are used for analysis. Matrices are devel-
oped for the distances, and the associated cost is as shown
in Table 1. The current practice in the company is to ship
directly to all depots from the plants, although they could
have three TPs, D7, D9, and D10 selected through greedy
heuristics, through which they could ship to the other
depots as shown in Figure 2.
Model implementation
The triangular functions were approximated by the deter-
ministic equivalents (quantile values for the stochastic
model and alpha cuts for the fuzzy model) to derive the
training rules during implementation. The stochastic model
is divided into quantiles of demand and cost value ranges
from which the class values are used for determining the
values required for ANN training process. The associated
linguistic values for ANFIS model were defined, and dif-
ferent alpha cut values were defined using the fuzzy rules
developed and the corresponding values of linguistic vari-
ables determined in consistence with the fuzzy decomposi-
tion theorems. ANN and ANFIS training of models were
done using the derived values from the available data set.
In solving the transshipment problem where product
flows from source through TPs for onward distribution to
the final destination, the deterministic model was solved
using Tora software 2.0. For the nondeterministic input
variables problem, popular solvers were not useful. The
ANN technique, using neural weight update algorithm, and
the ANFIS technique, which works based on the principle
of fuzzy inference rules, were used to determine the opti-
mal weight vectors that minimize the objective function of
the LP problem presented.
The ANFIS rule viewer is a simplified toolbox with
inbuilt algorithms for both the fuzzification of input para-
meters and the optimization of neural weights inside the
same toolbox. It presents the analyst with the input inter-
face to specify the necessary fuzzy parameters for each
variable involved and presents the output in a de-
fuzzified manner once the appropriate de-fuzzification
algorithm of choice has been specified.
The creative model
In implementing the ANN model, a feed-forward
model with backpropagation, having 4-neuron input
Table 1. Quantity of product available and unit cost of distribution.
S/No. Source/sink Layout Product availability; demand unit (weekly) Unit cost
1 Asejire Source/depot 1 135,255 S
1
to D
1
¼0
2 Ado-Ekiti Depot 2 11,215 S
1
to D
2
¼62
3 Ore Depot 3 20,125 S
1
to D
3
¼60
4 Akure Depot 4 18,540 S
1
to D
4
¼55
5 Abeokuta Depot 5 21,869 S
1
to D
5
¼50
6 Ijebu-Ode Depot 6 11,986 S
1
to D
6
¼45
7 Ibadan Depot 7 31,500 S
1
to D
7
¼30
8 Ondo Depot 8 8348 S
1
to D
8
¼48
9 Ife Depot 9 7126 S
1
to D
9
¼35
10 Ilesha Depot 10 4546 S
1
to D
10
¼40
Okwu and Adetunji 5
layer with linear combiner, 10-neuron middle layer, and a
single-neuron output layer was designed using the sigmoid
output transfer function. This is shown in Figure 1. The
model is trained with the MATLAB ANN toolbox using
the gradient descent algorithm. Mean squared error (MSE)
was selected as the performance function, and R
2
values
were examined throughout the training period.
The following notations are defined for exclusive use
within the ANN and ANFIS models.
U
k
is the linear combiner output due to input signal
W
kv
the connection weight from neuron vto neuron k
X
j
the signal flow from source/input through synaptic
weights to summing junction, that is, X
1
,X
2
,X
3
,...,
X
j
jthe index identifying the number of inputs
nthe index identifying destinations
kthe index identifying the processing elements
V
k
the linear combiner with bias to support input signal
Y
k
the total weighted sum of input to the kth processing
element
pthe index representing TPs (p¼1 for depot 7, p¼2for
depot 9, and p¼3 for depot 10)
W
ki
the weight or strength of connection from neuron, i
to k
X
ki
the demand input signal from depots ito TPs k
the learning rate of the neural network (NN) in the
backpropagation algorithm implementation
A typical standard neuron consists of set of synapses,
each characterized by weight or strength of connections. It
also comprises the bias or summing junction and the acti-
vation junction. The total sum of the neuron Ukis obtained
by linearly combining all the inputs, multiplied by the
appropriate weight or strength of connection so that in
mathematical form, a neuron Kis expressed as shown in
the below equation
Uk¼X
n
j¼1
xjwkj ð12Þ
Rewriting equation (12) for each transshipment location
(i.e. at each TP)
Upk ¼X
6
j¼1
xjwpkj 8pð13Þ
Introducing a bias (b
k
), then the combined input (V
k
) can
be expressed as
Vk¼X
3
p¼1
Upk þbk ð14Þ
Substituting the value of U
k
for each pin equation (13)
into equation (14), the combined input can be computed as
Vk¼X
3
p¼1X
6
j¼1
xjwpkj 8pð15Þ
Defining the output threshold function as a sigmoid
function ’ð:Þ¼ 1
1þex, then the total output can be
expressed for each p, and with Wkn from middle layer neu-
ron kto output layer neuron n, equation (12) can be written
as
Ypk ¼’ðwpknxnþbpk Þ¼ 1
1þewpknxnþbpn 8pð16Þ
Batch update was done to train the neurons, and the
general weight update rule for the NN is
wnew ¼wold Xxjð17Þ
Figure 2. Rough and final MSE variations for training, testing, and
validation data sets against epoch during the training. MSE: mean
squared error.
Figure 1. Structure of the MATLAB ANN model. ANN: artificial
neural network.
6International Journal of Engineering Business Management
where ¼ðYexp:mYpredÞwhen the transfer function is
linear, ¼Ypredð1Ypred ÞðYexp YpredÞwhen the trans-
fer function is sigmoid, Ypred represents the predicted ANN
value, and Yexp denotes the expected or targeted output
value.AccordingtoLotfanetal.,
24
equations (18) and
(19) are used to test the fitness during training
MSE ¼1
nX
n
m¼1
ðYexp:mYpred:mÞ2ð18Þ
R¼Xn
m¼1ðYpred:mypredÞðYexp:myexp Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Xn
m¼1ðYpred:mypredÞ2X
n
m¼1
ðYexp:myexpÞ2
s
0
B
B
B
B
@
1
C
C
C
C
A
ð19Þ
Data analysis, results, and discussion
Approach 1 (Using ANN)
Using a three-layer (input, hidden, and output) NN model as a
result of the quantitative and architectural characteristics of
the NN which has a close resemblance of the outbound man-
ufacturing system where the input denotes product available
at the source, and the route to different destinationsrepresents
weighted strength of the network. Table 2 shows the available
data set for network training, with cost and demand set as
input variables of Wand Xand the output which is the product
availability, Y,atdifferentTPs.Dis the demand of products at
various locations.In NN programming, it is necessary to train
the network for accurate value. The accuracy ofthe value is a
function of MSE and Rvalues. Our goal is to minimize these
errors. This was achieved by running series of iterations and
training continuously with many weighted input parameters
till a fairly minimal error is obtained.
Results and discussion. The goal of this section is to obtain
the optimum cost of product distribution in a bottling sys-
tem. The ANN was trained with four inputs, 10 hidden, and
an output layer (4–10–1). The network was trained many
times for effective evaluation of the accuracy of the model
using MSE and Rvalue to determine the degree of associ-
ation between the predicted and expected values. The sys-
tem of equation for MSE and Rvalue is established in
equations (18) and (19). Y
pred
and Y
exp
denote the predicted
and experimental values obtained for the NN model,
respectively. y
exp
and y
pred
also signify the average experi-
mental and predicted values, correspondingly. The perfor-
mance and best possible iteration result of the ANNs are
presented in Table 1. Figure 4 presents testing performance
with worthy MSE value of 1.57 10
5
.Bolditemsin
Table 3 also reveal good performance solution. Training
and performance values of this network were obtained at
1.000 and 0.80369, respectively.
Table 2. ANFIS training data.
S/No. Cost at TP7 Cost at TP9 Cost at TP10 D Y
K1
Y
K2
Y
K3
1 45 27 20 11,215 65,355 35,599 34,301
2 33 20 25 20,125
3 32 22 18 18,540
4 20 29 35 21,869
5 25 32 30 11,986
6 26 30 15 8348
ANFIS: adaptive neuro-fuzzy inference system; D: product demand; TP: transshipment point. Note:Y
K1
¼product available at transshipment point 1;
Y
K2
¼product available at transshipment point 2; Y
K3
¼product available at transshipment point 3.
Figure 3. Plot of ANN-predicted output against actual value for (a) training, (b) validation, (c) testing, and (d) target. ANN: artificial
neural network.
Okwu and Adetunji 7
Figure 2(a) is the MSE value for rough iteration in which
the best validation result was obtained at epoch 2 with the
minimum MSE value of 65 10
5
(see Table 3, no. 1 in
bold items).
In Figure 2(b), the training process was repeated for MSE
and the best validation result was obtained at epoch 5 with
minimum MSE of 1.57 10
5
. This performance is meritor-
ious compared to the MSE result obtained for rough iteration
(see Table 3, no. 11 in bold items). Figure 5 shows the plots of
the ANN-predicted outputs generated by the ANN model for
training, validation, testing, and target. The performance
value obtained is demonstrated in Table 4. A satisfactory
iteration, though not hundred percent, is obtained in Table 4.
Interpretation of the correlation coefficient (R). The correlation
coefficient specifies the degree of association or relationship
among some variables of interest. Generally speaking, a
correlation value of 0 is believed to be the absence of linear
relationship, while 1 implies perfect relationsip between
variables. There are rules for interpreting the R
2
value, but
a value greater than 0.7 and less than 1 can be regarded as
substantial to meritorious for a reasonably sized data set.
Continuous iteration (or refining) of weight parameters
was done to obtain a model with the best possible fit. The
iteration was performed several times to obtain the best
value as shown in Table 3 and Figure 3. Table 3 is a sum-
mary table showing the MSE and Rvalues for the itera-
tions. Apart from the 11th value in Table 3, other values
have high MSE value for training, validation, and testing.
The best solution to the programmed ANN system is
obtained at the 11th run with the minimal MSE value. The
ANN model at the first trial stage was trained with 50 data
points of which the best 15 are shown in Table 3. This was
done to test the effectiveness of the tool.
As shown in Figure 3(a) to (d), the Rvalues for
training, validation, and testing are: 1.000, 0.95504, and
0.77089, respectively, with performance value of
0.80369. Therefore, the ANN prediction for training,
validation, and testing are substantial and meritorious
in terms of correlation. The performance value for the
Table 3. Performance results of multilayer perceptron (MLP) network for different numbers of neurons in the hidden layer for testing
data set.
Rvalue
No. MSE Training Testing Validation Performance
1 (Rough) 6,514,260 0.9688 0.9990 0.5950 0.37153
2 2,940,735 0.4768 1.0000 1.0000 0.41513
3 3505 0.9994 0.5558 0.19681 0.63866
4 2800 1.0000 0.1035 0.5057 0.69111
5 2851 1.0000 0.11657 0.80388 0.68865
6 3586 1.0000 0.10346 0.5057 0.69111
7 3357 1.0000 0.55358 0.71808 0.56428
8 2676 0.6810 0.4638 0.81260 0.65247
9 2280 0.9821 0.7218 0.56580 0.73260
10 1950 0.9784 0.7136 0.6842 0.71216
11 (Final) 1.57 10
5
1.0000 0.77089 0.95504 0.80369
12 78 0.9999 0.4447 0.67610 0.69830
13 1900 0.9998 0.6669 0.85110 0.78962
14 3839 0.9999 0.7134 0.82341 0.79990
15 2348 0.9999 0.7614 0.71320 0.68530
MSE: mean squared error,R: regression coefficient; R
2
: average determination coefficient.
Figure 4. High-level fuzzy architecture for transshipment model.
8International Journal of Engineering Business Management
final iteration gave a satisfactory result as shown clearly
in Figure 9, where SD7 represents the Supply signal at
D7; SD9 is the Supply signal at D9; SD10 is the Supply
signal at D10; TP is the transshipment point; D is the
product demand; UTP7 is the Unit cost from TP7 to
depots; UTP9 is the Unit cost from TP9 to depots;
UTP10 is the Unit cost from TP10 to depots; and D
represents the Depots which are 10 in all.
Total cost of distribution at D7 ¼(45 0) þ(33 0) þ
(32 0) þ(20 21,869) þ(25 11,986) þ(26 0) ¼
737,030
Total cost of distribution at D9 ¼(27 8348) þ(20
20,125) ¼225,396 þ402,500 ¼627,896
Total cost of distribution at D10 ¼(20 2867) þ(18
18,540) þ(15 8348) ¼516,280
Totalcostoftransshipment¼Supply cost at D7 þ
Supply cost at D9 þSupply cost at D10 ¼737,030 þ
627,896 þ516,280 ¼1,881,206
Total cost of distribution in the entire system ¼Cost of
distribution from source to TPs plus cost of distribution
from TPs to final destination
¼X
n
i¼1X
m
k¼1
Cik Xik þX
n
k¼1X
m
j¼1
CkjXkj ¼2;289;327
þ1;881;206 ¼4;170;533
Therefore, total cost of distribution using the developed
ANN model ¼4,170,533.00
Approach 2 (Using ANFIS model)
In this section, fuzzy logic was used to find a delivery
pattern that would meet customer demand and minimize
delivery cost. Since there exists a source and 10 destina-
tions with three possible TPs and there are two imprecise
inputs, finding optimal solution requires continuous itera-
tions. Using ANFIS, transshipment routes were created and
analysis was carried out. Using the graphical user interface
of Simulink to generate input, an initial fuzzy model was
derived to trigger the modeling process. Membership func-
tions were established for the input signals.
The high-level fuzzy model logic is shown in Figure
4. The input variables were passed into the fuzzy toolbox
that utilizes the Sugeno class function for fuzzification.
The membership functions (mf) at the input and output
ends of the ANFIS logic were then defined so that the
appropriate rules can be developed for training as shown
in Figure 5. The triangular membership function (mf)
was selected for the inputmf variables and the Gaussian
for the outputmf supply signal. Examples of these can be
seen in Figure 6(a) to (d). Figure 8 is an example of
the three-dimensional (surface) views generated using
the ANFIS rule viewer. Example of the application of
Figure 5. ANFIS architecture with input–output membership functions. ANFIS: adaptive neuro-fuzzy inference system.
Table 4. Performance value for ANN modeling (final iteration).
Results Samples (%) MSE R
Training 68 1.56999 10
5
1.000000
Validation 16 — 0.955041
Testing 16 — 0.770892
ANN: artificial neural network; MSE: mean squared error; R: regression
coefficient.
Okwu and Adetunji 9
the rule viewer using discretization of the continuous
triangular inputs to take the relevant alpha cuts in order
to generate the fuzzy inputs is shown in Figure 7. Eighty
one training rules were developed. These rules are shown
in the Appendix.
Table 2 is the data set representing product demand at
source and the available products at the TPs 7, 9, and 10 for
onward distribution to the final destinations. Figure 8
shows the membership function plot for input variables in
terms of demand.
The final allocation is presented in Figures 9 and 10,
showing the quantity to be supplied from each TP to each of
the destinations using ANN and ANFIS, respectively. It
should be noted that each of the TPs supplied its own
Figure 6. Fuzzy Inference System (FIS) membership function plot for cost of distribution from TP7, TP9, and TP10 and demand at
various destinations. TP: transshipment point.
Figure 7. Rule viewer for input–output supply signals.
10 International Journal of Engineering Business Management
demands. The total distribution cost per week obtained (in
Nigerian Naira, NGN) is shown in Figures 9 and 10. The
solution produced using ANFIS is marginally cheaper than
the one produced using ANN and the company’s traditional
transshipment model. While the company’s traditional
solution produced NGN6,332,304 minimum cost per week,
ANN generated a minimum of NGN4,170,533. ANFIS,
however, yielded a minimum cost of 4,053,661, which is
Figure 8. Three-dimensional input–output surface plots.
Figure 9. ANN solution figures for supply signal at TPs 7, 9, and 10. ANN: artificial neural network; TP: transshipment point.
Figure 10. ANFIS optimal solution figures for supply signal at TPs 7, 9, and 10. ANFIS: adaptive neuro-fuzzy inference system; TP:
transshipment point.
Okwu and Adetunji 11
better than both the traditional technique of the company
and the ANN model solution.
Total cost from TP to final destination ¼1,764,334
Total cost of distribution from source to TPs ¼2,289,327
Total cost of distribution in the entire system
¼X
n
i¼1X
m
k¼1
Cik Xik þX
n
k¼1X
m
j¼1
CkjXkj ¼2;289;327
þ1;764;334 ¼4;053;661
Conclusion
In this article, ANN and ANFIS models have been applied
successfully to solve a transshipment problem with nondeter-
ministic demand and cost input variables. With data set
obtained from a bottling company, training was performed
using ANN fitting tool where the best iteration was consid-
ered based on MSE and Rvalues obtained. The model based
on the ANFIS logic was also implemented using the rule
viewer interface. This model is capable of adjusting para-
meters based on series of alpha cuts taken on the data set. Our
results seem to have followed the pattern reported in other
papers that have compared the performance of models with
nondeterministic inputs to those that used the mean values to
approximate such variables. The results suggest that the
ANFIS logic seems to have produced a slightly better solution
than that obtained using the ANN, and both are much better
than that obtained using the company’s traditional distribu-
tion algorithm. ANFIS is, however, only marginally better in
this context, and it is generally inconclusive in the literature if
one technique is better than the other.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect
to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
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APPENDIX 1 – FURTHER SAMPLES OF FUZZY VARIABLES
OUTPUT RESULT AT Transhipment Point 7
Okwu and Adetunji 13
OUTPUT VALUE AT Transhipment Point 9
OUTPUT VALUE AT Transhipment Point 10
14 International Journal of Engineering Business Management
APPENDIX II – ANFIS Linguistic Rules generated
(1) If the unit cost to D7 is in 1mf1 and unit cost to D9 is in
2mf1 and unit cost to D10 is in 3mf1 and demand is in 4mf1
then supply at D7 is out 1mf1.
(2) If the unit cost to D7 is in 1mf1 and unit cost to D9 is
in 2mf1 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf2.
(3) If the unit cost to D7 is in 1mf1 and unit cost to D9 is
in 2mf1 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf3.
(4) If the unit cost to D7 is in 1mf1 and unit cost to D9 is
in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf4.
(5) If the unit cost to D7 is in 1mf1 and unit cost to D9 is
in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf5.
(6) If the unit cost to D7 is in 1mf1 and unit cost to D9 is
in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf6.
(7) If the unit cost to D7 is in 1mf1 and unit cost to D9 is
in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf7.
(8) If the unit cost to D7 is in 1mf1 and unit cost to D9 is
in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf8.
(9) If the unit cost to D7 is in 1mf1 and unit cost to D9 is
in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf9.
(10) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf1 then supply at D7 is out 1mf10.
(11) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf11.
(12) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf12.
(13) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf13.
(14) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf14.
(15) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf15.
(16) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf16.
(17) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf17.
(18) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf18.
(19) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf1 then supply at D7 is out 1mf19.
(20) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf20.
(21) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf21.
(22) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf22.
(23) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf23.
(24) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf24.
(25) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf25.
(26) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf26.
(27) If the unit cost to D7 is in 1mf1 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf27.
(28) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf1 and demand is in
4mf1 then supply at D7 is out 1mf28.
(29) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf29.
(30) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf30.
(31) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf31.
(32) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf32.
(33) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf33.
(34) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf34.
(35) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf35.
(36) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf36.
Okwu and Adetunji 15
(37) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf1 then supply at D7 is out 1mf37.
(38) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf38.
(39) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf39.
(40) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf40.
(41) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf41.
(42) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf42.
(43) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf43.
(44) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf44.
(45) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf45.
(46) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf1 then supply at D7 is out 1mf46.
(47) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf47.
(48) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf48.
(49) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf49.
(50) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf50.
(51) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf51.
(52) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf52.
(53) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf53.
(54) If the unit cost to D7 is in 1mf2 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf54.
(55) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf1 and demand is in
4mf1 then supply at D7 is out 1mf55.
(56) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf56.
(57) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf57.
(58) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf58.
(59) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf59.
(60) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf60.
(61) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf61.
(62) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf1 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf62.
(63) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf63.
(64) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf1 then supply at D7 is out 1mf64.
(65) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf65.
(66) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf66.
(67) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf67.
(68) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf68.
(69) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf69.
(70) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf70.
(71) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf2 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf71.
(72) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf72.
16 International Journal of Engineering Business Management
(73) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf1 then supply at D7 is out 1mf73.
(74) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf2 then supply at D9 is out 1mf74.
(75) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf1 and demand is in
4mf3 then supply at D10 is out 1mf75.
(76) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf1 then supply at D7 is out 1mf76.
(77) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf2 then supply at D9 is out 1mf77.
(78) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf2 and demand is in
4mf3 then supply at D10 is out 1mf78.
(79) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf1 then supply at D7 is out 1mf79.
(80) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf2 then supply at D9 is out 1mf80.
(81) If the unit cost to D7 is in 1mf3 and unit cost to D9
is in 2mf3 and unit cost to D10 is in 3mf3 and demand is in
4mf3 then supply at D10 is out 1mf81.
Okwu and Adetunji 17
