Study on the commercial value evaluation of commercial photography with uncertain linguistic information

1. Introduction

Commercial photography is the most commonly used in advertising as a means of display of goods, which is different from other Photography categories, such as artistic and commercial dual identity [1, 2, 3, 4]. Commercial photography under the consumer culture has led to the formation of visual consumption patterns and consumer spending than the product itself. Meanwhile, economic development also adds to the commercial photography invisible power. As digital photography continues to mature and involved in today to commercial photography after the impact to commercial photography far beyond people’s imagination. Now, with technology, social progress, from the city we live to the village every day to see a huge number of colorful posters and advertising, history never appeared at any stage will focus on the visual image and so such intensive visual information. We are entering the era of the image. Commercial photography, as born in the business community and has a unique art form of photography categories, it has been more open style and more and more visual representation of the graphic art forms in society today the most impact of a mass media advertising [5, 6, 7, 8, 9, 10].

In the literature, we find a wide range of methods for decision making [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. However, under many conditions, for the real decision making problems, the decision information about alternatives is usually uncertain or fuzzy due to the increasing complexity of the socio-economic environment and the vagueness of inherent subjective nature of human think, thus, numerical values are inadequate or insufficient to model real-life decision problems. Indeed, human judgments including preference information may be stated in uncertain linguistic variables. In order to effectively avoid the loss and distortion of information in uncertain linguistic information processing process, Xu [39, 40, 41] proposed the uncertain linguistic representation and computational model which has a distinct advantage over other linguistic processing methods in accuracy and reliability. All of the existed uncertain linguistic aggregation operators [42, 43, 44, 45, 46] only consider situations where all the elements in the uncertain linguistic variables are independent.

To do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to uncertain linguistic variables and some operational laws of uncertain linguistic variables. In Section 3 we have developed some distance measures with uncertain linguistic information and study some desirable properties of the proposed measures. In Section 4, an illustrative example for evaluating the commercial value evaluation of commercial photography with uncertain linguistic information is pointed out. In Section 5, we conclude the paper and give some remarks.

2. Preliminaries

Let S = { s i | i = 1 , 2 , … , t } be a linguistic term set with odd cardinality. Any label, s i represents a possible value for a linguistic variable, and it should satisfy the following characteristics [47, 48, 49, 50]: (1) The set is ordered: $s_i>s_j$ , if $i>j$ ; (2) There is the negation operator: 𝑛𝑒𝑔 ⁢ ( s i ) = s j such that j = t + 1 - i ; (3) Max operator: max ⁡ ( s i , s j ) = s i , if s i ⩾ s j ; (4) Min operator: min ⁡ ( s i , s j ) = s i , if s i ⩽ s j . For example, S can be defined as

S = { s 1 = extremely poor , s 2 = very poor , s 3 = 𝑝𝑜𝑜𝑟 , s 4 = 𝑚𝑒𝑑𝑖𝑢𝑚 , s 5 = 𝑔𝑜𝑜𝑑 , s 6 = very good , s 7 = extremely good }

To preserve all the given information, the discrete term set S should be extended to a continuous term set S ¯ = { s a | s 1 ⩽ s a ⩽ s q , a ∈ [ 1 , q ] } , where q is a sufficiently large positive integer. If s a ∈ S , then s a is termed an original linguistic term, otherwise, s a is termed a virtual linguistic term. In general, the decision maker uses the original linguistic term to evaluate attributes and alternatives, and the virtual linguistic terms can only appear in calculation [47, 48, 49, 50].

In many real-world problems, the input linguistic information may not match any of the original linguistic labels, and they may be located between two of them. In such cases, Xu [47, 48] defined the uncertain linguistic variables and introduced some of their operational laws.

Definition 1 [47, 48]. Let s ∼ = [ s α , s β ] , where s α , s β ∈ S , s α and s β are the lower and the upper limits, respectively, we call s ∼ the uncertain linguistic variable. Let S ∼ be the set of all the uncertain linguistic variables.

Definition 2. Let s ∼ 1 = [ s α 1 , s β 1 ] and s ∼ 2 = [ s α 2 , s β 2 ] be two uncertain linguistic varibles, such that

d ⁢ ( s ∼ 1 , s ∼ 2 ) = 1 2 ⁢ ( | β 1 - β 2 | + | α 1 - α 2 | )(1)

is called an distance between uncertain linguistic s ∼ 1 and s ∼ 2 .

Let α = ( α 1 , α 2 , … , α n ) and β = ( β 1 , β 2 , … , β n ) be two collection of arguments, and ω = ( ω 1 , ω 2 , … , ω n ) T is the weight vector of the absolute difference values | α i - β i | ( i = 1 , 2 , … , n ) , where ω i ⩾ 0 , ∑ i = 1 n ω i = 1 , then the generalized weighted distance is defined as follows [51]:

𝑊𝐷 ⁢ ( α , β ) = ( ∑ i = 1 n ω i ⁢ | α i - β i | λ ) 1 / λ(2)

Motivated by the idea of the ordered weighted averaging operator [52], Xu and Chen [51] introduced a type of ordered weighted distance (OWD) measures:

$\text{𝑂𝑊𝐷}(\alpha ,\beta )={\left({\displaystyle \sum _{i=1}^n}{w}_i{\left|{\alpha }_{\sigma \left(i\right)}-{\beta }_{\sigma \left(i\right)}\right|}^{\lambda }\right)}^{1/\lambda },\lambda >0.$(3)

where ( σ ⁢ ( 1 ) , σ ⁢ ( 2 ) , … , σ ⁢ ( n ) ) is a permutation of ( 1 , 2 , … , n ) , such that

| α σ ⁢ ( i - 1 ) - β σ ⁢ ( i - 1 ) | ⩾ | α σ ⁢ ( i ) - β σ ⁢ ( i ) | ⁢ for all (4) i = 2 , … , n .

and w = ( w 1 , w 2 , … , w n ) T is the weighting vector associated with the OWD measures, that’s to say, the weighting vector of the ordered positions of the arguments | α i - β i | ( i = 1 , 2 , … , n ) , w j ⩾ 0 and ∑ j = 1 n w j = 1 .

It is clear that the OWD measures Eq. (5) emphasize the importance of ordered position of each argument | α i - β i | ( i = 1 , 2 , … , n ) rather than the importance of the argument itself.

Consider that weights represent different aspects in both the distance measures Eqs (3) and (5), in order to reflect the importance of both the argument | α i - β i | ( i = 1 , 2 , … , n ) and its ordered position, Xu [53] develop a new type of distance measures as below:

Definition 3 [53]. Let α = ( α 1 , α 2 , … , α n ) and β = ( β 1 , β 2 , … , β n ) be two collection of arguments, then we call

HWD(M,N) = ( ∑ i = 1 n w i ⁢ Δ ⁢ ( α σ ⁢ ( i ) - β σ ⁢ ( i ) ) λ ) 1 / λ(5)

a type of hybrid weighted distance (HWD) measures, where Δ ⁢ ( α σ ⁢ ( i ) - β σ ⁢ ( i ) ) is the i th largest of the weighted arguments Δ ⁢ ( α i - β i ) (here, Δ ⁢ ( α i - β i ) = n ⁢ ω i ⁢ | α i - β i | λ , ( i = 1 , 2 , … , n ) , w = ( w 1 , w 2 , … , w n ) T is the weighting vector associated with the HWD measures, w i ⩾ 0 and ∑ i = 1 n w i = 1 , ω = ( ω 1 , ω 2 , … , ω n ) T is the weight vector of the arguments | α i - β i | ( i = 1 , 2 , … , n ) , and ω i ⩾ 0 , ∑ i = 1 n ω i = 1 , and n is the balancing coefficient which plays a role of balance (in such a case, if the vector ω = ( ω 1 , ω 2 , … , ω n ) T approaches ( 1 / n , 1 / n , … , 1 / n ) , then n ω i | α i - β i | λ ( i = 1 , 2 , … , n ) approaches | α i - β i | λ ( i = 1 , 2 , … , n ) ) .

Definition 4. Let S ∼ = ( s ∼ 1 , s ∼ 2 , … , s ∼ n ) and S ∼ ′ = ( s ∼ 1 ′ , s ∼ 2 ′ , … , s ∼ n ′ ) be two sets of the uncertain linguistic varibles, and d ⁢ ( s ∼ j , s ∼ j ′ ) be the distance between s ∼ j = [ s α j , s β j ] and s ∼ j ′ = [ s α j , s β j ] , then

𝑈𝐿𝑊𝐷 ⁢ ( S ∼ , S ∼ ′ ) = ( ∑ j = 1 n ω j ⁢ d ⁢ ( s ∼ j , s ∼ j ′ ) λ ) 1 / λ(6)

is called an uncertain linguistic weighted distance (ULWD) between S ∼ and S ∼ ′ , where ω = ( ω 1 , ω 2 , … , ω n ) T be the weight vector of d ( s ∼ j , s ∼ j ′ ) ( j = 1 , 2 , … , n ) , and ω j ⩾ 0 , ∑ j = 1 n ω j = 1 , and $\lambda >0$ .

Specially, if λ = 1, then the ULWD measure is reduced to the uncertain linguistic weighted Hamming distance (ULWHD); If λ = 2, then the ULWD measure is reduced to the uncertain linguistic weighted Euclidean distance (ULWED) measure.

Definition 5. Let S ∼ = ( s ∼ 1 , s ∼ 2 , … , s ∼ n ) and S ∼ ′ = ( s ∼ 1 ′ , s ∼ 2 ′ , … , s ∼ n ′ ) be two sets of the uncertain linguistic varibles, and d ⁢ ( s ∼ j , s ∼ j ′ ) be the distance between s ∼ j = [ s α j , s β j ] and s ∼ j ′ = [ s α j , s β j ] , then

𝑈𝐿𝑂𝑊𝐷 ⁢ ( S ∼ , S ∼ ′ ) = ( ∑ j = 1 n w j ⁢ d ⁢ ( s ∼ σ ⁢ ( j ) , s ∼ σ ⁢ ( j ) ′ ) λ ) 1 / λ(7)

is called an uncertain linguistic ordered weighted distance (ULOWD) between S ∼ and S ∼ ′ , where ( σ ( 1 ) , σ ( 2 ) , … , σ ( n ) ) is a permutation of ( 1 , 2 , … , n ) , such that d ⁢ ( s ∼ σ ⁢ ( j - 1 ) , s ∼ σ ⁢ ( j - 1 ) ′ ) ⩾ d ⁢ ( s ∼ σ ⁢ ( j ) , s ∼ σ ⁢ ( j ) ′ ) for all i = 2 , … , n , where w = ( w 1 , w 2 , … , w n ) T is the associated weight vector such that w j ⩾ 0 and ∑ j = 1 n w j = 1, and $\lambda >0$ .

Specially, if λ = 1, then the ULOWD measure is reduced to the uncertain linguistic ordered weighted Hamming distance (ULOWHD); If λ = 2, then the ULOWD measure is reduced to the uncertain linguistic ordered weighted Euclidean distance (ULOWED) measure.

From the above definitions, we know that the ULWD measure emphasize the importance of given individual distances, while the ULOWD measure only emphasizes the importance of the ordered position of the given individual distances, it weights the ordered position of the given individual distances instead of weighting arguments themselves. Therefore, weights represent different aspects in both the ULWD and ULOWD measures. However, both the ULWD and ULOWD operator consider only one of them. To solve this drawback, in the following we shall propose an uncertain linguistic hybrid weighted distance (ULHWD) measure.

Definition 6. Let S ∼ = ( s ∼ 1 , s ∼ 2 , … , s ∼ n ) and S ∼ ′ = ( s ∼ 1 ′ , s ∼ 2 ′ , … , s ∼ n ′ ) be two sets of the uncertain linguistic varibles, then we call

𝑈𝐿𝐻𝑊𝐷 ⁢ ( S ∼ , S ∼ ′ ) (8) = ( ∑ j = 1 n w j ⁢ d ˙ ⁢ ( s ∼ σ ⁢ ( j ) , s ∼ σ ⁢ ( j ) ′ ) λ ) 1 ⁢ λ

The uncertain linguistic hybrid weighted distance (ULHWD) measure, where d ˙ ⁢ ( s ∼ σ ⁢ ( j ) , s ∼ σ ⁢ ( j ) ′ ) is the j th argest of the weighted arguments d ˙ ⁢ ( s ∼ j , s ∼ j ′ ) (where, d ˙ ⁢ ( s ∼ j , s ∼ j ′ ) = n ⁢ ω j ⁢ d ⁢ ( s ∼ j , s ∼ j ′ ) , j = 1 , 2 , … , n ) , w = ( w 1 , w 2 , … , w n ) T is the weighting vector associated with the ULHWD measures, w j ⩾ 0 and ∑ j = 1 n w j = 1 , ω = ( ω 1 , ω 2 , … , ω n ) T is the weight vector of the arguments d ( s ∼ j , s ∼ j ′ ) ( j = 1 , 2 , … , n ) , and ω j ⩾ 0 , ∑ j = 1 n ω j = 1, and n is the balancing coefficient which plays a role of balance (in such a case, if the vector ω = ( ω 1 , ω 2 , … , ω n ) T approaches ( 1 / n , 1 / n , … , 1 / n ) , then d ˙ ⁢ ( s ∼ j , s ∼ j ′ ) approaches d ( s ∼ j , s ∼ j ′ ) ) .

Moreover, in what follows, we discuss two special cases of the ULHWD measures:

Theorem 1. The GHFHWD measure is a special case of the GHFHWD measure.

Proof. Let w = ( 1 / n , 1 / n , … , 1 / n ) , then

𝑈𝐿𝐻𝑊𝐷 ⁢ ( S ∼ , S ∼ ′ ) = ( ∑ j = 1 n w j ⁢ d ˙ ⁢ ( s ∼ σ ⁢ ( j ) , s ∼ σ ⁢ ( j ) ′ ) λ ) 1 / λ = ( 1 n ⁢ ∑ i = 1 n d ˙ ⁢ ( s ∼ σ ⁢ ( j ) , s ∼ σ ⁢ ( j ) ′ ) λ ) 1 / λ = ( 1 n ⁢ ∑ j = 1 n n ⁢ ω j ⁢ d ⁢ ( s ∼ σ ⁢ ( j ) , s ∼ σ ⁢ ( j ) ′ ) λ ) 1 / λ = ( ∑ j = 1 n ω j ⁢ d ⁢ ( s ∼ σ ⁢ ( j ) , s ∼ σ ⁢ ( j ) ′ ) λ ) 1 / λ = 𝑈𝐿𝑊𝐷 ⁢ ( S ∼ , S ∼ ′ )

Which completes the proof of Theorem 1.

Theorem 2. The ULOWD measure is a special case of the ULHWD measure.

Proof. Let ω = ( 1 / n , 1 / n , … , 1 / n ) , then

d ˙ ⁢ ( s ∼ j , s ∼ j ′ ) = n ⁢ ω i ⁢ d ⁢ ( s ∼ j , s ∼ j ′ ) = d ⁢ ( s ∼ j , s ∼ j ′ )

This completes the proof of Theorem 2.

From Definition 6 and the above theorems, we know that:

(1)

The ULHWD measure first weights the given arguments, and then reorders the weighted arguments in descending order and weights these ordered arguments by the ULHWD weights, and finally aggregates all the weighted arguments into a collective one.

In the 21st century, our country’s commercial photography began to get rapid development, but because the commercial photography career starts late in our country, the technical equipment backward, and there is a large gap compared with abroad. As a new industry, many subjective and objective factors make it development is weak in many ways. Reform and opening up has brought the development of economy and people thought liberation, has brought the great development of China’s commercial photography also. Today, commercial photography technology and foreign equipment has no obvious difference, but in the transmission of commodity information and advertising creative still exist very big difference. In the context of globalization, many enterprises to go abroad, more and more foreign firms will come to China. How to meet the needs of international customers, show the development of China’s commercial photography level, promote the great development of the domestic economy is we need to solve the problem. In this section, we present an empirical case study of for evaluating the commercial value evaluation of commercial photography. The project’s aim is to select the best commercial photography modes. Five possible commercial photography modes A i ( i = 1 , 2 , 3 , 4 , 5 ) are evaluated. Three decision makers (experts) form a committee to act as decision makers. The company must take a decision according to the following four attributes: G 1 is the technical standards for commercial photography; G 2 is the scale of the commercial photography market; G 3 is the commercial photography customer experience value; G 4 is the commercial photography market development potential. The five possible commercial photography modes A i ( i = 1 , 2 , … , 5 ) are to be evaluated using the linguistic term set S by the decision makers under the above four general characteristics, and construct the decision matrix as shown in Table 1. If we suppose that w = ( 0.2 , 0.3 , 0.3 , 0.2 ) , λ = 1 and the positive ideal alternative A + = { [ S 7 , S 7 ] , [ S 7 , S 7 ] , [ S 7 , S 7 ] , [ S 7 , S 7 ] } , then we can get

Then, we utilize the ULHWD measure to get the most desirable commercial photography mode.

Step 5. According to the aggregating results shown in Table 2 and the ULHWD measure, the ordering of the alternatives are shown in Table 2. Note that $>$ means “preferred to”. As we can see from Table 2, depending on the aggregation operators used, the ordering of the alternatives may be different, and thus, the investor can properly select the desirable commercial photography modes according to his interest and the actual needs.

In this paper, we have developed the uncertain linguistic hybrid weighted distance (ULHWD) measure. We have proved both g uncertain linguistic weighted distance (ULWD) measure and the uncertain linguistic ordered weighted distance (ULOWD) measure are the special case of the ULHWD measure. The ULHWD measure first weights the given arguments, and then reorders the weighted arguments in descending order and weights these ordered arguments by the ULHWD weights, and finally aggregates all the weighted arguments into a collective one. Obviously, the ULHWD measure generalizes both the ULWD measure and ULOWD measure, and reflects the importance degrees of both the given argument and the ordered position of the argument. Furthermore, the ULHWD measure can relieve the influence of unfair arguments on the decision results by using the ULHWD weights to assign low weights to those “false” or “biased” ones. Finally, based on the ULHWD measure, we have proposed a practical method for evaluating the commercial value evaluation of commercial photography with uncertain linguistic information. In the future, we shall continue working in the application of the ULHWD measure to other domains [54, 55, 56, 57, 58, 59, 60, 61, 62, 63].