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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.

References

1. Birge JR and Louveaux F. Introduction to stochastic pro-

gramming. Booth school of business, University of Chicago:

USA: Springer Series in operations Research, Springer, 1997.

2. Klir GJ and Yuan B. Fuzzy sets and fuzzy logic, theory and

applications. New Jersey: Prentice Hall, 1995.

3. Gaines BR. Stochastic and fuzzy logics. Elect Lett 1975; 11:

188–189.

4. He Z, Wen X, Liu H, et al. A comparative study of artificial

neural network, adaptive neuro fuzzy inference system and

support vector machine for forecasting river flow in the semi-

arid mountain region. J Hydrol 2014; 509: 379–386.

5. Anari PL, Darani HS, and Nafarzadegan AR. Application of

ANN and ANFIS models for estimating total infiltration rate in

an arid rangeland ecosystem. Res J Environ Sci 2011; 5(3):

236–247.

6. Do GH and Chen JF. A comparative study of hierarchical

ANFIS and ANN in predicting student academic perfor-

mance. WSEAS Trans Inf Sci Appl 2013; 10(12): 396–405.

7. Kamali R and Binesh AR. A comparison of neural networks

and adaptive neuro-fuzzy inference systems for the prediction

of water diffusion through carbon nanotubes. Micro Nano

2013; 14: 575–581.

8. Sredanovic B and Cica D. Comparative study of ANN and

ANFIS prediction models for turning process in different

cooling and lubricating conditions. SAE Int J Mater Manuf

2015; 5: 49–64.

9. Azeez D, Ali MAM, Gan KB, et al. Comparison of adaptive

neuro-fuzzy inference system and artificial neutral networks

model to categorize patients in the emergency department.

SpringerPlus 2013; 2: 416.

10. Esen H. ANN and ANFIS models for performance evaluation

of a vertical ground source heat pump system. Expert Syst

Appl 2010; 37(12): 8134–8147.

11. Vieira JAB, Dias FM, and Mota MM. Comparison between

artificial neural networks and neurofuzzy systems in model-

ing and control: a case study. IFAC J 2003: 265–273.

12. Kisi A and Ay M. Comparison of ANN and ANFIS tech-

niques in modelling dissolved oxygen. In: Sixteenth interna-

tional water technology conference, IWTC-16, Istanbul,

Turkey, 2012; 1–10.

13. Farahmand AR, Manshouri M, Liaghat A, et al. Comparison

of kriging, ANN and ANFIS models for spatial and temporal

distribution modelling of groundwater contaminants. J Food

Agri Environ 2010; 8(3–4): 1146–1155.

14. Li Y and Chen D. A learning-based comprehensive evalua-

tion model for traffic data quality in intelligent transportation

systems. Multi Tools Appl 2016; 75: 11683–11698.

15. Azadeh A and Zarrin M. An intelligent framework for pro-

ductivity assessment and analysis of human resource from

resilience engineering, motivational factors, HSE and ergo-

nomics perspectives. Saf Sci 2016; 89: 55–71.

16. Barak S and Sadegh SS. Forecasting energy consumption

using ensemble ARIMA–ANFIS hybrid algorithm. Elect

Power Energy Syst 2016; 82: 92–104.

17. Mansouri I, Ozbakkaloglu T, Kisi O, et al. Predicting beha-

vior of FRP-confined concrete using neuro fuzzy, neural net-

work, multivariate adaptive regression splines and M5 model

tree techniques. Mater Struct 2016; 49: 4319–4334.

18. Mostafa MM and El-Masry AA. Oil price forecasting using

gene expression programming and artificial neural networks.

Econ Model 2016; 54: 40–53.

19. Oduro SD, Ha QP, and Duc H. Vehicular emissions predic-

tion with CART-BMARS hybrid models. Trans Res D 2016;

49: 188–202.

20. Augustina D, Lee CKM, and Piplani R. A review: mathemat-

ical models for cross docking planning. Int J Eng Bus Manag

2010; 2(2): 47–54.

21. Zaki SA, Mousa AA, Geneedi HM, et al. Efficient multi-

objective genetic algorithm for solving transportation,

assignment, and transshipment problems. Appl Math 2012;

3: 92–99.

12 International Journal of Engineering Business Management

22. Wisittipanich W and Hengmeechai P. A multi-objective dif-

ferential evolution for just-in-time door assignment and truck

scheduling in multi-door cross docking problems. Ind Eng

Manag Syst 2015; 14(3): 299–311.

23. Cota PM, Gimenez BMR, Arau

´jo DPM, et al. Time-indexed

formulation and polynomial time heuristic for a multi-dock

truck scheduling problem in a cross-docking centre. Comput

Ind Eng 2016; 95: 135–143.

24. Lotfan SR, Ghiasi A, Fallah M, et al. ANN-based modeling

and reducing dual-fuel engine’s challenging emissions by

multi-objective evolutionary algorithm NSGA-II. J Appl

Energy 2016; 175: 91–99.

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