# Optimum Intervention in Transportation Networks Using Multimodal System under Fuzzy Stochastic Environment

Authors: Gurupada Maity; Vincent F. Yu; Sankar Kumar Roy
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

Multimodal transport refers to the transportation of goods under a single contract but performed with at least two different modes of transport. This research designs a new method for solving the Transportation Problem (TP) by introducing multimodal transport systems under fuzzy-stochastic environment, which we refer to as Fuzzy-Stochastic Multimodal Transportation Problem (FSMMTP). An algorithm is developed to reduce FSMMTP to a deterministic TP, which is mainly based onα-cut of fuzzy numbers, and uses a signed distance function based on the mean expectation of the fuzzy-stochastic cost parameters. We derive the optimal solution as well as optimal selection of mode for transporting the goods in our proposed model. A numerical example justifies the effectiveness of our proposed study. The paper ends with a conclusion and an outlook to future studies.


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## Body

## 1. Introduction

In the earlier days in operations research, the Transportation Problem (TP) was mainly used to minimize the total transportation cost for distributing items from sources to numerous destinations. In most daily-life problems, there are several decisions to be made such as fixing the cost of goods, profit for sellers, and multiple objective functions, which can be generated by TP. Nowadays, in competitive market scenarios, minimizing the transportation cost in a business economy and in government policies is one of the most important matters.Multimodal transport, which is also known as combined transport, allows to transport goods under a single contract, but it is performed with at least two modes of transport. The carrier is liable (in the usual sense) for the whole conveyance even though several/different modes of transport are considered like road, sea, and train. The carrier does not have to possess all the means of transport, and in practice usually this is not valid. The carrier often contracts subcarriers, which are known as actual carriers in common language. Under the situation in which a single type of commodity is required to be transported to the destinations by more than a single type of carrier in a transportation situation, the different types of carriers are termed as Multimodal Transport Operators. In a transportation network, if there exists at least onesupplementary origin, then it is described as MMTP. To accommodate the real-life transportation problem, it is not always possible to fulfill the demand of the customers at the destination points using a single mode of transportation. There are sometimes restrictions for transporting the goods, and so it is required to consider multimodes of transportation from different nodes. Thus, the transportation is not a simple TP but becomes MMTP.In real-world situations, sometimes data are not precisely available. This imprecision may occur in a stochastic or nonstochastic (i.e., fuzzy) sense or for both fuzzy and stochastic environments. Here, a fuzzy-stochastic environment on a cost parameter is considered for solving the Multimodal Transportation Problem (MMTP) and selecting the mode of transportation corresponding to the optimal solution under the prescribed uncertain situations. Probability and mathematical expectations have often been used during the formulation of stochastic models to deal quantitatively with random data. We incorporate a simple and effective way for defuzzification and derandomization of the fuzzy-stochastic cost parameter of our proposed method. We use the concept ofα-cut of the fuzzy numbers, take the expectation separately on both the lower and upper α-cuts, and then calculate the mean expectation with the help of signed distance. Thereafter, we solve the reduced Linear Programming Problem (LPP) using the simplex method to derive the mode of transportation as well as optimal solution.Now, we present the main contributions of our study as follows:Design FSMMTP, a new class of TP under the consideration of multimodal transport systemsIncorporate uncertainty through fuzzy-stochastic cost parametersIntroduce a new algorithm to solve FSMMTPOn solving FSMMTP, we establish the superiority of our approach compared with classical TPThe rest of the paper is set out as follows. In the next section (Section2), the review of related research is presented. Section 3 considers the preliminary concepts related to fuzzy and random numbers. The detailed problem background is presented in Section 4. Section 5 designs a new mathematical model for the MMTP under fuzzy-stochastic environment. Section 6 discusses the solution procedure. To show the application of proposed mathematical model of MMTP, Section 7 considers a numerical example including a subsection (c.f., Section 7.1) which depicts the result and discussion regarding the proposed study. Finally, Section 8 describes the conclusions with avenues for future study.


## 2. Review of Related Research

The traditional transportation model was first initiated by Kantorovich [1], who prescribed an incomplete algorithm for calculating the solution of the transportation problem. Hitchcock [2] studied the problem of minimizing the cost of product distribution from several warehouses to a number of purchasers. Various studies on transportation safety planning were developed by Ergun et al. [3], Luathep et al. [4], Zhi-Chun et al. [5], and Norouzi et al. [6]. Recently, the multiobjective transportation problem has been discussed in different directions by several researchers such as Ehrgott et al. [7], Gupta et al. [8], Maity et al. [9], Castellanos and Fruett [10], Gupta et al. [11], and Roy et al. [12].There are several research studies available in the literature to accommodate the transportation problem through the unimodal system, such as Nanry and Wesley [13] who presented a detailed study for solving the uni-modal TP. Catalani [14] considered a statistical study to improve intermodal freight transport in Italy, by using road-ship and road-train transports. Qu and Chen [15] posed MMTP as a Multicriteria Decision-Making (MCDM) problem. Recently, Maity et al. [16] studied on MMTP and solved an optimization problem involving neural network.Several research works (cf. [17, 18]) are available in the literature that analyze the transportation problem, but to the best of our knowledge, until now no one has used multimodal transport systems in uncertain environments for solving TP. Li and Lai [19] presented a fuzzy compromise programming approach to MOTPs. In order to measure a fuzzy event, Zadeh [20] proposed the concept of a possibility measure. Since then, the possibility theory has been studied by many researchers such as Zimmermann [21], Dubois and Prade [22], and Liu [23]. Although the possibility measure has been widely used, it has no self-duality property. However, a self-dual measure is absolutely needed in both theory and practice. A study on bi-level MOTP under fuzziness has been incorporated by Mardanya et al. [24], while Ebrahimnejad et al. [25] examined the shortest path routing problem under a fuzzy environment. The multiobjective multi-item transportation problem has been analyzed through just-in-time by Mardanya et al. [26]. Several researchers analyzed TP in stochastic (cf. [17, 27]) and fuzzy (cf. [21, 28, 29]) environments. Xu and Huang [30] pointed out that the time windows in a vehicle routing problem with soft time windows contains both fuzzy and stochastic information simultaneously. The concept of fuzzy-random variable was introduced by Kwakernaak [31].The contributions of various researchers related to this field are shown in Table1. To the best of our knowledge, no attempt has been made so far to solve MMTP under fuzzy-stochastic environment. Therefore, in this study, we oppose FSMMTP and justify its utility considering a real-life problem.Table 1
Contributions of different authors related to TP under uncertainty.
Author(s)Crisp costFuzzy costStochastic costFuzzy-stochastic costSingle mode of transportMultiple mode of transportKantorovich [1]√Puri and Ralescu [32]√Mahapatra et al. [17]√√Mathur et al. [29]√√Maity et al. [33]√√Maity et al. [16]√√Khalifa et al. [34]√√Biswas et al. [35]√√Mardanya et al. [24]√√Chhibber et al. [36]√√Our paper√√√√


## 3. Preliminary Concepts

Consider a triangular fuzzy numberA˜=a1,a2,a3. An approximation on Triangular Fuzzy Number (TFN) A˜ is presented in many different ways. According to the proposition of Kaufmann and Gupta [37], the approximated value of A˜ is denoted by apA˜ and is defined as apA˜=a1+2a2+a3/4.The algebraic operations on TFNs are as follows:(i)
Addition. The addition of two TFNs A˜=a1,a2,a3 and B˜=b1,b2,b3 using maximin convolution on fuzzy sets is defined as follows:(1)μA˜⊕B˜z=∨μA˜x∧μB˜y:z=x+y.Therefore,A˜⊕B˜=a1,a2,a3+b1,b2,b3=a1+b1,a2+b2,a3+b3 and  A˜⊖B˜=a1,a2,a3−b1,b2,b3=a1−b3,a2−b2,a3−b1.Here, the symbols⊕ and ⊖ are addition and subtraction of fuzzy numbers, respectively.(ii)
Scalar Multiplication. Multiply by a scalar k with TFN, A˜ whose membership function is μA˜x which gives a TFN that has the following membership function: μkA˜z1=∨kμA˜x,z1=kx. Therefore,(2)kA˜=ka1,ka2,ka3,ifk≥0,ka3,ka2,ka1,ifk<0.(iii)
Signed Distance between Two Fuzzy Numbers. According to the concept of Yao and Wu [28], if two fuzzy numbers a˜ and b˜ are represented by ∪α∈0,1aαL,aαU and ∪α∈0,1bαL,bαU, then the signed distance between a˜ and b˜ is the distance between the midpoints Maα=1/2aαL,aαU of aαL,aαU and Mbα=1/2bαL,bαU of bαL,bαU over the set of α values in 0,1, which is defined as follows:  da˜,b˜=1/1−0∫01Maα−Mbαdα=1/2∫01aαL+aαU−bαL−bαUdα.(iv)
Fuzzy Random Variable. The underlying ideas and definitions incorporating the theory of fuzzy random variables are discussed by Kwakernaak [31] and Puri and Ralescu [32]. Considering a real number a, we define a fuzzy number a˜ with the membership function a˜x∈0,1∀x∈ℝ such that a˜a=1 and 0≤a˜x≤1∀x≠a (cf. the membership function has a unique maximum at x=a). Usually, we denote a˜ as a fuzzy number induced by the real number a. Let F be the set of all fuzzy real numbers. The α-cut of a˜∈F is denoted by aα and is defined as aα=x∈ℝ:a˜x≥α,α∈0,1=aαL,aαU (say). Thus, aαL=minx∈ℝ:a˜x≥α,,α∈0,1 and aαU=maxx∈ℝ:a˜x≥α,α∈0,1. Furthermore, addition and scalar multiplication on F are defined by a+bα=aα+bα and λaα =  λaα for any λ∈ℝ and α∈0,1. Thereafter, we define a function ρ:F×F⟶ℝ as follows:     ρa˜,b˜=1/2∫01aαL−bαL2+aαU−bαU2dα, a˜,b˜∈F, which forms a metric on F. Again, the ordered pair F,ρ forms a complete metric space.The Fuzzy Random Variable forms a Borel measurable functionX^˜:S,Ω,P↦F,ρ, where S is considered as the sample space, Ω is a Borel measurable subsets of S, and P refers to a probability measure.Proposition 1.
(see Kwakernaak [31]). Consider a closed fuzzy-valued function X^˜:S,Ω,P↦F,ρ. Here, X^˜ is a fuzzy random variable if and only if XαL and XαU are random variables for all α∈0,1.(v)
Fuzzy Probability Density Function. The fuzzy probability distribution function f˜x^ of X^˜ is the set of probability distribution functions of all originals X^ of X^˜ with the membership values μf˜x^.Corresponding to a fuzzy valuex˜ of fuzzy random variable X^˜, the α-level set is defined as x˜α=xαL,xαU, where xαL and xαU are considered as the observations of XαL and XαU, respectively. Since XαLw=xαL and XαUw=xαU are monotonically increasing and monotonically decreasing functions, respectively, for all α∈0,1, i.e., xαL≤x1L and x1U≤xαU∀α∈0,1, they are hence continuous with respect to α for any fixed value of w. Again, for x∈xαL,xαU, there exists β≥α such that x=xβL or x=xβU. Therefore, any x∈xαL,xαU can be associated with a pdf f˜x^ with x^. Again, using the extension principle of fuzzy set theory (see Zadeh [20]), the membership function of the fuzzy probability density function f˜x^ of X^˜ is defined as follows:(3)μf˜x^z=supw∈x^:f˜x^=zμx^w,whereμx^˜w is the membership function of fuzzy random variable x^˜ of X^˜.(vi)
Expectation of Fuzzy Random Variable (Xu and Huang [30]). The expectation of fuzzy random variable X^˜ is a unique fuzzy number EX^˜, which is defined as     EX^˜=∪α∈0,1EXαL,XαU=∪α∈0,1∫−∞∞xαLfαLxdx∫−∞∞xαUfαUxdx.For a discrete fuzzy number, the expectation can be defined through the following formula:EX^˜=∑i=1∞xi˜PX^=xi˜.


## 4. Problem Background

The termmultimodal is defined for several modes of transport in a TP, and as a whole, it is referred to as the multimodal transportation problem. Goods can be transported via several modes, and people also use different modes of transportation for their journey. Therefore, multiple modes of transportation are considered as follows:(i)
Roadway/highway automobiles (including taxi), truck, motorcycle, and so on(ii)
Passenger rail and traditional freight train, bus, car, and so on(iii)
Air-passenger service and freight transportation(iv)
Water-ferry, barge, transatlantic vessel, cruise ship, and so on(v)
Bicycling, nonmotorized-walking, and so onUnder the consideration that in a transportation system, there exist multiple types of transportation vehicles and goods are transported from origins to destinations by use of multiple steps using different types of vehicles, then MMTP is used to formulate and solve the mathematical problem. There are important perspectives in MMTP such as follows:(i)
To increase economic productivity and effectiveness, thereby enhancing the nation’s competitiveness globally(ii)
To decrease total transportation costs by allowing each mode to be used for the portion of the trip in which it is the best suited(iii)
To generate higher returns from public and private investments(iv)
To reduce overcrowding and burden of a transportation management(v)
To improve mobility for the elderly, isolated, disabled, and economically disadvantages(vi)
To reduce energy consumption and apply in sustainable development (cf. Das et al. [38])To represent the MMTP, we introduce the following useful definitions.Definition 1.
(Starting Origin (STO)).In a transportation problem, the sources that have the capacity to supply the goods only but there are no capacity to gather the goods treated as Starting Origins.Definition 2.
(Final Destination (FD)).In a transportation problem, the destinations that have the capacity of gathering the goods only but no such capacity to supply the goods are considered as Final Destinations.
It is not possible to supply the goods according to the requirement of thefinal destinations from the starting origins because of the vehicle capacity/multiple routes of transport. In that case, some destination nodes are required that have the capacity of supplying and receiving the goods simultaneously. These nodes are known as supplementary origins.Definition 3.
(Supplementary Origin (SO)).In a transportation problem, the destinations that have the capacity of collecting the goods as well as the capacity of delivering the goods are denoted as Supplementary Origins.
In Figure1, O1, O2, …, Om1 are the ground origins; SO11, SO12, …, SO1m2 are the supplementary origins of label 1, and D1, D2,…, Dn1 are referred as the final destinations.
In a transportation network, if there exists at least onesupplementary origin, then it is described as MMTP. To accommodate the real-life transportation problem, it is not always possible to fulfill the demand of the customers at the destination points using a single mode of transportation. There are sometimes restrictions for transporting the goods, and so it is required to consider multimodes of transportation from different nodes. Thus, the transportation is not a simple TP but becomes MMTP.
This study also considers real-life practical uncertain phenomena and incorporates a fuzzy-stochastic cost parameter in MMTP, which is termed as the Fuzzy-Stochastic Multimodal Transportation Problem (FSMMTP). When there are several agents of transportation corresponding to a particular mode of transportation with different probabilities of selection by a transportation agent, then corresponding to each mode of transportation, there is some ambiguous information on the transportation cost. Hence, the selection of an agent for transportation in MMTP is considered to be stochastic in nature, and again each agent provides a fuzzy cost to complete the transportation due to the seasonal effect. As a whole, the cost parameter in MMTP is formulated in fuzzy-stochastic environment. The mathematical model of the proposed FSMMTP is described in detail in the next section.Figure 1
Graphical representation of MMTP.


## 5. Mathematical Model

A fuzzy-stochastic transportation problem is a typical one in which the main objective is to minimize the total transportation cost and is defined as follows.Model 1.
(4)minimizez=∑i=1m∑j=1nC^˜ijxij,subject to∑j=1nxij≤aii=1,2,…,m,∑i=1mxij≥bjj=1,2,…,n,xij≥0∀iandj,where C^˜iji=1,2,…,m;j=1,2,…,n is the fuzzy-stochastic transportation cost per unit commodity from the ith origin to the jth destination. Here, aii=1,2,…,m and bjj=1,2,…,n are availability and demand at the ith origin and at the jth destination, respectively, and ∑i=1mai≥∑j=1nbj is the feasibility condition.
The presence of SO in a fuzzy-stochastic TP reduces Model1 to FSMMTP. To formulate the mathematical model of FSMMTP, we use the following notations (Table 2).
The complete FSMMTP model (c.f., Maity et al. [16]) is described as follows.Table 2
Different notations for designed mathematical model.
m1:Number ofStarting Origins (STOs)n1:Number ofFinal Destinations (FDs)mt:Number ofSupplementary Origins (SOs) at t−1th level, t=2,3,…,rr:Number of labels for originsai1:Availability of goods atith origin of STOait:Availability of goods atith origin of t−1th level SO, t=2,3,…,rbj:Demand atjth node of FDα11:Carrying capacity of a vehicle from STO to FDC^˜ij11:Transportation cost per unit commodity fromith origin to jth destination from STO to FDC^˜ij1t:Transportation cost per unit commodity for transportation fromith origin to jth destination from SO of t−1th level to FD where t=2,3,…,rC^˜ijs1:Transportation cost per unit commodity for transportation fromith origin to jth destination from STO to SO of s−1th level, s=2,3,…,rC^˜ijst:Transportation cost per unit commodity for transportation fromith node of TO of t−1th level to jth node of TO of s−1th level, t=2,3,…,r−1; s=2,3,…,r with t<sα1t:Carrying capacity of vehicle from SO oft−1th level to FDαst:Carrying capacity of vehicle from SO oft−1th level, t=2,3,…,r, to SO of s−1th label, s=r,r−1,…,2z1:Objective function for minimizing the transportation cost to the final destination from STO and all SOzi:Objective function for minimizing the transportation cost tor−i+1th label destination points from STO and all SO of r−ith label, i=2,3,…,r−1xij11:Decision variable of transportation fromith origin of STO to jth destination of FDxij1t:Decision variable of transportation fromith origin to jth destination from SO of t−1tht=2,3,…,r level to FDxijs1:Decision variable of transportation fromith origin to jth destination from STO to SO of s−1th level, s=2,3,…,rxijst:Decision variable of transportation fromith node of TO of t−1th level to jth node of TO of s−1th level, t=2,3,…,r−1; s=2,3,…,r with t<sModel 2.
(5)minimizez=z1+z2+⋯+zr,z1=∑i=1m1∑j=1n1α11C^˜ij11xij11+∑i=1m2∑j=1n1α12C^˜ij12xij12+⋯+∑i=1mr∑j=1n1α1rC^˜ij1rxij1r,z2=∑i=1m1∑j=1mrαr1C^˜ijr1xijr1+∑i=1m2∑j=1mrαr2C^˜ijr2xijr2+⋯+∑i=1mr−1∑j=1mrαrr−1C^˜ijrr−1xijrr−1,⋮zr=∑i=1m1∑j=1m2α21C^˜ij21xij21,subject to the constraints representing storing capacity at STO and SO of all labels:(6)∑j=1n1α11xij11+∑j=1mrαr1xijr1+⋯+∑j=1m2α21xij21≤ai1i=1,2,…,m1,∑j=1n1α12xij12+∑j=1mrαr2xijr2+⋯+∑j=1m3α32xij32≤ai2i=1,2,…,m2,∑j=1n1α13xij13+∑j=1mrαr3xijr3+⋯+∑j=1m4α43xij43≤ai3i=1,2,…,m3,⋮∑j=1n1α1rxij1r≤airi=1,2,…,mr,the constraints regarding least demands at FD:(7)∑i=1m1α11xij11+∑i=1m2α12xij12+⋯+∑i=1mrα1rxij1r≥bjj=1,2,…,n1,and the constraints regarding storing and distributing of goods at the nodes of, SO of all labels:(8)∑j=1n1α12xtj12+∑j=1mrαr2xtjr2+⋯+∑j=1m3α32xtj32≤∑i=1m1α21xit21t=1,2,…,m2,∑j=1n1α13xtj13+∑j=1mrαr3xtjr3+⋯+∑j=1m4α43xtj43≤∑i=1m1α31xit31+∑i=1m2α32xit32t=1,2,…,m3,⋮∑j=1n1α1rxtj1r≤∑i=1m1αr1xitr1+∑i=1m2αr2xitr2+⋯+∑i=1mr−1αrr−1xitrr−1≤artt=1,2,…,mr,xijps≥0∀i,j,sandp.
To obtain the feasible solution of Model2, it is again essential to satisfy that the amount of goods required at the nodes of FD be less or equal to the sum of their availability at the nodes of STO. Therefore, the feasibility condition of Model 2 is considered as ∑i=1m1ai1≥∑j=1n1bj.
In Model2, the number of decision variables is m1×m2×⋯×mr×n1.
The feasible region of Model2 is constructed by considering the following assumptions:(i)
There arem1 availability constraints (6) for the ground origins.(ii)
There aren1 number of demand constraints (6) for the final destinations.(iii)
There are restrictions of storing items in the supplementary origins, and so we introducem2+m3+⋯+mr number of inequalities from (6) to (7).(iv)
Again, the delivered amount of goods from the supplementary origins does not exceed the supplied amount of goods to the respective supplementary origins. To do this, we introducem2+m3+⋯+mr number of inequalities from (8).
Model2 thus consists of m1×m2×⋯×mr×n1 variables and 2m2+m3+⋯+mr+m1+n1 constraints along with the nonnegativity conditions. It is noted that, if there does not exist any path between two nodes in the proposed FSMMTP, then we consider the value of decision variable to be 0 and remove the variable form the proposed model, which reduces the number of variables in the mathematical model. In the next section, we present the solution procedure for solving FSMMTP.


## 6. Solution Procedure

The formulated FSMMTP consists of fuzzy-stochastic cost parameters, and so the objective function is now in a nondeterministic form. First, we discuss the reduction of fuzzy-stochastic cost parameters to crisp parameters, and thereafter we present an algorithm to solve FSMMTP.Reduction of fuzzy-stochastic parameter into crisp form is as follows.Proposition 2.
Assume an exponential functionfx=Ae−|x−μ|,x≥0, where A is a constant; then, f is a probability density function if A=1/2−e−μ; here, μ>0 is a real constant.Proof.
For continuous case, the probability density functionf satisfies ∫−∞∞fxdx=1.
Here,fx=Ae−x−μ,x≥0.
Therefore,(9)A∫0∞e−x−μdx=1,orA∫0μex−μdx+∫μ∞e−x+μdx=1,orAex−μ0μ−e−x+μμ∞=1,orA=11−e−μ+1=12−e−μ.
This completes the proof of the proposition.
LetC^˜ijt be the fuzzy-stochastic unit transportation cost from i-th source to j-th destination in t-th label of FSMMTP. These costs can be represented in the following way.
LetC^˜ijt follow a fuzzy-stochastic density function fC^˜ijt with C^˜ijt≥0, which has the membership function μC^˜ijt:ℝ⟶0,1, and the membership function μC^˜ijt is defined as follows:(10)μC^˜ijtx=x−C^ijt+d1d1,ifC^ijt−d1≤x≤C^ijtC^ijt+d2−xd2,ifC^ijt≤x≤C^ijt+d20,if         elsewhere.
Here,d1 and d2 are left and right deviations with respect to the mean of the fuzzy-stochastic cost parameter. The probability density function can be taken according to the decision maker’s choice, which may be Normal, Gamma, Weibull, and so on. For many real-life situations in TP, the DM may consider the previous record of transportation cost, and from them the DM can easily calculate the mean cost. It is also easy to formulate the corresponding exponential function. In this paper, we consider the exponential distribution as follows:(11)gCijt=12−e−μijt2e−Cijt−μijt.
The mean corresponding to the pdf can then be calculated as follows:(12)12−e−μijt2μijt+e−μijt.
Again, the variance of the pdf is defined as1/2−e−μijt2e−2μijt−2μijt2+4μijt+8e−μijt+8.
The objective function of Model2 is not in deterministic form as the fuzzy-stochastic cost components are present in it. Hence, we need to convert the objective function into deterministic form by reducing the fuzzy-stochastic cost parameter into a crisp parameter in the following way.
First, we treat the fuzzy nature of the cost parameter and defuzzify it by considering theα-cut of the cost parameter with the corresponding probability distribution. Second, we take an expectation to reduce it to the corresponding crisp parameter. Following the concept of Yao et al. [28], the membership function μC^˜ijrtx of the fuzzy probability density function fC^˜ijrt corresponding to C^˜ijrt is obtained by the extension principle. The α-cut of C^˜ijrt and fC^˜ijrt is, respectively, C^˜ijrtα=CijrtαL,CijrtαU, where CijrtαL=Cijrt−1−αd1, CijrtαU=Cijrt+1−αd2, and fC^˜ijrtα=fCijrtLα,fCijrtUα, where fCijrtLα and fCijrtLα can be obtained by using equation (11).(13)ECijrtLα=∫−∞∞CijrtLαfCijrtLαdCijrt∫0∞Cijrt−1−αd1fCijrtLαdCijrt,ECijrtUα=∫−∞∞CijrtUαfCijrtUαdCijrt=∫0∞Cijrt+1−αd2fCijrtUαdCijrt.
Using the signed distance function, we reduce the objective function with the fuzzy-stochastic cost to a crisp objective function described as follows:(14)Ez∗=Ez1∗+Ez2∗+⋯+Ezr∗whereEz1∗=∑i=1m1∑j=1n1α11xij1112∫01ECij11Lα+ECij11Uαdα+∑i=1m2∑j=1n1α12xij1212∫01ECij12Lα+ECij12Uαdα+⋯+∑i=1mr∑j=1n1α1rxij1r12∫01ECij1rLα+ECij1rUαdα,Ez2∗=∑i=1m1∑j=1mrα21xij2112∫01ECij21Lα+ECij21Uαdα+∑i=1m2∑j=1mrα22xij2212∫01ECij22Lα+ECij22Uαdα+…+∑i=1mr−1∑j=1mrα2r−1xij2r−112∫01ECij2r−1Lα+ECij2r−1Uαdα⋮Ezr∗=∑i=1m1∑j=1m2αr1xijr112∫01ECijr1Lα+ECijr1Uαdα.
In particular, let us consider the fuzzy probability density function:(15)fC˜ijt=1/2−μijte−C˜ijt−μijt,∀C˜ijt≥0.
We then have(16)fCijtLα=12−μijte−Cijt−μijt,Cijt≥1−αd1,fCijtUα=12−μijte−Cijt−μijt,Cijt≥−1−αd2,ECijtLα=12−μijtlimB⟶∞∫1−αd1BCijt−1−αd1e−Cijt−μijtdCijt=12−μijt∫1−αd1μijtCijt−1−αd1eCijt−μijtdCijt+12−μijtlimB⟶∞∫μijtBCijt−1−αd1e−Cijt−μijtdCijt=12−μijt2μijt−21−αd1+e−μijt−1−αd1ECijtUα=12−μijtlimB⟶∞∫1−αd2BCijt+1−αd2e−Cijt−μijtdCijt=12−μijt∫1−αd2μijtCijt+1−αd2eCijt−μijtdCijt+12−μijtlimB⟶∞∫μijtBCijt+1−αd1e−Cijt−μijtdCijt=12−μijt2μijt+21−αd2+e−μijt+1−αd2.
The integral in the expression ofz1∗,z2∗,…,zr∗ now reduces to the following form:(17)∫01ECijrtLα+ECijrrUαdα=12−μijt∫012μijt−21−αd1+e−μijt−1−αd1+2μijt+21−αd2+e−μijt+1−αd2dα=12−μijt4μijt−d1−d2+e−μijted1−1d1+e1−d2d2.
We thus present an algorithm to solve our proposed FSMMTP problem (Algorithm 1).Algorithm 1: Steps for solving FSMMTP.
(i)
Step 1: according to the given problem, first set up the level of the supplementary and other nodes of the decision making problem.(ii)
Step 2: consider the fuzzy-stochastic cost with respective membership functions and distribution functions for each transportation route by the decision maker’s choice.(iii)
Step 3: construct the objective function corresponding to each level of transportation along with the main objective function and formulate the restrictions as per choice of the DM, described in the constraints (6)–(8) of Model 2.(iv)
Step 4: reduce FSMMTP to crisp TP using Section 5.(v)
Step 5: solve the deterministic model obtained in Step 4 by using a simplex algorithm, and in case of a number of variables, one can use MATLAB or LINGO software to solve the problem.(vi)
Step 6: from the obtained solution, we get the optimal value of the objective function as well as the optimal selection transportation mode. Thus, DM can get information about the amount of goods run over the supplementary origins throughout the procedure.(vii)
Step 7: stop.


## 7. Numerical Example

A numerical example is presented here to justify the utility of FSMMTP. A reputed Mineral Water Production Company (MWPC) has three factories producing mineral water at three locations S1, S2, and S3 (here treated as the ground origins). There are two cities, D1 and D2 (here taken as final destinations), with serious drinking water problems. MWPC supplies drinking water to these two cities. MWPC has three storage sites, namely, B1, B2 (considered as the first-level supplementary origins), and C1 (considered as the second-level supplementary origins). There is only one way to supply water from B1 or B2 to C1. There are different modes of transportation from origins to destinations. Company wants to determine the transportation plan with the minimum transportation cost. The graphical representation of the numerical example is shown in Figure2.Figure 2
Graphical representation of the numerical example.The traditional approach of TP cannot provide any such mathematical model to solve the proposed problem. To solve the problem, we here formulate the mathematical model known as FSMMTP.The following notations and assumptions are considered to formulate the mathematical model of FSMMTP:(i)
The decision variables for transporting the goods are as follows:When S1, S2, and S3 to D1 and D2 are considered asxij11, use ship-way with vehicle capacity α11=500 unitWhen B1 and B2 to D1 and D2 are taken asxij21, use rail-way with vehicle capacity α12=100 unitWhen C1 to D1 and D2 are considered asxij31, use road-way without any vehicle capacity restriction, i.e., α13=1 unitWhen S1, S2, and S3 to C1 are considered asxij12, vehicle restriction α21=250 unitWhen B1 and B2 to C1 are considered asxij22, there is no vehicle restrictionWhen S1, S2, and S3 to B1 and B2 are considered asxij13, there is not any vehicle restriction(ii)
The optimal feasible region of FSMMTP corresponding the numerical example consists by twelve constraints.The supply capacity at the ground origins S1, S2, and S3 is introduced by three constraints. The demand at the final destinations D1 and D2 is considered by three constraints. Storing capacity at the supplementary origins B1, B2, and C1 provides three constraints. Amount of goods distributed from the supplementary origins B1, B2, and C1 does not exceed the amount of storing items and produces three constraints. Hence, the number of constraints in MMTP of the numerical example is 12.Since the seasonal effects of the cost parameters (e.g., production costs and carrying costs) are uncertain, MWPC considered fuzzy-stochastic costs, which follow a triangular fuzzy exponential distribution. Therefore, MWPC sets the expected cost as the exponential mean (i.e.,μijt) of fuzzy-stochastic cost parameters for different modes of transportation, which as presented in Tables 3 to 8. MWPC also assumes that d1=3 and d2=4.Table 3
Exponential mean cost from S1, S2, and S3 to D1 and D2 (in $).
D1D2S114.1212.25S214.2517.20S313.1012.20Table 4
Exponential mean cost from B1 and B2 to D1 and D2 (in $).
D1D2B17.259.85B28.206.12Table 5
Exponential mean cost from C1 to D1 and D2 (in $).
D1D2C15.404.25Table 6
Exponential mean cost from S1, S2, and S3 to C1 (in $).
C1A110.15A211.25Table 7
Exponential mean cost from B1 and B2 to C1 (in $).
C1S17.20S28.15S36.25Table 8
Exponential mean cost from S1, S2, and S3 to B1 and B2 (in $).
B1B2S14.145.25S26.205.15S37.256.15The availability of goods at each Ground Origin S1, S2, and S3 is 1250, 1250, and 1250 units, respectively. The maximum capacity of storing at the Supplementary Origins B1, B2, and C1 is 1100 units, 1200 units, and 1000 units, respectively. The demands at the destinations D1 and D2 are 1755 and 1975, respectively. Using the procedure described in Section5, we obtain the mathematical model as follows.Model 3.
(18)minimizez1+z2+z3,z1=50014.37x1111+12.50x1211+14.50x2111+17.45x2211+13.35x3111+12.45x3211+1007.50x1112+10.10x1212+8.45x2112+6.37x2212+5.65x1113+4.51x1213,z2=25010.4x1121+10.50x2121+7.45x3121+8.40x1122+6.50x2122,z3=4.40x1131+5.51x1231+6.45x2131+5.51x2231+7.50x3131+6.40x3231,500x1111+x1211+100x1121+250x1131+x1231≤1250,500x2111+x2211+100x2121+250x2131+x2231≤1250,500x3111+x3211+100x3121+250x3131+x3231≤1250,500x1111+x2111+x3111+100x1112+x2112+x1113≥1755,500x1211+x2211+x3211+100x1212+x2212+x1213≥1975,x1131+x2131+x3131≤1100,x1231+x2231≤1200,250x1121+x2121+x3121+x1122+x2122≤1000,x1131+x2131+x3131≤100x1112+x2112+x1122+x2122,x1231+x2231+x3231≤100x1212+x2212+x1222+x2222,250x1121+x2121++x3121+x1122+x2122≤x1113+x1213,xijsp≥0all are taken integers;∀i,j,s,p.
Model3 is simply a LPP that can be solved through any simplex algorithm. As Model 3 contains a large number of variables, we use LINGO software to obtain the solution of Model 3.


### 7.1. Result and Discussion

The minimum value of the objective function is $46, 082. The optimal solution of Model3 is shown in Tables 9–12.Table 9
The amounts of transported goods to final destinations D1 and D2.
Variablex1111x1211x2111x2211x3111x3211x1112x1212x2112x2212x1113x1213Value01×500001×50008×1000012×100455275Table 10
The amounts of transported goods to supplementary origin C1.
Variablex1121x2121x3121x1122x2122Value003×25000Table 11
The amounts of transported goods to supplementary origins B1 and B2.
Variablex1131x1231x2131x2231x3131x3231Value750050120000Table 12
The amounts of transported goods stored at all supplementary origins.
NodeB1B2C1D1D2Value800120073017551975Solution of the numerical example in absence of supplementary origins is as follows.In the absence of supplementary origins (cf. B1, B2, and C1), the problems become a classical TP. Then, the mathematical mode is as follows:Model 4.
(19)minimizez1,z1=50014.37x1111+12.50x1211+14.50x2111+17.45x2211+13.35x3111+12.45x3211,500x1111+x1211≤1250,500x2111+x2211≤1250,500x3111+x3211≤1250,500x1111+x2111+x3111≥1755,500x1211+x2211+x3211≥1975,xijsp≥0.
The minimum value of the objective function is $49, 495. The optimal solution of Model4 is shown in Table 13.
However, the company would have to pay the transportation cost for the following amount of goods (see Table14) because the carrying cost of vehicle is not reduced by the empty space in the vehicle.
As a result, the total transportation cost significantly exceeds the computed cost $49, 495. Furthermore, sometimes the transportation of goods between the nodes is limited due to vehicle capacity. To minimize the transportation costs, different appropriate types of vehicles need to be used at different nodes.
To validate the proposed mathematical model of MMTP, we describe various possibilities for the numerical example as follows:(i)
Consider that the routes among the supply points S1, S2, and S3 to destination points D1 and D2 are sea-way. Delivering the goods is thus made via ship. Obviously, a large amount of goods can be delivered by ship, and the amount is 500 units. In that situation, if there are no other nodes available like B1, B2, and C1, then the formulated TP is a classical TP. In this case, we see that there exists a feasible solution of the proposed problem, but the transportation cost is not minimized. In each destination the minimum requirements are 1755 units and 1975 units of goods, which indicate at least two ships are required for delivering the goods in each node D1 and D2. Thus, traditional TP is not enough to give a definite conclusion without considering the supplementary origins as we considered in our proposed study.(ii)
We again assume that there is a connection through rail-way between B1 and B2 to D1 and D2. The capacity of transports in each time by rail-way is large, and so we consider that a single transport it needs 100 units of goods. In that situation, we solve the problem without considering the supplementary origin C1 (i.e., using the value for the variables as “0” for those that are taken for C1) and add the total transportation cost of Dollar47, 604; clearly, this is a larger cost than the original solution presented by our method. The amount of transported goods to nodes D1 and D2, respectively, is 2000 units and 1800 units. The amount of goods supplied at the supplementary origins B1 and B2 is 1100 units and 1200 units, respectively.(iii)
In a similar way, if we formulate the mathematical model without considering the supplementary origins B1 and C1 or B2 and C1, then the transportation cost will increase.(iv)
Another important factor in the study is that it considers transportation parameters as fuzzy-stochastic types which accommodate the real situations of MMTP where data are not given in a crisp way. The fuzzy-stochastic parameters are handled in our study and which will be helpful in real-life uncertain multimodal transportation systems.
According to our discussion, we have analyzed that the introduction of a multimodal system in TP is very much essential to reduce the transportation cost for delivering goods. However, in the classical TP, this is not possible. Because of that, we have printed a way to handle the situation in a multimodal transportation system.Table 13
The quantity of goods transported to final destinations D1 and D2.
Variablex1111x1211x2111x2211x3111x3211Value02.5×5002.46×50001.05×5001.45×500Table 14
Transportation costs need to be paid for the goods transported to final destinations D1 and D2.
Variablex1111x1211x2111x2211x3111x3211Value01500150005001000


## 7.1. Result and Discussion

The minimum value of the objective function is $46, 082. The optimal solution of Model3 is shown in Tables 9–12.Table 9
The amounts of transported goods to final destinations D1 and D2.
Variablex1111x1211x2111x2211x3111x3211x1112x1212x2112x2212x1113x1213Value01×500001×50008×1000012×100455275Table 10
The amounts of transported goods to supplementary origin C1.
Variablex1121x2121x3121x1122x2122Value003×25000Table 11
The amounts of transported goods to supplementary origins B1 and B2.
Variablex1131x1231x2131x2231x3131x3231Value750050120000Table 12
The amounts of transported goods stored at all supplementary origins.
NodeB1B2C1D1D2Value800120073017551975Solution of the numerical example in absence of supplementary origins is as follows.In the absence of supplementary origins (cf. B1, B2, and C1), the problems become a classical TP. Then, the mathematical mode is as follows:Model 4.
(19)minimizez1,z1=50014.37x1111+12.50x1211+14.50x2111+17.45x2211+13.35x3111+12.45x3211,500x1111+x1211≤1250,500x2111+x2211≤1250,500x3111+x3211≤1250,500x1111+x2111+x3111≥1755,500x1211+x2211+x3211≥1975,xijsp≥0.
The minimum value of the objective function is $49, 495. The optimal solution of Model4 is shown in Table 13.
However, the company would have to pay the transportation cost for the following amount of goods (see Table14) because the carrying cost of vehicle is not reduced by the empty space in the vehicle.
As a result, the total transportation cost significantly exceeds the computed cost $49, 495. Furthermore, sometimes the transportation of goods between the nodes is limited due to vehicle capacity. To minimize the transportation costs, different appropriate types of vehicles need to be used at different nodes.
To validate the proposed mathematical model of MMTP, we describe various possibilities for the numerical example as follows:(i)
Consider that the routes among the supply points S1, S2, and S3 to destination points D1 and D2 are sea-way. Delivering the goods is thus made via ship. Obviously, a large amount of goods can be delivered by ship, and the amount is 500 units. In that situation, if there are no other nodes available like B1, B2, and C1, then the formulated TP is a classical TP. In this case, we see that there exists a feasible solution of the proposed problem, but the transportation cost is not minimized. In each destination the minimum requirements are 1755 units and 1975 units of goods, which indicate at least two ships are required for delivering the goods in each node D1 and D2. Thus, traditional TP is not enough to give a definite conclusion without considering the supplementary origins as we considered in our proposed study.(ii)
We again assume that there is a connection through rail-way between B1 and B2 to D1 and D2. The capacity of transports in each time by rail-way is large, and so we consider that a single transport it needs 100 units of goods. In that situation, we solve the problem without considering the supplementary origin C1 (i.e., using the value for the variables as “0” for those that are taken for C1) and add the total transportation cost of Dollar47, 604; clearly, this is a larger cost than the original solution presented by our method. The amount of transported goods to nodes D1 and D2, respectively, is 2000 units and 1800 units. The amount of goods supplied at the supplementary origins B1 and B2 is 1100 units and 1200 units, respectively.(iii)
In a similar way, if we formulate the mathematical model without considering the supplementary origins B1 and C1 or B2 and C1, then the transportation cost will increase.(iv)
Another important factor in the study is that it considers transportation parameters as fuzzy-stochastic types which accommodate the real situations of MMTP where data are not given in a crisp way. The fuzzy-stochastic parameters are handled in our study and which will be helpful in real-life uncertain multimodal transportation systems.
According to our discussion, we have analyzed that the introduction of a multimodal system in TP is very much essential to reduce the transportation cost for delivering goods. However, in the classical TP, this is not possible. Because of that, we have printed a way to handle the situation in a multimodal transportation system.Table 13
The quantity of goods transported to final destinations D1 and D2.
Variablex1111x1211x2111x2211x3111x3211Value02.5×5002.46×50001.05×5001.45×500Table 14
Transportation costs need to be paid for the goods transported to final destinations D1 and D2.
Variablex1111x1211x2111x2211x3111x3211Value01500150005001000


## 8. Conclusion and Outlook for Future Study

This study has introduced the multimodal system into TP under fuzzy-stochastic environment and has significantly analyzed it through real-life decision-making problems. Situations may occur in a transportation system due to the presence of multiple-mode transportation in which the traditional TP fails to formulate a mathematical model that finds the lowest cost transportation route. The proposed FSMMTP may be applied to formulate the mathematical model under multiple-mode transportation, and its solution suggests the selection of mode of transportation as well as the optimal solution of the problem. By considering real-life situations, we have accommodated the fuzzy-stochastic cost parameter in MMTP and we have discussed a solution procedure to calculate the optimal value as well as optimal mode of transportation.In a future scope of study, multimodal transportation planning should be integrated such as networks, stations, user information, and fare payment systems under different uncertain environments. The proposed model of FSMMTP can thus be used for selecting the modes in a variety of transportation improvement policies such as mobility management strategies, pricing reforms, and smart growth land use policies.



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