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Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade AN UPGRADED APPROACH TO SOLVE FUZZY TRANSPORTATION PROBLEMS
An Upgraded Approach to Solve Fuzzy Transportation
Problems
Kaushik A Joshi
Department of First Year Engineering Prof Ram Meghe College of Engineering and Management Badnera-Amravati (MS India)
Kirankumar L. Bondar
P. G. Department of Mathematics, Govt Vidarbh Institute of Science and Humanities, Amravati.
klbondar [email protected]
Pandit U. Chopade
Research Supervisor, Department of Mathematics, D.S.M's Arts Commerce and Science College ,Jintur. [email protected]
Abstract
TP has many applications and applications and applications to reduce costs. A good algorithm has been developed to adjust the TP in the context of all given parameters, namely the supply, demand and TC team one, well. However, in real applications, there are many different situations due to uncertainty. It is therefore important to study PT in an uncertain environment. In this paper, an updated procedure is proposed to fix FTP where all parameters represents the non-triangular FN. The first is to use a non-trivial assembly to convert FTP to an LP with FC and net resistance. The second is to use a new vending system to turn the problem-solving lab into a three-wire lab. The value of a well-updated system is assessed compared to existing systems from an application model. The results obtained show that the updated method proposed in this study is simpler and more efficient than some existing methods commonly used in literature.
Keywords: Fuzzy Transportation Problem, Fuzzy Numbers, Solid Transportation Problems, Linear Programming Problem.
Abbreviations
Transportation Problem : TP
Transportation Cost : TC
Fuzzy Transportation Problem : FTP
Fuzzy Number : FN
Linear Programming : LP
Fuzzy Cost : FC
Linear Programming Problem : LPP
Fuzzy Linear Programming : FLP
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade AN UPGRADED APPROACH TO SOLVE FUZZY TRANSPORTATION PROBLEMS
Membership Function
Objective Function
Solid Transportation Problem
Fuzzy Variables
Goal Programming
Optimal Solution
Initial Basic Feasible Solution
Fuzzy Supply
Fuzzy Demand
Fuzzy Quantity
Objective Value
MF
OF
STP
FV
GP
OS
IBFS
FS
FD
FQ
OV
1. Introduction
TP is an important LPP installed network that appears in many situations and should receive a lot of attention in documentation. The main idea of this problem is to find the minimum TC of the product to meet the requirements in the destination by using the resources at the beginning. TP can be used for a variety of situations such as planning, production, investment, plant location, product management, project management and many more [26]. In general, TP is handled assuming TC and the price and rejection values are directly related, i.e. around the network. However, in most cases, the decision maker has no information about the TP rate. As [35] explained, the following factors may affect the state of fuzziness in TPs: (a) the decision-maker lacks sufficient information about the TC unit of the transport function and therefore the TC does not n ', (b) there may be some kind of misconception about the demand for a new product on the market, (c) there may be uncertainty about the availability of the product from the source or importer due to time constraints. Also, in the fast, there are many different conditions due to uncertainty, such as changing weather conditions, oil prices, and traffic conditions. It is therefore important to study TP in an uncertain environment. Since TP is actually LPP, the easiest way is to add the current FLP format to FTP [4, 19, 22, 25, 33, 39, 44, 49]. However, some of the existing systems provide only a clever solution that represents agreement in the case of nonlinear data [3, 28-29, 32, 36].
FTP studies are also available, [27] demonstrated that the solution obtained by FLP always worked and developed in the previous studies, FLP in several of the best ways to fix TP. [13] developed an algorithm to adjust TPs where supply and demand are solid foundations with linear or triangular MF. [46] used FSs to fix the TP and the required by the parametric program. Their approach provides a solution that meets the highest objectives and objectives at the same time. [34] discussed the TP type and FC rate and turned the problem into a TP bicriterion on OF net. Their system provides only a good solution based on a good solution of the changed problems. [5] proposed a free STP of trapezoidal FN representing transportation cost, requirement and authority. They put together a parametric system to find a non-trivial solution. [21] devised a method based on the extension principle to obtain unnecessary OV via FTP as well as FC and FS rates as well as the required number. The LP system requires several objectives to solve the FTP problem. Liu incorporated a similar mechanism to regulate nonlinear STP [12]. [42] showed two ways to reduce the FTP cost of supply chain and the required are trapezoidal FNs. They used a parametric method to find a non-invasive solution with the aim of reducing the TC concentration in both systems. [50] studied STP bicriterion in the stochastic parameter and built three mathematical models for the problem, including the expected GP value, the unpredictable GP block, and the GP-based space. [37] proposed a STP of a specific load in an unfamiliar environment, where immediate and unpaid prices, supply standards and requirements, as well as transportation capacity were FV. [18] demonstrated a non-
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71)
TRANSPORTATION PROBLEMS_Volume 17, December 2022
invasive GP approach to solving the generation / transfer configuration problem embedded in a number of unimportant objectives in an uncertain environment. [14] devised a new GP-based system to streamline FTP and FC. [23] proposed several ways to fix TP in a non-invasive network. [17] used algorithm cells to solve TPs in random numbers. [43] produced FTP, using trapezoidal FNs and developed a modified fuzzy distribution system to achieve the OS in FN format. [24] introduced a new algorithm, a zero-sum algorithm for finding seamless FTP operating systems with critical parameters. [30] devised a strategy for reducing TC as well as travel time when demand, supply and TC per minute are available in FN. [40] developed a new FLPP-based system to find the FTP OS. They developed a new system based on the quality function to fix FTP on the TC, product supply and demand are fully represented by trapezoidal [38]. After that, [11] introduced a similar algorithm to fix the same type of FTP assuming that the decision maker was unsure of the correct TC values but there was nothing wrong with that involved in products and demand. [20] introduced a simpler algorithm to solve an FTP problem that was simpler and easier to understand than the method suggested in [45]. In addition, a new mechanism was developed to locate a non-invasive OS that had no transport problems on the new trapezoidal FN representation of [48]. [1] devised a systematic approach to fix all types of FTP, both increasing or decreasing OF. [15] proposed a new complex strategy for the consideration and choice of investing in intelligent travel systems. A unique set of many STPs with nonsense penalties, sources, requirements and liability was developed and implemented by [31]. FLP-based messaging systems for FS, request, and travel capabilities have been included in this report. [6] developed a fuzzy version of the Vogel method and MODI to achieve a non-invasive IBFS and a viable solution that could be done, one by one, without translating them into classic TP. Furthermore, [52] developed algorithms to find the OS of FTP, where supply, demand and price are all FN. Their algorithm provides the decision maker a better solution compared to current systems. [16] used examples to show that their system will not always lead to a useless OS. [41] discussed the psychological analysis of FTP. [7] demonstrated an ingenious GP approach to resolving TP with multiple objectives and intermediate costs. Furthermore, [10] compared two working TPs with 2 fuzzy parameters where unit TCs, the specific charge and the first problem in TC units, charge , supply and demand and the second problem are type FV 2. [9] focused on the generation of PTS OS in a nonlinear environment, which assumes that all parameters are type 2 FV due to lack of transparency.
In this study, the following contributions were made by developing a new destruction method to mimic and modify FTPs: (a) Unlike some current methods, it is assumed that the arrival, desire and group TC values are negative triangular FNs. (b) The shortcomings of the existing FTP method are briefly discussed. (c) How to manage FTP is divided into two main parts. The first is to convert FTP to LPP with FC blocking on the net. The second is to develop a new sales strategy to convert the resulting LPP into three TP net. (d) The computational compression of the required method is greatly reduced compared to some conventional methods commonly used in literature.
2. Preliminaries
In this section, we examine some important concepts related to the fuzzy set concept, which will be used in other scripts [2].
First Definition: A FN is a convex normalized fuzzy set v * of real row R; the MF who goes on and on.
Second Definition: FN v = (v1f v2, v3) is said to be FN triangular if its MF is given as follows:
qv*( y) = ' Vi < y < V2
V2 - V1
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71) TRANSPORTATION PROBLEMS_Volume 17, December 2022
v, - y (A1)
dv*(y) = v2 < y < v3
V3 - v2 Qv* (y) = 0, elsewhere
FN v = (vlr v2, v3) can also mean v * = (m), v(d), v(v)J . In this case, the point (v) d of the
triangle FN v * gives the highest point 6V* (y), i.e., 6V* (vd) = 1, this is also called v * triangular FN core. Furthermore, (v)l and (v)v are individually spaced supporting the bottom and top of the triangular FN v *. All FN settings are defined as triangular TF (R).
Third Definition: It is said that FN triangular v = (v1f v2, v3) is the positive FN triangular if vl > 0. All these FN triangular curves are defined as all TF (R) *.
Forth Definition: It is said that the two triangular FNs v = (vv v2, v3) and u = (uv u2, u3) are equal, v * = u *, v1 = u1, v2 = u2 if only and v3 = u3.
Fifth Definition: Let v = (vv v2, v3) and u = (ul, u2, u3)be two negative triangular FNs in l e R. Then the mathematical function is given in v * and u * by:
i) l > 0, Iv* = (lvx, lv2, lv3 )
ii) l < 0, Iv * = (lv3, lv2 , lv1 )
iii) v *XU * = (vj + Uj, v2 + u2, v3 + u3)
iv)v*xu* = (v1u1, v2u2, v3u3)
3. FTP
In a typical TP, the decision maker is thought to have accurate information about the magnitude of the problem. In real-world applications, TC standards, product supply and demand may not be fully understood due to uncontrolled events. To deal with such situations, a fuzzy program is put on paper to fix TP. FTP, by a decision maker who is unsure of the exact principles of TC, supply and demand, can be developed [51] as follows:
n o
X yjk
j=1 k=1
Subject to:
o
^ yj = j =1'2'3'...'n
k=1 o
^ y*k = ek k =1,2,3.....0
k=1
yjk eTF(R), j = 1,2,...,n, k = 1,2,...,o. (A2)
where, n = total number of points, o = total number of places, tj = FS of product and jth origin, ek =
FD of product and place kth where, djk = fuzzy TC per minute product from jth from kth point and
*
yjk = FQ of the product to be transferred from jth point from kth to reduce the total TC concentration. Sixth Definition: Eq.A2 is considered a balanced FTP, if:
n o
£ t=I
j=1 k=1
Elsewhere, it is known as unbalanced FTP.
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade AN UPGRADED APPROACH TO SOLVE FUZZY TRANSPORTATION PROBLEMS
First Remark: Without a general hiccup, we think the Eq.A2 is balanced one.
Second Remark: Since the negative number of products in the negative TC is insignificant, it is assumed that all parts of the FTP are not negative triangular FNs. About 2 words, take it as djk, tj,
ek and y.k and all represent it djk = (djk) m, (djk) d, (djk) v, tj = (tj) m, (tj) d, (tj) v. , ek = (ek) m,
* * * *
(ek) d, (ek) v and y^k = ( y^k) m, ( y^k) d, ( y jk) v, one. So from description 5, Eq.A2 can be interpreted as follows:
n o /
\m / \m
££((djk)m(yß)m,(j) (yß),(jJ(yß)l
j=1 k=1 (
Subject to Constraints:
J[(yjk )m, (yjk f, (yjk J 01 = ((ekT, (ek i, (ek)V), k =12,3,...,0 £ ^ yjk)m , (yjk) , ( yjkJ ) = ^ )m '(t i '(t J ^ ' j = !'2'3....."
(yjk) - (yjk) ^ j =123...' n k =123...'0 (yjk) - (yjk) ^o'j=1'2'3'•••'nk =12'3'•••'0
(yjk) > 0,j = 1,2,3,..., n; k = 1,2,3,..., o Similarly, with respect to definition 4, Eq.A3 can be rewritten as follows:
ZZl (j)" ()■ IZZI (j) () }ZZ[( j ) (i
j=l k=1 ^ J j=l k=1 ^ J j=l k=1 ^
Subject to Constraints:
Zl (y J
j=1v
Z ((j)
j=1v
Z ((* f
j=1v
Z ((*)"
k=1 V
Z (()'
k=1 V
Z f ()'
fc =1 i \d
--(tj) , j = 1,2,3,...,n
~-(tj) , j = 1,2,3,...,n
--(tj) , j = 1,2,3,...,n
/ \m
--(tj) , k = 1,2,3,...,o
--(tj) , k = 1,2,3,...,o
--(tj) , k = 1,2,3,...,o
(A3)
(A3.0)
(yjk) - (yjk) ^ oj=12'3' •••'n;k=12'3' •••'0
(A3.1)
(A3.2)
(A3.3)
(A3.4)
(A3.5)
(A3.6) (A3.7)
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71) TRANSPORTATION PROBLEMS_Volume 17, December 2022
(Vjk)- (yjk) ^ o i = 1,2,3,»vn k = h2,3,...,o (A3.g)
(yjk J" > 0, j = 1,2,3,..., n; k = 1,2,3,..., o (A3.9)
Also, for the Eq.A2 equilibrium inhibitors, they use Forth Definition. In the next section, new ways to overcome these weaknesses are suggested.
4. Newly Developed Approach It is useful to point out that we can think of Eq.A3.0 as LPP has several objectives having OF:
t t ((j T ()" ) t t ((d, )d (y k )d )i t ((dk )■ (y„ )v
j=1 k=1 ^ J j=1 k=1 ^ J j=1 k=1 ^
based on the operating space of Eq.A.3.0. Also, blockchain (Eq.A3.7), (Eq.A3.8) and (Eq.A3.9) in Eq.A3.0 (and Eq.A2 and Eq.A3) are guaranteed only in the system d exploitation must have been a negative triangular FN. This means that without these constraints, the potential gap of Eq.A3.0 is separated by a series of changes (yk) m, (yk) d and (yk) v. Therefore, we remove these barriers from
the scope of the A3.0 scale and solve the underlying problem that makes these barriers satisfactory. This confirms that the OS was obtained as a negative FN triangular. Regarding the above discussion, we first find out the operating system of this net problem:
n o /
mmZZ[(djk)m (yjk J
j=l k=1 [
Subject to Constraints:
£(( yjk T\=(tj T' j=1,2,3.....n
j=1 V 0
i ( (yjk )m 0=(tj T, k=1,2,3.....°
k=1 ^ 0
(yjk) > 0,j = 1,2,3,..., n; k = 1,2,3,..., o
(A4) (A4.1) (A4.2) (A4.3)
As we have seen, the problem (Eq.A4) is pure TP that can be solved using standard travel simplex algorithm. The OS of this issue is a left-hander of the non-essential OS of Eq.A4.2. Note that the Eq.A4.3 bandwidth guarantees an unobtrusive OS of the Eq.A2 as a negative FN. Now assuming that (y) m = (yk) on * 1 is the OS of the problem (Eq.A4), we solve the following problem to enter
the OS of Eq. A2:
minjr [(j )d (y.k f | (A5)
Subject to Constraints:
j=1 k=1
Z(yjk)d =(tj)d,j =1'2'3,...,n ( )
j=1
\d i \d (A5.2)
L(yjk) =Vk) ' k = 1,2,3,...,0
k=1
(yjk) ^(yjk) ' j =1,23'•••'n k =123'•••'0 (A5.3)
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71)
TRANSPORTATION PROBLEMS_Volume 17, December 2022
Note that the barrier (Eq.A5.3) ensures that the fuzzy OS of the Eq.A2 is not lower than its left. Furthermore, it is clear that Eq.A5 is a limited TP that can be corrected using the method provided
by [24]. Finally, assuming that (y)d = (y^k) d on * 1 are the OS of the problem (Eq.A5), we solve the following problem to get the exact point of the fuzzy OS of the Eq .A2:
minjr f (djk J (yjk J j ( )
j=1 k=1 ^ s
Subject to Constraints:
V ( Y (t Y ■ 1 2 3 (A6-1)
Z (yjk ) = (fj ) 'J = 1'2'3'...'n
j=1
V ( V U Y k 1 2 3 (A6.2)
k=1
(yjk) ^ (yjk)' i = 1,2,3,...,k = l2,3,...,o (A6.3)
As we have seen, this problem is also limited TP can be solved by using the method given in [8]. Moreover, the barrier (Eq.A6.3) supports that the correct position of the fuzzy OS of Eq.A2 is greater than or equal to its center. Therefore, the OS of the problem of Eq.A4, Eq.A5 and Eq.A6 ensure that the base OS of Eq.A2 is a triangular FN that is not negative. In summary, if (y) m, (y) d and (y) v are SO of network problems Eq.A4, Eq.A5 and Eq.A6, respectively, then y = (y) m, (y) d, (y) v would be the base OS of Eq.A2. Finally, the optimal value of the Eq.A2 problem is obtained by adding y and d x y as follows:
d * y = i £ [ (djk )m, (djk )d, (djk J (yjk )m, (yjk )d, (yjk)
j=1 k=1V 0 V
= £ t (Nm (yjk)m ,(djk)d (Vjk)d, (dk)) (yjkj
j=l k=1 (
5. Merits of Newly Proposed Method
In this section, the benefits of an expected FTP processing method are defined.
□ Not only will the required system to implement FTP and TC representing FN be implemented in the principle of donations and requests as existing numbers, but it can also be used to configure FTP on all FN-enabled devices.
□ The OS looks like a negative FN, that is, there are no negative components either in the FQ product or in the non-core TC.
□ Using different methods such as northwest corner system, minimum price system and Vogel fuzzy approximation system to find the IBFS of the problem (Eq.A4), (Eq.A5) and (Eq.A6) lead to the same total CT.
□ The main advantage of the proposed method is that solving the Eq.A2 problem (Eq.A4), (Eq.A5) and (Eq.A6) is relatively large compared to the problem (Eq.A3. 0) by a. mathematical considerations, regarding the number of barriers to change. There is a direct relationship between the conventional complexity of LPP and the number of barriers to change. In particular, since the memory size required to place the backup in the simplex algorithm is square of the number of constraints, reducing the number of constraints of the LP type is very important from a mathematical point of view. Thus, the reduction in the number of inhibitions and the variability of the LP type leads to a reduction in the complexity of the LP type modified by
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71) TRANSPORTATION PROBLEMS_Volume 17, December 2022
simplex algorithms and key methods such as Khachian's ellipsoid algorithm and projective
algorithms.
Compare the resistance and variability of the problem (Eq.A4), (Eq.A5) and (Eq.A6) the problem (Eq.A3.0). Problem (Eq.A4) has no barrier (n + o) and no change, while problem (Eq.A3.0) has 3 (n + o) +2 no barrier and 3 no change. This indicates that the problem (Eq.A4) has 2 (n + o + no) barriers and 2 less change than the problem (Eq.A3.0), so the use of the problem (Eq.A4) is strong the economy in terms of the problem (Eq. .A3.0) is based on mathematical concepts. There is a similar comparison between the resistance and exchange rate (Eq.A5) and (Eq.A6) and (Eq.A3.0). It is noteworthy that the constraints (Eq.A5.3) and (Eq.A6.3) agree that the variance (yk )d and (yk )v are limited. The simplex bounded system also uses these blockchains in the same way as the simplex bounded block (yk )m>0. This means that these barriers do not increase the number of barriers immediately. Therefore, regarding the above discussion, we recommend using problem (Eq.A4), (Eq.A5), and (Eq.A6) instead of problem (Eq.A3.0) to solve Eq.A2 from a mathematical point of view.
6. Applications of Newly Proposed Method
In this section, the model application is analyzed using the recommended method and evaluating the obtained results.
First Example: One company has two sources P1 and P2, as well as three E1, E2 and E3 sources; TC fuzzy for quantity of products from jth source to where kth is djk where,
djk = [(20,30,40)(60,70,90)(90,100,110)(70,80,90)(85,95,115)(35,45,55)]
The FS of the product in the first and second stages are (80, 100, 130) and (50, 70, 100), respectively. FD of the product in the first place, second and third are (40, 50, 60), (30, 40, 50) and (70, 90, 120), respectively. The company wants to determine the FQ of the product that needs to be shipped from any start to anywhere so that the total TC value is minimal. This problem can be solved with the following FTP:
min (20,30,40) y11 x (60,70,90) y12 x (90,100,110) y13 x (70,80,90) y21 x (90,100,120) y22 x (40,50,60) y23 Subject to Constraints: y11 x y12 x y13 = (80,100,130), y21 x y22 x y23 = (50,70,100), y11 x y21 = (40,50,70),
Vl2 X y22 =(30,40,50), yi3 X y23 =(65,85,115), yU, Vl2' Vl3' V21' V22' VT3 eTF (R). ^
Total FS = (125, 165, 225) = total FD so FTP is appropriate. Regarding the problem (Eq.A6), this problem can be translated into the following FTP:
Min 20 (yn) m + 60 (y12) m + 90 (y13) m + 70 (y21) m + 85 (y22) m + 35 (y23) m, 30 (y11) d + 70 (y12) d + 100 (y13) d + 80 (y21) d + 95 (y22) d + 45 (y23) d, 40 (y 11)v + 90 (yl2)v + 11° (yl3) v + 9° (y2l)v +II5 (y22) v + 55 (y23)v
Subject to Constraints:
(Vii)m + (yi2)m + (yi3)m = 80, (yii)d + (yi2)d + (yi3)d =100' (yii)v + (yi2)v + (yi3)v =130,
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AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71) TRANSPORTATION PROBLEMS_Volume 17, December 2022
(V21) m + (V22 ) m + (V23 ) m = 50, (y21) d + (y22 ) d + (V23 ) d = 70, (V21) V + (^22 ) V + (V23 ) V = 100,
(Vi1 ) m + (y21) m = 40, (yil) d + (y21) d = 50, (yil ) V + (y21) V = 70, (yi2 ) m + (y22 ) m = 30,
(yi2 ) d + (V22 ) d = 40, (yi2 ) V + (y22 ) V = 50, (yi3 ) m + (y23 ) m = 65, (Vi3 ) d + (V23 ) d = 85,
(yi3)v + (y23) V = 115,
(Vjk)d - (Vjk) m > 0 (yjk) v - (Vjk)d > 0 (Vjk) m > 0 j =1 2k =1 2 3
In this case, regarding the problem (Eq.A4), we will first fix the following TP net to get the left side of Eq.A7:
Min 20 (y11) m + 60 (y12) m + 90 (y13) m + 70 (y21) m + 85 (y22) m + 35 (y23) m, Subject to Constraints:
(yn ) m + (yi2 ) m + (yi3 ) m = 80, (y21) m + (y22 ) m + (y23 ) m = 50, (yil) m + (y21) m = 40,
(A9
(Vi2 ) m + (y22 ) m = 30, (yi3 ) m + (y23 ) m = 65, (yil) m' (yi2 ) m' (yi3 ) m' (y21) (Vi! ) (VI3 ) m ^ 0 )
Fix TP (Eq.A9) using the standard transportation simplex algorithm that delivers the best OS and OV:
(y11)m = 40, (y12) m = 30, (y13) m = 20, (y21) m = 5, (y22) m = 5, (y23) m = 50, (d xy)m = 5030. (A10)
Now based on the OS (Eq.A10) and related to the problem (Eq.A5), we are setting up a limited LPP to access the free OS space of FTP (Eq.A7):
Min 30 (y11) d + 70 (y12) d + 100 (y13) d + 80 (y21) d + 95 (y22) d + 45 (y23) d
Subject to Constraints:
(yn ) d + (yi2 ) d + (yi3 ) d = 100, (y21) d + (y22 ) d + (y23 ) d = 70, (yil ) d + (y21) d = 50, (yi2 ) d + (V22 ) d = 40, (yi3 ) d + (y23 ) d = 85,
(y11) d > 40, (y12) d > 30, (y13) d > 20, (y21)d > 0, (y22) d > 0, (y23 ) d > 50.
Classical TP can be corrected using the method provided above. OS and OV (Eq.A11) problems are found as follows:
(y11) d = 50, (y12) d = 40, (y13) d = 20, (y21) d = 5, (y22) d = 5, (y23) d = 70,(d x y)d = 7930. (A12)
Also, regarding OS (Eq.A12) and Troubleshooting (Eq.A6), we solve the following bounded LPP to find the true fuzzy OS core of Eq.A7:
Min 40 (y11) v + 90 (y12) v + 130 (y13) v + 90 (y21) v + 115 (y22) v + 55 (y23) v Subject to Constraints: (Vl1) V + (yi2 ) V + (yi3 ) V = 130, (y21) v + (y22 ) v + (y23 ) v = 100'
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AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71) TRANSPORTATION PROBLEMS_Volume 17, December 2022
(Vl1) v + (V21) V = 70, (yi2 ) v + (y22 ) V = 50, (yi3 ) V + (y23 ) V = 115, (A13)
(yu ) V > 50, (yi2 ) v > 40, (yi3 ) V > 2° (y2i ) V > 5, (y22 ) v > 5, (y^ ) v > 70.
OS and OV (Eq.A11) problems are found as follows:
(y11) v = 70, (y12) v = 50, (y13) v = 20, (y21) v = 5, (y22) v = 5, (y23) v = 100, (d x y )v = 12930. (A14)
Finally, based on OS (Eq.A10), (Eq.A12) and (Eq.A14), the OS and OV of Eq.A7 are offered as follows:
y = Vu =( yii)(yii)d' (yii)v = (40, 50, 70); y = yn =( yi2)(yi2)d' (yi2)v = (30, 40, 50) y = yi3 =( yi3) m,( yi3) d,( yi3)v = (20, 20, 20); y = yn =( y2i) m,( yn) d,( yn)v = (5' 5' 5)
y = y22 =(y22 ) (y22 ) d' (y22 ) v = (5' 5' 5) ^ H = V23 =(H23 ) m> (H23 ) d' (H23 ) V = (50, 70, 100)
dx y = x yjk = (5030,7930,12930) (A15)
j=1 k=1
The minimum TC minimum can be converted to the default OS. Using the required method, the minimum nonlinear TC is w = (5030,7930,12930) which can be translated as follows:
□ The minimum TC total is 5030 minutes.
□ The maximum possible TC is 7930 minutes.
□ The minimum travel time is 12930 minutes.
This result indicates that the minimum TC minimum will be more than 5030 minutes and less than 12930 minutes and the maximum chance is that the minimum TC total is 7930 minutes. It can be seen that there is no negative aspect of the OS that fuzzy got in the fuzzy TC collection, as you consider the existing system, if we apply the general distribution system fuzzy switch to find the OS without issue. An Eq.A6 with the help of IFBFS, obtained based on generalized fuzzy northwest corner system, then we get the following fuzzy OS and there is a negative part in the FQ of the product y which should be changed by originally from third place and therefore no physical meaning.
y = yn =( yu)(yu)d' (yu)v = (40, 50, 70); y = yu =( yi2)(yi2)d' (yu)v = (30, 40, 50) y = yi3 =( yi3) m,( yi3) d,( yi3)v = (20, 20, 20); y = yii =( y2i) m,( yn) d,( yn)v = (5' 5' 5) ) y = y22 =( y22 ) m'{ y22 ) d,( y22 ) V = (5' 5' 5)
Also, the levels y13 and fuzzy OS (Eq.A15) and (Eq.A16) are equal, i.e. R (-40, 20, 70) = R (20, 20, 20)
= 20, where obviously. (-40, 20, 70) * (20, 20, 20) and this indicates another setback of the method, which used the degree function to adjust the FTP. Finally, to get the base OS of Eq.A7 based on the problem (Eq.A3.0), we will fix the following LPPs as follows:
Min ( 20 ( y11) m + 55 ( y11) d + 40 ( y11) v^4 + (60 (y12 ) m +135 (y12 ) d + 90 ( y12 ) v^4 + (90 ( yi3 ) m +195 (y ) d + 110 ( ) v ^4 + (70 ( y^ ) m +155 (y2X) d + 90 (y^ ) v )/4 (A17)
+ (85 ( y22 ) m +185 ( y22 ) d + 115 (y22 ) v)/4 + (35 (y23 ) m + 85 ( y23 ) d + 55 ( y23 ) v)/4
Subject to Constraints of Problem (Eq.A8).
The classic LPP provides an unparalleled OS of Eq.A7 that is compatible with the rare OS (Eq.A15) obtained on the basis of the system we recommended. However, there are two main reasons why
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71)
TRANSPORTATION PROBLEMS_Volume 17, December 2022
we consider the proposed method as follows:
□ The classical LPP (Eq.A17) applied to fix FTP (Eq.A7) is not a modified LPP, while the problem (Eq.A9), (Eq.A11) and (Eq.A13) fix FTP (Eq. .A7) as classic TPs. To fix TPs, a tabular method is preferred over the LP method [11] so it may be recommended to use the expected method instead of the current method to fix FTPs.
□ The primitive LPP (Eq.A17) applied to fix FTP (Eq.A7) has 27 obstacles and 18 variables, while the problem (Eq.A9) has 5 obstacles and 6 variables. There is a similar comparison between rates of prevention and change problems (Eq.A11) and (Eq.A13) and (Eq.A17). Therefore, using the problem (Eq.A9), (Eq.A11) and (Eq.A13) to fix FTP (Eq.A7) is a big deal compared to the problem (Eq.A17) based on the assumption of math. , about the number of restrictions and changes. In summary, it is better to use the methods we have recommended than the existing methods to resolve FTP from a conventional view.
7. Conclusion
A large number of TPs at different levels of sophistication have been documented in the paper. However, some of these models have a small global application because common TPs take net data for TC, requesting and query values. Unlike conventional TPs, we analyzed inaccurate data on real TP and developed a simpler method and corrected these weaknesses in the form described. In the FTP discussed in this study, non-negative triangular FNs represent all aspects of the problem. In this article, we are trying to create some important LPPs to fix FTP, which have fewer restrictions and changes compared to other existing LPPs. In particular, the proposed method can be useful in large-scale applications. Since the proposed system is based on the ancient travel simplex algorithm, it is easy to learn and apply the required system to get a real-time FTP operating system in the application world. One of the advantages of the recommended methods is that the fuzzy OS acquired and the best advantage are the negative triangular FNs. Finally, we believe that there are many more studies that need to be further explored. Some of these points are discussed below:
□ The recommended method works well in fixing FTP because all invalid parameters are represented as triangular FNs. The consolidation of this system to find the free OS of TP and trapezoidal FN will be an interesting research project in the future.
□ STP assesses supply, demand and transportation to meet transportation needs effectively. Therefore, research on the topic to create the process required to obtain non-trivial OV from nonlinear STP when price, supply and required quantities in non-triangular FNs travel capacity are negative, as leave it at another checkpoint.
□
References
[1] Agatz, N., Bouman, P., & Schmidt, M. (2018). Optimization approaches for the traveling salesman problem with drone. Transportation Science, 52(4), 965-981.
[2] Ahmadvand, S., & Pishvaee, M. S. (2018). An efficient method for kidney allocation problem: a credibility-based fuzzy common weights data envelopment analysis approach. Health care management science, 21(4), 587-603.
[3] Babaee Tirkolaee, E., Mahdavi, I., Seyyed Esfahani, M. M., & Weber, G.-W. (2020). A hybrid augmented ant colony optimization for the multi-trip capacitated arc routing problem under fuzzy demands for urban solid waste management. Waste Management & Research, 38(2), 156-172.
[4] Baykasoglu, A., & Subulan, K. (2017). Constrained fuzzy arithmetic approach to fuzzy transportation problems with fuzzy decision variables. Expert Systems with Applications, 81, 193-222.
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71) TRANSPORTATION PROBLEMS_Volume 17, December 2022
[5] Bharati, S. K., & Singh, S. (2018). Transportation problem under interval-valued intuitionistic fuzzy environment. International Journal of Fuzzy Systems, 20(5), 1511-1522.
[6] Castillo, O., & Amador-Angulo, L. (2018). A generalized type-2 fuzzy logic approach for dynamic parameter adaptation in bee colony optimization applied to fuzzy controller design. Information Sciences, 460, 476-496.
[7] Chen, D. (2017). Research on traffic flow prediction in the big data environment based on the improved RBF neural network. IEEE Transactions on Industrial Informatics, 13(4), 2000-2008.
[8] Chen, J., Gui, P., Ding, T., Na, S., & Zhou, Y. (2019). Optimization of transportation routing problem for fresh food by improved ant colony algorithm based on tabu search. Sustainability, 11(23), 6584.
[9] Christi, M. (2017). Solutions of fuzzy transportation problem using best candidates method and different ranking techniques. International Journal of Mathematical and Computational Sciences, 11(4), 182-187.
[10] Dahooie, J. H., Zavadskas, E. K., Firoozfar, H., Vanaki, A., Mohammadi, N., & Brauers, W. K. M. (2019). An improved fuzzy MULTIMOORA approach for multi-criteria decision making based on objective weighting method (CCSD) and its application to technological forecasting method selection. Engineering Applications of Artificial Intelligence, 79, 114-128.
[11] Darbari, J. D., Kannan, D., Agarwal, V., & Jha, P. (2019). Fuzzy criteria programming approach for optimising the TBL performance of closed loop supply chain network design problem. Annals of operations research, 273(1), 693-738.
[12] Das, S. K., & Roy, S. K. (2019). Effect of variable carbon emission in a multi-objective transportation-p-facility location problem under neutrosophic environment. Computers & Industrial Engineering, 132, 311-324.
[13] Dell'Orco, M., Marinelli, M., & Altieri, M. G. (2017). Solving the gate assignment problem through the fuzzy bee colony optimization. Transportation Research Part C: Emerging Technologies, 80, 424-438.
[14] Deng, W., Xu, J., Zhao, H., & Song, Y. (2020). A novel gate resource allocation method using improved PSO-based QEA. IEEE Transactions on Intelligent Transportation Systems.23, 1737 - 1745.
[15] Deveci, M., Canitez, F., & Gokagar, I. (2018). WASPAS and TOPSIS based interval type-2 fuzzy MCDM method for a selection of a car sharing station. Sustainable cities and society, 41, 777791.
[16] Deveci, M., Pamucar, D., & Gokasar, I. (2021). Fuzzy Power Heronian function based CoCoSo method for the advantage prioritization of autonomous vehicles in real-time traffic management. Sustainable cities and society. 69, 102846.
[17] Dozic, S., Lutovac, T., & Kalic, M. (2018). Fuzzy AHP approach to passenger aircraft type selection. Journal of Air Transport Management, 68, 165-175.
[18] Du, J., Li, X., Yu, L., Dan, R., & Zhou, J. (2017). Multi-depot vehicle routing problem for hazardous materials transportation: A fuzzy bilevel programming. Information Sciences, 399, 201-218.
[19] Ebrahimnejad, A., & Verdegay, J. L. (2018). A new approach for solving fully intuitionistic fuzzy transportation problems. Fuzzy optimization and decision making, 17(4), 447-474.
[20] Emeg, §., & Akkaya, G. (2018). Stochastic AHP and fuzzy VIKOR approach for warehouse location selection problem. Journal of Enterprise Information Management, 31(6), 950-962.
[21] Fakhrzad, M. B., Goodarzian, F., & Golmohammadi, A. (2019). Addressing a fixed charge transportation problem with multi-route and different capacities by novel hybrid meta-heuristics. Journal of Industrial and Systems Engineering, 12(1), 167-184.
[22] Gao, D., Wang, G.-G., & Pedrycz, W. (2020). Solving fuzzy job-shop scheduling problem using DE algorithm improved by a selection mechanism. IEEE Transactions on Fuzzy Systems, 28(12), 3265-3275.
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71) TRANSPORTATION PROBLEMS_Volume 17, December 2022
[23] Goli, A., Tirkolaee, E. B., & Aydin, N. S. (2021). Fuzzy integrated cell formation and production scheduling considering automated guided vehicles and human factors. IEEE Transactions on Fuzzy Systems, 29(12), 3686-3695.
[24] Huang, W., Liu, Y., Zhang, Y., Zhang, R., Xu, M., De Dieu, G. J., . . . Shuai, B. (2020). Fault Tree and Fuzzy DS Evidential Reasoning combined approach: An application in railway dangerous goods transportation system accident analysis. Information Sciences, 520, 117-129.
[25] Juman, Z., & Nawarathne, N. (2019). An efficient alternative approach to solve a transportation problem. Ceylon journal of Science, 48(1), 19-29.
[26] Kumar, R., Edalatpanah, S., Jha, S., & Singh, R. (2019). A Pythagorean fuzzy approach to the transportation problem. Complex & intelligent systems, 5(2), 255-263.
[27] Li, J.-q., Deng, J.-w., Li, C.-y., Han, Y.-y., Tian, J., Zhang, B., & Wang, C.-g. (2020). An improved Jaya algorithm for solving the flexible job shop scheduling problem with transportation and setup times. Knowledge-Based Systems, 200, 106032.
[28] Li, J.-q., Liu, Z.-m., Li, C., & Zheng, Z.-x. (2020). Improved artificial immune system algorithm for type-2 fuzzy flexible job shop scheduling problem. IEEE Transactions on Fuzzy Systems, 29(11), 3234-3248.
[29] Li, Y., Soleimani, H., & Zohal, M. (2019). An improved ant colony optimization algorithm for the multi-depot green vehicle routing problem with multiple objectives. Journal of cleaner production, 227, 1161-1172.
[30] Ma, C. (2018). Network optimisation design of hazmat based on multi-objective genetic algorithm under the uncertain environment. International Journal of Bio-Inspired Computation, 12(4), 236-244.
[31] Ma, C., Hao, W., Pan, F., & Xiang, W. (2018). Road screening and distribution route multi-objective robust optimization for hazardous materials based on neural network and genetic algorithm. PLoS One, 13(6), e0198931.
[32] Mahajan, S., & Gupta, S. (2021). On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions. Annals of operations research, 296(1), 211-241.
[33] Maity, G., Mardanya, D., Roy, S. K., & Weber, G.-W. (2019). A new approach for solving dual-hesitant fuzzy transportation problem with restrictions. Sadhana, 44(4), 1-11.
[34] Mishra, A. R., Rani, P., & Pandey, K. (2022). Fermatean fuzzy CRITIC-EDAS approach for the selection of sustainable third-party reverse logistics providers using improved generalized score function. Journal of Ambient Intelligence and Humanized Computing,13, 295-311.
[35] Mitlif, R. J., Rasheed, M., & Shihab, S. (2020). An Optimal Algorithm for a Fuzzy Transportation Problem. Journal of Southwest Jiaotong University, 55(3).
[36] Mohammed, M. A., Abd Ghani, M. K., Hamed, R. I., Mostafa, S. A., Ahmad, M. S., & Ibrahim, D. A. (2017). Solving vehicle routing problem by using improved genetic algorithm for optimal solution. Journal of computational science, 21, 255-262.
[37] Mohammed, M. A., Abd Ghani, M. K., Hamed, R. I., Mostafa, S. A., Ibrahim, D. A., Jameel, H. K., & Alallah, A. H. (2017). Solving vehicle routing problem by using improved K-nearest neighbor algorithm for best solution. Journal of computational science, 21, 232-240.
[38] Niroomand, S., Mahmoodirad, A., Heydari, A., Kardani, F., & Hadi-Vencheh, A. (2017). An extension principle based solution approach for shortest path problem with fuzzy arc lengths. i, 17(2), 395-411.
[39] Pratihar, J., Kumar, R., Edalatpanah, S., & Dey, A. (2021). Modified Vogel's approximation method for transportation problem under uncertain environment. Complex & intelligent systems, 7(1), 29-40.
[40] Qiao, W., & Yang, Z. (2019). An improved dolphin swarm algorithm based on Kernel Fuzzy C-means in the application of solving the optimal problems of large-scale function. Ieee Access, 8, 2073-2089.
Kaushik A. Joshi, Kirankumar L. Bondar, Pandit U.Chopade
AN UPGRADED APPROACH TO SOLVE FUZZY RT&A, No 4 (71) TRANSPORTATION PROBLEMS_Volume 17, December 2022
[41] Rabbani, M., Hosseini-Mokhallesun, S. A. A., Ordibazar, A. H., & Farrokhi-Asl, H. (2020). A hybrid robust possibilistic approach for a sustainable supply chain location-allocation network design. International Journal of Systems Science: Operations & Logistics, 7(1), 60-75.
[42] RASHEED, M., SHIHAB, S., RASHID, T., & Ali, Y. H. (2021). Experimental Results for a Nonlinear Equation Using Improved Newton-Raphson Estimation Method. Journal of Al-Qadisiyah for Computer Science and Mathematics, 13(1), 193-200.
[43] Roy, S. K., Ebrahimnejad, A., Verdegay, J. L., & Das, S. (2018). New approach for solving intuitionistic fuzzy multi-objective transportation problem. Sadhana, 43(1), 1-12.
[44] Sadeghi-Moghaddam, S., Hajiaghaei-Keshteli, M., & Mahmoodjanloo, M. (2019). New approaches in metaheuristics to solve the fixed charge transportation problem in a fuzzy environment. Neural computing and applications, 31(1), 477-497.
[45] Sanei, M., Mahmoodirad, A., Niroomand, S., Jamalian, A., & Gelareh, S. (2017). Step fixed-charge solid transportation problem: a Lagrangian relaxation heuristic approach. Computational and Applied Mathematics, 36(3), 1217-1237.
[46] Sun, Y., Hrusovsky, M., Zhang, C., & Lang, M. (2018). A time-dependent fuzzy programming approach for the green multimodal routing problem with rail service capacity uncertainty and road traffic congestion. Complexity, 2018.
[47] Tian, G., Hao, N., Zhou, M., Pedrycz, W., Zhang, C., Ma, F., & Li, Z. (2019). Fuzzy grey choquet integral for evaluation of multicriteria decision making problems with interactive and qualitative indices. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51(3), 1855-1868.
[48] Tian, Z.-P., Zhang, H.-Y., Wang, J.-Q., & Wang, T.-L. (2018). Green supplier selection using improved TOPSIS and best-worst method under intuitionistic fuzzy environment. Informatica, 29(4), 773-800.
[49] Türk, S., Deveci, M., Özcan, E., Canitez, F., & John, R. (2021). Interval type-2 fuzzy sets improved by Simulated Annealing for locating the electric charging stations. Information Sciences, 547, 641-666.
[50] Wu, T., Liu, X., & Liu, F. (2018). An interval type-2 fuzzy TOPSIS model for large scale group decision making problems with social network information. Information Sciences, 432, 392-410.
[51] Xiao, L., Dridi, M., Hajjam El Hassani, A., Fei, H., & Lin, W. (2018). An improved cuckoo search for a patient transportation problem with consideration of reducing transport emissions. Sustainability, 10(3), 793.
[52] Yu, K., Lin, L., Alazab, M., Tan, L., & Gu, B. (2020). Deep learning-based traffic safety solution for a mixture of autonomous and manual vehicles in a 5G-enabled intelligent transportation system. IEEE Transactions on Intelligent Transportation Systems, 22(7), 4337-4347.
[53] Zhang, G., Hu, Y., Sun, J., & Zhang, W. (2020). An improved genetic algorithm for the flexible job shop scheduling problem with multiple time constraints. Swarm and Evolutionary Computation, 54, 100664.
