selection.

In the literature, Vendor Selection Problem (VSP) was studied by Dickson [1]. First, Zadeh [2] proposed the philosophy of fuzzy sets. Fuzzy decision-making was developed by Zimmermann [3], where they introduced fuzzy programming and LPP with multiple objective functions. Later, several researchers worked on various applications based on fuzzy uncertainty. Dubois and Prade [4] investigated the theory and applications of fuzzy uncertainty. Kaufmann and Gupta [5] studied several fuzzy mathematical models with their applications to engineering and management sciences. Maleki et al. [6] proposed a very effective method to solve a LPP including the fuzzy variables and the comparison of fuzzy numbers.

Many authors [7]–[9], studied the problems where all the parameters are uncertain. Lai and Hwang [10] assumed that the parameters might be characterized by triangular possibility distribution; then, they proposed an auxiliary model, which was solved by applying the multi-objective LPP.

Weber and Current [11] studied the multiple objective methods for solving the VSP. Several authors used fuzzy multi-objective approaches to VSP, like [12]–[15], studied the manufacturing delivery performance problem with an application to supply chain management, further investigated by Carvalho and Costa [16]. Several researchers studied the modified S-curve membership function in VSP, for example, Vasant, M. [17], Madroñero [18], Torabi and Hassini [19] presented a novel method based on possibilistic programming to solve a multi-objective supply chain model. Selim and Ozkarahan [20] addressed a supply chain model integrated with network design, which an interactive fuzzy goal programming method solved. Many papers integrated fuzzy and stochastic uncertainty with VSP to determine the performance of suppliers and choose the suitable supplier are those of [21]–[24], and many more. Several papers have been published in green supplier selection in the last few decades. For instance, Jain et al. [25] studied a buyer-initiated decision-making process with the inclusion of the green supplier selection concept.

Luthra et al. [22] investigated a method to solve the sustainable VSP further extended by Chen and Zou [26]. Ehsan et al. [10] studied the green VSP using a MODM algorithm. Ghaniabadi and Mazinani [27] presented a dynamic lot size model for multiple suppliers. Turk et al. [28] studied the multi-objective model in inventory management with supplier selection. Aouadni et al. [29] presented a detailed survey on VSP as well as order allocation problems. This study investigates the approximation of close interval for piecewise quadratic fuzzy to solve the VSP. The remaining portion of the work is arranged as follows. Section 2 recalls some basic concepts needed. Section 3 introduces VSP as an LPP. Section 4 formulates the piecewise quadratic fuzzy VSP. Section 5 provides a numerical example to support and validate the proposed methodology. In the end, a few concluding remarks are reported in Section 6.

In this section, some basic terms and results are recalled.

Definition 1 ([30]). A piecewise quadratic fuzzy number (PQFN) is denoted by a ̃PQ = (a1, a2, a3, a4, a5), where a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 are reals, is defined by its membership function a ̃ as follows (Fig. 1).

PQ

0, x < a1, (x − a1) , a1 ≤ x ≤ a2, ( )2 2 a2 − a1 (x − a3) + 1, a2 ≤ x ≤ a3, ( )2 2 a3 − a2

μa ̃ = 1 1 (1)

PQ 2

(x − a3) + 1, a3 ≤ x ≤ a4, ( )2 2 a4 − a3 (x − a5) , a4 ≤ x ≤ a5, ( )2 2 a5 − a4 0, x > a5.

The interval of confidence at level α for the PQFN is defined as (a ̃PQ) = Lα(a ̃PQ) = [a1 + 2(a2 − a1)α, a5 − 2(a5 − a4)α] for all α ∈ [0,1].

L U ̃

Definition 2 ([30]). An interval approximation [A] = [aα, aα] of a PQFN A is referred to as a closed interval approximation if aα = inf{x ∈ ℜ: μ̃ ≥ 0.5}, and aα = sup{x ∈ ℜ: μ̃ ≥ 0.5}.

Definition 3 ([30]). The associated ordinary numbers of PQFN corresponding to [A] = [aα, aα ] is [A] =

L U aα+aα

[a , a ] is equal to A ̂ = .

Definition 4 ([30]). Suppose that a ̃ = (a , a , a , a , a ) and b ̃ = (b , b , b , b , b ) be two PQFNs. Then

PQ 1 2 3 4 5 PQ 1 2 3 4 5

I. Addition: a ̃ ⊕ b ̃ = (a + b , a + a , a + b , a + b , a + b ).

PQ PQ 1 1 2 2 3 3 4 4 5 5

II. Scalar multiplication: α. a ̃PQ = (α a1, α a2, α a3, α a4, α αa5), α > 0.

L U L U

Definition 5 ([30]). Let [A] = [(aα) , (aα) ], and [B] = [(bα) , (bα) ] be two interval approximations of PQFN. Then, two define the arithmetic operations as follows:

L L U U

I. Addition: [A] ⊕ [B] = [A] ⊕ [B] = [aα + bα, aα + bα].

L U U L

II. Subtraction: [A] ⊖ [B] = [aα − bα, aα − bα].

[α aα, α aα], α > 0,

III. Scalar multiplication: α [A] = {

[α aα, α aα], α < 0.

U L L U L L U U

aαbα+aαbα aαbα+aαbα

IV. Multiplication: [A] ⊙ [B] = [ , ].

2aα 2aα L U

[ , ] , [B] > 0 and bα + bα ≠ 0,

L U L U

[A] bα+ bα bα+ bα

V. Division: = {

[ ] U L B 2aα 2aα

[ , ] , [B] < 0 and bα + bα ≠ 0.

L U L U

bα+ bα bα+ bα

VI. The order relations (≈, ≅, ≲, ≳) are defined as

L L U U

I. Fuzzy equal: [A] ≈ [B] if aα = bα and aα = bα.

L U L U

II. Fuzzy equivalent: [A] ≅ [B] if aα + aα = bα + bα.

L L U U L U L U

III. [A] ≲ [B] if aα ≤ bα and aα ≤ bα , or aα + aα ≤ bα + bα .

L U U L

IV. [A] ≳ [B] if aα ≥ bα and aα ≥ bα.

A VSP is defined as a LPP for minimizing price under the restriction of quality performance measures, services, and lead time is provided as [21]

min ( ) = ∑ pixi.

∑ qixi ≥ Q,

∑ sixi ≥ S,

∑ lixi ≤ L,

∑ xi = 1,

xi ≥ 0 for all .

Where xi: fraction of demand allocated to vendor i.

pi: price of item i.

qi: quality level of item i.

li: lead time of item i.

si: level of service of item i.

Q: required overall quality level.

L: lead time level.

S: service level.

Consider the following PQFVS using the close interval approximation as min ( ) = ∑ [pi]xi.

∑ [qi]xi ≥ [Q],

∑ [si]xi ≥ [S] ,

∑ [li]xi ≤ [L],

∑ xi = 1,

xi ≥ 0 for all .

Where [li] = [(li)α, (li)α ], [pi] = [(pi)α, (pi)α], [qi] = [(qi)α, (qi)α], [si] = [(si)α, (si)α ], [L] = [Lα, Lα], [Q] = [Qα, Qα], [S] = [Sα, Sα ] ∈ P(ℜ).

P(ℜ )=set of all close interval approximations on ℜ.

Definition 6. Any xj which satisfies the constraints in Problem (2) is referred to as a feasible solution. Let G be the set of all feasible solutions of Problem (1). We claim that x ∈ G is said to be an optimal solution only when [p]x ≤ [p]x for all x ∈ G.

Lemma 1. Problems (1) and (2) are equivalent.

Proof: let G1 and G2 denote the respective two feasible solution sets of Problems (1) and (2). Then, an element x ∈ G1 if and only if

∑ [qi]xi ≥ [Q],

∑ [si]xi ≥ [S],

∑ [li]xi ≤ [L],

∑ xi = 1.

If and only if

L U L U

∑ [(qi)α, (qi)α]xi ≥ [Qα, Qα],

L U L U

∑ [(si)α, (si)α]xi ≥ [Sα, Sα ],

L U L U

∑ [(li)α, (li)α]xi ≤ [Lα, Lα],

∑ xi = 1.

If and only if

L U L U

∑ [(qi)αxI, (qi)αxi] ≥ [Qα, Qα],

L U L U

∑ [(si)αxi, (si)αxi] ≥ [Sα, Sα ],

L U L U

∑ [(li)αxI, (li)αxi] ≤ [Lα, Lα],

∑ xi = 1.

If and only if m L U L U

(qi)α + (qi)α Qα + Qα ∑ ( )xi ≥ ( ), I=1 2 2

m L U L U

(si)α + (si)α Sα + Sα ∑ ( ) xi ≥ ( ), i=1 2 2

m L U L U

(li)α + (li)α Lα + Lα ∑ ( ) xi ≤ ( ), I=1 2 2 ∑ xi = 1.

If and only if

∑ qixi ≥ Q ,

∑ sixi ≥ S,

∑ lixi ≤ L,

∑ xi = 1.

If and only if x ∈ G2.

Hence G1 ≅ G2.

Let X be the optimal feasible solution of Problem (2), then we obtain.

[P]X ≤ [P]X for all X ∈ G1.

If and only if ∑ [pi]xi ≤ ∑ [pi]xi for all xj ∈ G1,

i=1 i=1

If and only if

m L U m L U

(pi)α + (pi)α (pi)α + (pi)α ∑ ( ) xi ≤ ∑ ( ) xi, i=1 2 i=1 2 If and only if ∑ pixi ≤ ∑ pixi .

i=1 i=1

Thus, we have X is an optimal feasible solution of the crisp VSP as given by (1).

Referring to the interval arithmetic operations [31] Problem (2) is converted into 2 inequalities as follows:

L U U L

∑ (qi)αxi ≥ Qα , ∑ (qi)αxi ≥ Qα ,

i=1 i=1 L U U L

∑ (si)αxi ≥ Sα , ∑ (si)αxi ≥ Sα,

i=1 i=1 L L U U

∑ (li)αxi ≤ Lα , ∑ (li)αxi ≤ Lα,

i=1 i=1

∑ xi = 1,

xi ≥ 0 for all .

Let Gi be a set of solutions to the i inequality and define.

2m+1 2m+1

G = ⋃i=1 Gi and G = ⋂i=1 Gi.

m (i) (i) m (i) (i) m (i) (i)

Definition 7. Suppose that: ∑i=1[q , q ]xi ≥ [Q1, Q2], ∑i=1[s , s ]xi ≥ [S1, S2] and ∑i=1[l , l ]xi ≥

1 2 1 2 1 2 m (i) m (i) m (i)

[L1, L2]. Then inequalities ∑i=1 q xi ≥ Q, ∑i=1 s xi ≥ S and ∑i=1 l xi ≥ L are called the characteristic

m (i) (i) m (i) (i) m (i) (i)

formulas of ∑i=1[q , q ]xi ≥ [Q1, Q2], ∑i=1[s , s ]xi ≥ [S1, S2] and ∑i=1[l , l ] ≥ [L1, L2]; respectively.

1 2 1 2 1 2 (i) (i) (i) (i) (i) (i) (i) (i) (i)

Here, q ∈ [q , q ], s ∈ [s , s ], and l ∈ [l , l ].

1 2 1 2 1 2 m (i) (i) m (i) (i)

Definition 8. For each inequality ∑i=1[q , q ]xi ≥ [Q1, Q2], ∑i=1[s , s ]xi ≥

1 2 1 2 m (i) (i)

[S1, S2] and ∑i=1[l , l ]xi ≥ [L1, L2] if there is one characteristic formula such that its solution set is similar to G or G, in this case, we say that the characteristic formulas as maximum and minimum values range inequalities.

m (i) (i) m (i) (i)

Proposition 1. Let the following inequalities ∑i=1[q , q ]xi ≥ [Q1, Q2], ∑i=1[s , s ]xi ≥

1 2 1 2 m (i) (i) m L U m U

[S1, S2] and ∑i=1[l , l ]xi ≥ [L1, L2]. Then, we have ∑i=1(qi)αxi ≥ Qα, ∑i=1(qi)αxi ≥

L m L U m U L m L L m U U

Qα , ∑i=1(si)αxi ≥ Sα , ∑i=1(si)αxi ≥ Sα, ∑i=1(li)αxi ≤ Lα , ∑i=1(li)αxi ≤ Lα are maximum and minimum values range inequalities for these constraint conditions, respectively.

By the arithmetic operations of intervals, Problem (2) is reduced into the following problems as

min ( ) = ∑ (pi)α xi.

∑ (qi)αxi ≥ Qα,

∑ (qi)αxi ≥ Sα ,

∑ (li)αxi ≤ Lα,

∑ xi = 1,

xi ≥ 0 for all i.

Now, we obtain the following optimization problem:

min ( ) = ∑ (pi)α xi.

∑ (qi)αxi ≥ Qα,

∑ (qi)αxi ≥ Sα ,

∑ (qi)αxi ≥ Sα ,

∑ (li)αxi ≤ Lα,

∑ xi = 1,

xi ≥ 0 for all i.

Suppose that the optimal solutions to Problems (3) and (4) are

∗ ∗ ∗ ∗ ∗∗ ∗∗ ∗∗ ∗∗

x1, x2, … , xm; Z1, x1 , x2 , … , xm; Z2 .

Thus, the optimal solution to the close interval approximation Problem (2) is ∗ ∗∗

minZ(x) = [Z1, Z2 ].

∗ ∗∗

x1 [x , x ] x ∗ ∗∗ 2 [x2, x1 ]

= . (5)

[x ] ∗ ∗∗ m [[xm, xm ]]

Consider the following piecewise quadratic fuzzy VSP as in Table 1.

The close interval approximations for the problem are presented in Table 2.

Applying Problem (2) to these data minZ(x) = [9, 11]x1 ⊕ [10, 11.5]x2 ⊕ [10.5, 12]x3.

[0.80, 0.95]x1 ⊕ [0.87, 0.97]x2 ⊕ [0.92, 1.02]x3 ≥ [0.80, 0.85], [0.80, 0.95]x1 ⊕ [0.92, 1.00]x2 ⊕ [0.90, 1.00]x3 ≥ [0.75, 0.80], [20, 35]x1 ⊕ [20, 30]x2 ⊕ [25, 27] 3 ≤ [20, 30], x1 + x2 + x3 = 1.0, x1, x2, x3 ≥ 0.

The interval Problem (6) is reduced to the following two LPPs as follows:

minZ1(x) = 9x1 + 10x2 + 10.5x3.

0.95x1 + 0.97x2 + 1.02x3 ≥ 0.80, 0.95x1 + 1.00x2 + 1.00x3 ≥ 0.75, 35x1 + 30x2 + 27 3 ≤ 30, x1 + x2 + x3 = 1.0, x1, x2, x3 ≥ 0.

Equivalently, we have the following problem.

minZ2(x) = 11x1 + 11.5x2 + 12x3.

0.80x1 + 0.87x2 + 0.92x3 ≥ 0.85, 0.80x1 + 0.92x2 + 0.90x3 ≥ 0.80, 20x1 + 20x2 + 25 3 ≤ 20, x1 + x2 + x3 = 1.0, x1, x2, x3 ≥ 0.

The solutions to the two Problems (7) and (8) are given below:

∗ ∗∗

1 [x1, x1 ] [0.2857, 0.375]

∗ ∗∗ ∗ ∗∗

[ 2] = [[x2, x2 ]] = [ [0, 0.7143] ] and minZ(x) = [Z1, Z2 ] = [9.9375, 11.3517].

∗ ∗∗

3 [x3, x3 ] [0, 0.625]

The VSP as an application of fully fuzzy LPP has been studied. The VSP with close interval approximation has been approached by taking the minimum and maximum value inequalities with the constraints reduced into two classical LPPs. The presented model possesses great future scope for development. One may consider the time value coefficients for more realistic objective function calculations. In addition, the model can be extended in the form of a multi-stage stochastic programming approach. As another future research direction, one may consider some more efficient algorithms to demonstrate the suggested model.

The authors declare no conflicts of interest to report regarding the present study.