A note on “Transportation problem under interval-valued intuitionistic fuzzy environment”

1 Introduction

Bharti and Singh [1, Sec. 4] proposed a method for solving such IVIF-TPs in which the availability of the product at each source, the demand at each destination and the cost for transporting one unit of product the product from a source to a destination are represented as IVTIFNs. Whereas, the quantity of the product that should be supplied from a source to a destination is represented as a real number. In this method [1, Sec. 4], firstly, the considered interval-valued triangular intuitionistic fuzzy transportation problem (IVIF-TP) is represented in tabular form. Then, using a function (named as expected value), each parameter of the considered IVIF-TP, represented by an interval-valued triangular intuitionistic fuzzy number (IVTIFN), is transformed into a real number. Using this transformation, the considered IVIF-TP is transformed into a crisp transportation problem (TP). Finally, the transformed crisp TP is solved to find the optimal solution of the considered IVIF-TP and hence, the optimal IVIF transportation cost of the considered IVIF-TP.

In this note, an IVIF-TP is solved by Bharti and Singh’s method [1, Sec. 4] and shown that more than one IVTIFNs, representing the optimal IVIF transportation cost are obtained, which is mathematically incorrect as the obtained distinct IVTIFNs has different physical meanings. Also, it is pointed out that to resolve this flaw of Bharti and Singh’s method [1, Sec. 4] may be considered as a challenging open research problem.

2 Bharti and Singh’s method

To show that on solving an IVIF-TP by Bharti and Singh’s method [1, Sec. 4] more than one IVTIFNs, representing the optimal IVIF transportation cost, may be obtained, there is need to discuss Bharti and Singh’s method [1, Sec. 4]. Therefore, the same method is discussed in this section.

Bharti and Singh [1, Sec. 4] proposed the following method for solving IVIF-TPs.

Step 1. Write the tabular representation of the considered IVIF-TP as shown in Table 1.

Where,

The IVTIFN c ˜ ij = { ( c ij , 1 U , c ij , 1 L , c ij , 2 , c ij , 3 L , c ij , 3 U ) , ( c ij , 1 ′ L , c ij , 1 ′ U , c ij , 2 , c ij , 3 ′ U , c ij , 3 ′ L ) } represents the IVIF transportation cost for transporting the unit quantity of the product from the i ^th source (S _i )to the j ^th destination (D _j ).

Step 2. Transform the IVIF-TP, represented by Table 1 into a crisp TP represented by Table 2.

where, EV ( c ˜ ij ) = c ij , 1 U + c ij , 1 L + c ij , 1 ′ L + c ij , 1 ′ U + 8 c ij , 2 + c ij , 3 L + c ij , 3 U + c ij , 3 ′ U + c ij , 3 ′ L 16 , EV ( a ˜ ij ) = a i , 1 U + a i , 1 L + a i , 1 ′ L + a i , 1 ′ U + 8 a i , 2 + a i , 3 L + a i , 3 U + a i , 3 ′ U + ac i , 3 ′ L 16 , EV ( b ˜ j ) = b j , 1 U + b j , 1 L + b j , 1 ′ L + b j , 1 ′ U + 8 b j , 2 + b j , 3 L + b j , 3 U + b j , 3 ′ U + b j , 3 ′ L 16 .

Step 3. Find the optimal solution {x _ij }of the crisp TP represented by Table 2.

Step 4. Using the optimal solution {x _ij }, obtained in Step 3, find the IVIF transportation cost ∑ i = 1 m ∑ j = 1 n { ( c ij , 1 U x ij , c ij , 1 L x ij , c ij , 2 x ij , c ij , 3 L x ij , c ij , 3 U x ij ) , ( c ij , 1 ′ L x ij , c ij , 1 ′ U x ij , c ij , 2 x ij , c ij , 3 ′ U x ij , c ij , 3 ′ L x ij ) }

In this section, the IVIF-TP, represented by Table 3 is solved by Bharti and Singh’s method [1, Sec. 4] and shown that two distinct IVTIFNs ( { ( 1680 , 2280 , 3120 , 3480 , 4080 ) , ( 480 , 1080 , 3120 , 4680 , 5280 ) } ) and { ( 1590 , 2490 , 3060 , 3600 , 4080 ) ( 780 , 1200 , 3060 , 4440 , 5430 ) } , representing the optimal IVIF transportation cost are obtained, which is mathematically incorrect as the physical meaning of these IVTIFNs are different.

Using Bharti and Singh’s method [1, Sec. 4], discussed in Section 2, the optimal solution and optimal IVIF transportation cost of the IVIF-TP, represented by Table 3, can be obtained as follows.

Step 1. Using Step 1 of Bharti and Singh’s method [1, Sec. 4], the IVIF-TP, represented by Table 3, can be transformed into a crisp TP represented by Table 4.

Step 2. On solving the crisp TP, represented by Table 4, the following two optimal basic feasible solutions are obtained x₁₁ = 30, x₁₂ = 0, x₂₁ = 0, x₂₂ = 30and x₁₁ = 0, x₁₂ = 30, x₂₁ = 30, x₂₂ = 0.

Step 3. Using Step 3 of Bharti and Singh’s method [1, Sec. 4]

The IVTIFN, representing the optimal IVIF transportation cost, corresponding to first optimal basic feasible solution is { ( 1680 , 2280 , 3120 , 3480 , 4080 ) , ( 480 , 1080 , 3120 , 4680 , 5280 ) } .

The IVTIFN, representing the optimal IVIF transportation cost, corresponding to second optimal basic feasible solution is { ( 1590 , 2490 , 3060 , 3600 , 4080 ) ( 780 , 1200 , 3060 , 4440 , 5430 ) } .

Hence, two distinct IVTIFNs { ( 1680 , 2280 , 3120 , 3480 , 4080 ) , ( 480 , 1080 , 3120 , 4680 , 5280 ) } and { ( 1590 , 2490 , 3060 , 3600 , 4080 ) ( 780 , 1200 , 3060 , 4440 , 5430 ) } , representing the optimal IVIF transportation cost of the IVIF-TP represented by Table 3, which is mathematically incorrect as physical meaning of these TIFNs are different.

Remark. It is pertinent to mention that Bharti and Singh [1, Sec. 4] have claimed that if A ˜ 1 = { ( a 1 U , a 1 L , a 2 , a 3 L , a 3 U ) , ( b 1 L , b 1 U , a 2 , b 3 U , b 3 L ) } and A ˜ 2 = { ( c 1 U , c 1 L , c 2 , c 3 L , c 3 U ) , ( d 1 L , d 1 U , c 2 , d 3 U , d 3 L ) } are two IVTIFNs then A ˜ 1 + A ˜ 2 = { ( a 1 U + c 1 U , a 1 L + c 1 L , 2 a 2 , a 3 L + c 3 L , a 3 U + c 3 U ) , ( b 1 L + d 1 L , b 1 U + d 1 U , 2 a 2 , b 3 U + d 3 U , b 3 L + d 3 L ) } . Whereas, in actual case, A ˜ 1 + A ˜ 2 = { ( a 1 U + c 1 U , a 1 L + c 1 L , a 2 + c 2 , a 3 L + c 3 L , a 3 U + c 3 U ) , ( b 1 L + d 1 L , b 1 U + d 1 U , a 2 + c 2 , b 3 U + d 3 U , b 3 L + d 3 L ) }

It is shown that on solving an IVIF-TP by the existing method [1, Sec. 4] more than one IVTIFN, representing the optimal IVIF transportation cost, may be obtained, which is mathematically incorrect. Therefore, it is inappropriate to use Bharti and Singh’s method [1, Sec. 4], for solving real life IVIF-TPs. To resolve this flaw of Bharti and Singh’s method [1, Sec. 4] may be considered as a challenging open research problem