Fuzzy system approaches to negotiation pricing decision support

Abstract

With the emergence of customisation services, business-to-business price negotiation plays an increasingly important role in economic and management science. Negotiation pricing aims to provide different customers with products/services that perfectly meet their requirements, with the “right" price. In general, pricing managers are responsible for identifying the “right" negotiation price with the goal of maintaining good customer relationship, while maximising profits for companies. However, efficiently and effectively determining the “right" negotiation price boundary is not a simple task; it is often complicated, time-consuming and costly to reach a consensus as the task needs to take a wide variety of pricing factors into consideration, ranging from operation costs, customers’ needs to negotiation behaviours. This paper proposes a systematic fuzzy system (FS) approach, for the first time, to provide negotiation price boundary by learning from available historical records, with a goal to release the burden of pricing managers. In addition, when the number of involved influencing factors increases, conventional FS approach easily suffers from the curse of dimensionality. To combat this problem, a novel method, simplified FS with single input and single output modules (SFS-SISOM), is also introduced in this paper to handle high-dimensional negotiation pricing problems. The utility and applicability of this research is illustrated by three experimental datasets that vary from both data dimensionality and the number of training records. The experimental results obtained from two approaches have been compared and analysed based on different aspects, including interpretability, accuracy, generality and applicability.

Keywords

Negotiation pricing decision support system fuzzy systems curse of dimensionality

1 Introduction

Pricing problems have attracted increasing attention throughout the last few decades. As pointed out in [19], there are three main classes of pricing problems, namely retailing pricing, bid/auction pricing and negotiation pricing. Many studies have been carried out in retail and auction/bid pricing decision support and have been successfully employed to a wide range of application domains [3 , 11]. However, negotiation pricing support is still at its very early stage. Though empirical results [23] suggest that buyer’s preferences contributes to the achievement of better outcomes, negotiation pricing should not only concern the buyer attributes and behaviours, but quantitatively and automatically take the influencing factors from seller’s perspective into account as well. This is a very important but under-addressed issue because nowadays buyers’ preference or personal details/preferences are not available or deleted due to growing data privacy concerns. Hence, this work mainly concerns the negotiation pricing problems, particularly from the seller point of view, to provide decision support for pricing managers.

Negotiation pricing decision making is a very complicated process which involves human negotiation behaviour, trading environment information and product/service information to identify the “right" negotiation price for the purpose of benefiting the company. Currently, pricing managers are responsible for setting or reviewing the negotiation price. However, this is not an easy task. Setting the proposed price too low may lead to minimise the company’s profits. Conversely, setting the price too high may result in loss of customers and cause higher inventory maintenance costs. Thus, effective and efficient negotiation pricing decision support is highly desirable by companies to stay competitive, balance supply and demand in dynamic trading conditions and improve customer relationship.

Different approaches (e.g. expert systems and game theory) have been developed to assist pricing managers in providing reasonable negotiation price. Although some approaches illustrate promising results in certain specific problem domains, most of existing methods are framed either by heavily relying on experts’ knowledge and/or static negotiation strategy or making too strong, or even unrealistic, assumptions and hypotheses [1 , 21]. This makes difficult to apply such methods in diverse real-world negotiation pricing problems. Additionally, negotiation pricing concerns a wide variety of influencing factors. In reality, some of these factors involve value, noisy and imprecise information. They are not always represented by numerical values. Dealing with such uncertain information within negotiation pricing is of particular interest to this work. Further, transparency and interpretability is crucial to decision support systems. If the end user could have a clear understanding of how the suggested results are derived, the system tends to be more trustful and reliable for use.

Fuzzy set theory [5, 29] has established itself as a leading tool due to its capability in handling uncertain information and preserving transparency and interpretability in modelling. This paper proposes a systematic approach, for the first time, to provide negotiation decision support under various circumstances by using fuzzy set theory. Given a set of historical records, mathematical relationships between influencing factors and proposed price could be built by learning from available data, whilst accounting for experts’ knowledge. The resulting predictive model can then be used to predict the will-to-pay price for unseen transactions.

Standard fuzzy system (FS) performs well in dealing with low dimensional problems with good data coverage (sufficient training samples). However, since standard FS aims to simulate the entire problem space at a time, the number of required fuzzy rules and parameters increase exponentially with the number of involved variables. This makes standard FS easily suffers from the curse of dimensionality problem. Hence, a novel method, simplified FS with single input and single output modules (SFS-SISOM), is also proposed in this paper to tackle such dimensionality problem. This paper also investigates and discusses the performance, particularly the applicability, of these two different FS approaches.

The remainder of this paper is organised as follows. Section 2 reviews the existing approaches for negotiation pricing decision support. The existing drawbacks and challenges are discussed. Section 3 proposes a systematic method which employs both standard FS and SFS-SISOM approaches, for the first time, to provide negotiation pricing decision support. The applicability and utility of proposed methods is demonstrated in Section 4 and the derived results with respect to some important properties are discussed. The final section concludes the paper and points out further work directions.

2 Negotiation pricing

Undoubtedly, negotiation is a crucial activity in business transactions. In general, negotiation pricing problem arises when a customer would like to purchase a large quantity of products (e.g. wholesales) and/or expensive items (e.g. property, car and system). Negotiation pricing aims to identify a mutually beneficial price for both seller and buyer. It is often achieved by iterative one-to-one interactions. An effective negotiation pricing decision support system may provide the following benefits:

Maximise the companys’ profitability

Maintain good customer relationship

Balance supply and demand

Improve inventory management

Enhance management supplier relationship

Research of negotiation pricing problem dates back to early 1920s [18]. Most existing work focuses on identifying and analysing the underlying relations between influencing factors and the proposed price. The work of [2] investigates the relationship between car price and buyer attributes (e.g. income, race and gender). A controlled experiment is conducted by inviting participants, who have similar attributes (e.g. age, education background, appearance and even negotiation strategy), to visit and negotiate the price of the same car mode with different car retails in Chicago, USA. The agreed prices are collected and analysed. In the same problem domain, Goldberg [12] uses a different methodology, ordinary least square algorithm, to calculate the coefficient of individual inference factor based on a historical transaction report. The results obtained from [2] and [12] are quite different.

Apart from the work of [2, 12], some psychologists investigate the human negotiation behaviour model in business transactions. However, due to the complexity of the negotiate process itself, very few concrete results have been emerged. In the past decade, with the rapid development of computational intelligence and data mining techniques, negotiation decision support systems become appealing. Initial work of negotiation decision support [14, 15] mainly acts as a communication platform to accelerate the negotiation process, whereas very limited intelligence support is provided. Though empirical results [23] suggest that buyer’s preferences contributes to the achievement of better outcomes, negotiation pricing should not only concern the buyer attributes and behaviours, but quantitatively and automatically take the influencing factors from sellers’ perspective into account as well. The reason that this issue has not been properly addressed is due to that the users’ preferences or personal details are not available for data privacy concern.

Until recent years, several negotiation support systems [1 , 16] start to employ data mining techniques, ranging from geographical information, rule-based reasoning, heuristic approach, neural network, joint learning theory, case-based reasoning to game theory, and automated linguistic analysis technologies [22], to provide negotiation decision support. However, such systems still have some drawbacks:

First, majority of the existing systems rely on experts’ knowledge and/or static negotiation strategy to generate decision-making rules. It has been pointed out in [21] that only under the circumstance that the problem domain is certain, small and loosely coupled, then the knowledge can be captured in manual methods (e.g. interview and observation). Also, small changes of trading conditions may cause substantial expert interventions to modify the corresponding rules with an effort to reflect the new conditions. Further, although the experts’ knowledge plays an important role in system performance, it is difficult to validate and assess the quality of knowledge captured from experts.

Second, the existing systems aim to provide negotiation decision support to both seller and buyer sides. No system is exclusively designed for the seller side to maximise company profits. Third, some strong, even unrealistic, assumptions and hypotheses have been made in some existing systems. For example, systems using game theory [13] often assume that it is possible to use the maximisation utility function to model negotiation behaviour and suppose all negotiators would act reasonable and strategically to optimise their outcome based on precisely defined and symmetrically available information. However, such assumptions are difficult to achieve in reality. It is often that different participants have different information sources, biases and objectives for different cases, this may results in diverse negotiation behaviour, even for the same participant. Additionally, the requirement of symmetrical available information is hardly to happen, since it is difficult to access opponent’s private and sensitive data.

For challenges, in many cases, the influencing factors are difficult to evaluate. For example, it is not straightforward to identify the future profit of a certain customer. Further, some factors are not able to be quantified by using precise numerical values. Second, the number of influencing factors could be large. Since pricing manager requires to understand the influencing factors from vendor side (e.g. inventory maintenance costs, delivery costs and profit margin) and buyer side (e.g. financial ability and requests). Importantly, the analysis of negotiation behaviour is required to build upon repeated interactions. The negotiation behaviour is customised and would be different from customer to customer. The company may not have sufficient historical transaction records for each customer, especially new customer, available for analysis. According to the authors’ knowledge, there is no such research effort aiming to systematically resolve this challenge.

To sum up, providing negotiation pricing decision support is not an easy task. This is evident in aforementioned existing drawbacks and challenges. Due to recent advances in machine learning techniques, there is an increasing interest of applying machine learning methods to negotiation pricing decision support. Different from existing approaches which highly depend on experts’ knowledge and captures them in manually ways, computational intelligence and soft computing aims to discovering patterns and behaviours from historical data and human knowledge. Thus, it seems appropriate to overcome the existing limitations and handle the problem at hand.

3 Fuzzy system approaches

Today, fuzzy set theory [5, 29] has become an increasingly popular methodology for representing and handling uncertain information and has proven useful in many applications, such as control engineering [7, 31], natural language processing [32] and intelligent decision support systems [8 , 17]. The merits of utilising fuzzy sets for modelling uncertainty, representing subjective human knowledge, and as a means of emulating human reasoning processes, have been discussed by many authors [6, 28]. In this theory, the ambiguity and ambivalence of a certain concept are measured by using possibility distributions which have been defined and described in the literature (e.g. [6, 27]).

Different from the retail pricing and auction pricing, the negotiation pricing is one to one pricing, that is, offers the right price to the right customer. Therefore, the negotiation pricing problem is in fact to identify, for each individual customer, the right price for him/her. If it is assumed that the influencing factors to a customer to accept an offered price is X = (x ₁, ⋯ , x _n) and the right price for the particular customer is y, then this negotiation pricing problem is transferred to the problem of identifying the following (mathematical) relationship between the right negotiation price offer y and the influencing factors X = (x ₁, ⋯ , x _n): $y = P (X) = P (x_{1}, \dots, x_{n})$ (1)

If function P(X) can be identified, then it can be used to predict what is the right negotiation price to be offered to a new customer based on his/her particular values of the influencing factors. In the above influencing factors and negotiation price relationship, there are several special features which make fuzzy systems particular useful: firstly, many influencing factors are fuzzy by nature, such as how useful a service or product to a customer, how strong the desire to buy the service or product, what strength the financial ability is, and etc; secondly, the values of these influencing factors often can not be obtained accurately but only some rough figures are obtained by communication with the customers, hence high uncertainties inherited; thirdly, the exact formula of P (X) is unknown and have to be learned from all available information such as historical data of previous negotiation cases and sale managers’ experiences. For this reason, this paper proposes the fuzzy system method which is the first systematic approach to learn the one-to-one negotiation price and the basis for a new negotiation pricing support system.

There are two most widely used fuzzy systems: Mamdani’s [25] and Sugeno’s [20] models. In this work, Mamdani’s model is employed to build MISO (multi-input single output) fuzzy inference system. This is because in Sugenos model, the conclusion parts in the if-then rules are linear functions which make the meaning as a negotiation price less clearly and difficult to justify by the pricing managers; second, more number of parameters are required. As aforementioned, the number of influencing factors could be large. Standard FS may easily suffer from the curse of dimensionality. To combat this problem, a novel method, SFS-SISOM approach, is also introduced in this paper to handle high-dimensional negotiation pricing problems.

3.1 Standard fuzzy system approach

3.1.1 Standard fuzzy system

Suppose that f (X) is a MISO standard FS from input space $U = U_{1} \times U_{2} \times \dots \times U_{n} \subset ℝ^{n}$ to output space $V \subset ℝ$ and n is the number of input variables. Given n variables (x _j ∈ U _j (j = 1, 2, ⋯ , n)) and one output variable (y ∈ V), the index set can be defined as: $I = {i_{1}, \dots, i_{n} | i_{j} = 1, 2, \dots, N_{j}; j = 1, 2, \dots, n},$ (2) where N _j is the number of fuzzy sets of the j ^th input variable. The fuzzy rules can be represented in the form of:

$\begin{matrix} R_{i_{1} \dots i_{n}} : \\ If x_{1} is A_{i_{1}}^{1} and x_{2} is A_{i_{2}}^{2} and \dots and x_{n} is \\ A_{i_{n}}^{n} then \hat{y} is y_{i_{1} \dots i_{n}} . \end{matrix}$ (3)

The fuzzy sets of the j ^th input variable are defined as $A_{i_{j}}^{j} (j = 1, 2, \dots, n)$ and y _{i₁,⋯,i_n} is the parameter of corresponding fuzzy rule, which is supposed to be learned from learning algorithm. The total number of required fuzzy rules is $K = \prod_{j = 1}^{n} N_{j}$ .

Suppose that the (numeric) inputs of standard FS are X = (x ₁, x ₂, ⋯ , x _n) and its corresponding output is $\hat{y}$ . When using the Centre-Average defuzzifier to derive the crisp output, the standard FS is written as: $\hat{y} = f (X) = \frac{\sum_{i_{1} \dots i_{n} \in I} y_{i_{1} \dots i_{n}} (\prod_{j = 1}^{n} μ_{i_{j}}^{j} (x_{j}))}{\sum_{i_{1} \dots i_{n} \in I} (\prod_{j = 1}^{n} μ_{i_{j}}^{j} (x_{j}))}$ (4) where i ₁ i ₂ ⋯ i _n is the index of rule. Given the input value x _j for the j ^th input variable, the membership grade of the $i_{j}^{th}$ fuzzy set in the j ^th input variable is . In this work, triangular membership functions, as depicted in Fig. 1, is employed to capture uncertain information in both standard FS and SFS-SISOM. As shown in Fig. 1, the edge of one triangular membership function does not go across the centroid of its neighbouring membership functions. In so doing, given an input, at most two fuzzy rules would be fired at a time. This helps to preserve the model interpretability and reduce the number of required fuzzy rules.

Hence, given input value x _j, the corresponding membership grade is:

$\begin{matrix} μ_{i_{j}}^{j} (x_{j}) \\ = {\begin{matrix} (x_{j} - c^{i_{j} - 1}) / (c^{i_{j}} - c^{i_{j} - 1}) & x_{j} \in [c^{i_{j} - 1}, c^{i_{j}}] \\ (c^{i_{j} + 1} - x_{j}) / (c^{i_{j} + 1} - c^{i_{j}}) & x_{j} \in [c^{i_{j}}, c^{i_{j} + 1}] \\ 0 & otherwise . \end{matrix} \end{matrix}$ (5)

In addition, for the selected membership functions, it can be derived that $\sum_{i_{1} \dots i_{n} \in I} (\prod_{j = 1}^{n} μ_{i_{j}}^{j} (x_{j})) = 1$ (6)

Then, Equation (4) can be rewritten as: $\hat{y} = f (X) = \sum_{i_{1} \dots i_{n} \in I} (\prod_{j = 1}^{n} μ_{i_{j}}^{j} (x_{j})) y_{i_{1} \dots i_{n}}$ (7)

3.1.2 Recursive least square learning algorithm

Least square learning aims to minimise the sum of errors between modelled values and observed values for all learning samples. Its capability to derive the global optimal results is well recognised, thus recursive least square learning algorithm [26] is employed in this work to optimise the FS parameters. The flow chart of the learning algorithm is illustrated in Fig. 2, the details are explained as below:

Step 1: Construct the input matrix, parameter vector and output vector

The input matrix can be represented as in Equation (8), where $A^{i_{1} \dots i_{n}} (X) = \prod_{j = 1}^{n} μ_{i_{j}}^{j} (x_{j})$ and M is the total number of instance in the training dataset. Standard FS in Equation (7) can be rewritten as: $I = [\begin{matrix} A^{1 \dots 1} (X_{1}) \dots A^{N_{1} 1 \dots 1} (X_{1}) & A^{12 \dots 1} (X_{1}) \dots A^{N_{1} 2 \dots 1} (X_{1}) \dots A^{1 N_{2} \dots N_{n}} (X_{1}) & \dots A^{N_{1} N_{2} \dots N_{n}} (X_{1}) \\ ⋮ \\ A^{1 \dots 1} (X_{r}) \dots A^{N_{1} 1 \dots 1} (X_{r}) & A^{12 \dots 1} (X_{r}) \dots A^{N_{1} 2 \dots 1} (X_{r}) \dots A^{1 N_{2} \dots N_{n}} (X_{r}) & \dots A^{N_{1} N_{2} \dots N_{n}} (X_{r}) \\ ⋮ \\ A^{1 \dots 1} (X_{M}) \dots A^{N_{1} 1 \dots 1} (X_{M}) & A^{12 \dots 1} (X_{M}) \dots A^{N_{1} 2 \dots 1} (X_{M}) \dots A^{1 N_{2} \dots N_{n}} (X_{M}) & \dots A^{N_{1} N_{2} \dots N_{n}} (X_{M}) \end{matrix}]$ (8) $\hat{y} = f (X) = \sum_{i_{1} \dots i_{n} \in I} A^{i_{1} \dots i_{n}} (X) y_{i_{1} \dots i_{n}}$ (9)

The input matrix consists of M rows and K columns, in which $K = \prod_{j = 1}^{n} N_{j}$ indicates the required number of fuzzy rules. As shown in Equation (4), y _{i₁⋯i_n} are the parameters to be specified and the parameter vector is written as:

$\begin{matrix} θ = (y_{1 \dots 1}, y_{N_{1} \dots 1}, y_{12 \dots 1}, \dots, y_{N_{1} 2 \dots 1}, \dots, \\ y_{1 N_{2} \dots N_{n}}, \dots, y_{N_{1} N_{2} \dots N_{n}})^{T} \end{matrix}$ (10)

The output vector is represented as: $Y = ({\hat{y}}_{1}, \dots, {\hat{y}}_{r}, \dots, {\hat{y}}_{M})^{T}$ (11) As a result, the standard FS can be simply written as: $Y = I_{SFS} θ$ (12) where Y is an M-by-1 matrix, I is a M-by-K matrix and θ is a K-by-1 matrix.

Step 2: Initialise parameters and start learning

The parameter vector θ can be initialised by experts or uses default values. This allows taking experts’ knowledge into consideration, if available. Otherwise, the default values can be arbitrarily selected from output domain and commonly θ (0) is set to be 0.

Apart from the parameter vector, a disturbance matrix P needs to be initialised. Initially, P (0) = σB where σ, termed as disturbance parameter in this work, is a large constant (e.g. 100000) and B is an identity matrix. The role of the disturbance parameter σ includes 1) assist in constructing the disturbance matrix P to start the learning process (see Equations (13–15)); 2) by choosing different values of σ, it can trade-off the fitting between to the initial parameter and to the historical data.

In this recursive learning process, two index parameters t and r are employed to track and control the iterations. The number of maximum iterations is predefined as T, t is used to track the iteration number and r indicates the row/instance number. To start with, both t and r are set to 1.

Step 3: Update parameter vector

Recursive least square (RLS) learning algorithm has been widely applied to diverse problem domains and it is not new for fuzzy system approach. According to [26], RLS algorithm can be described in the following steps: $K (r) = P (r - 1) {(I_{r})}^{'} {[I_{r} P (r - 1) {(I_{r})}^{'} + 1]}^{- 1}$ (13) $θ (r) = θ (r - 1) + K (r) [Y_{r} - I_{r} θ (r - 1)]$ (14)

$\begin{array}{l} P (r) = P (r - 1) - P (r - 1) (I r)^{'} \\ [I_{r} P (r - 1) (I_{r})^{'} + 1] - 1 I_{r} P (r - 1) \end{array}$ (15)

In Equations (13) –(15), θ (0) and P (0) are initialised in Step 2. The objective of RLS is to merge the global optimal parameter vector θ.

Step 4: Check terminal condition

The recursive learning process will be terminated either by reaching the maximum iteration number T or deriving a predefined satisfaction error rate. Within an iteration t < T, if r < M (which is the total number of instances) then the algorithm goes back to Step 3 and set r = r + 1 until r = M. When the current iteration finishes, set t = t + 1 and reset r = 1. This enables to repeat step 3 until either terminal condition meets. The final output of the RLS algorithm would be the updated parameter vector θ, which can be further used to predict the outputs of testing dataset or unseen samples.

3.2 Simplified fuzzy system with single input and single output modules approach

Standard FS relates all combinations of input variables by using fuzzy intersection operations in the inference. It could be useful in negotiation pricing, when the number of input variables and required fuzzy sets are not large. However, it is not good at handling high-dimensional problems. This section introduces a novel linear fuzzy inference method, SFS-SISOM, to overcome the dimensionality problem.

3.2.1 SFS-SISOM

In SFS-SISOM, fuzzy rules relate individual input variable to output variable, respectively. Such rules can be represented in the form of:

R _{i
₁}: If x ₁ is $A_{i_{1}}^{1}$ then $\hat{y}$ is y _{i
₁}

R _{i
₂}: If x ₂ is $A_{i_{2}}^{2}$ then $\hat{y}$ is y _{i
₂}

R _{i
_n}: If x _n is $A_{i_{n}}^{n}$ then $\hat{y}$ is y _{i
_n}

where y _{i
_n} is the central point of the corresponding fuzzy set of the output variable and it is the parameter tends to be obtained from learning algorithm. In terms of the number of required fuzzy rules, standard FS needs $K = \prod_{j = 1}^{n} N_{j}$ (N _j is the number of fuzzy sets in the j ^th input variable) to cover the input space, whereas SFS-SISOM only requires $S = \sum_{j = 1}^{n} N_{j}$ . In most cases, S is much smaller than K.

In principle, SFS-SISOM can be viewed as a hierarchical fuzzy system, which consists of a number of SISO (single input single output) standard FSs. The output of each SISO standard FS can be writtenas: $f_{i_{j}} (x_{j}) = \sum_{S = 1}^{N_{j}} μ_{i_{j}}^{j} (x_{j}) y_{i_{j}},$ (16) where $u_{i_{j}}^{j} (x_{j})$ is the output of the $i_{j}^{th}$ fuzzy set in the j ^th input variable, when given input value x _j.

Take the financial ability input variable for example, the fuzzy rule set, as depicted in Fig. 3, can be described as:

IF the financial ability is good, THEN the discount is small.

IF the financial ability is average, THEN the discount is average.

IF the financial ability is bad, THEN the discount is large.

Mathematically speaking, the output of SFS-SISOM is an aggregation of the output derived from individual SISO standard FS, and that is: $\hat{y} = \sum_{j = 1}^{n} w_{j} f_{i_{j}} (x_{j}) = \sum_{j = 1}^{n} w_{j} (\sum_{S = 1}^{N_{j}} μ_{i_{j}}^{j} (x_{j}) y_{i_{j}})$ (17) In general, w _i indicates weight of individual sub FSs. In the context of this work, w _i could also be interpreted as the importance weight of each input variable.

Combining with Equation (16), Equation (17) can be rewritten as:

$\begin{matrix} \hat{y} & = & \sum_{j = 1}^{n} w_{j} (\sum_{s = 1}^{N_{j}} μ_{i_{j}}^{j} (x_{j}) y_{i_{j}}) \\ = & \sum_{j = 1}^{n} \sum_{s = 1}^{N_{j}} w_{j} y_{i_{j}} μ_{i_{j}}^{j} (x_{j}) \\ = & \sum_{j = 1}^{n} \sum_{s = 1}^{N_{j}} C_{i_{j}} μ_{i_{j}}^{j} (x_{j}) \end{matrix}$ (18)

A new parameter, C _{i
_j} which combines parameters w _j and y _{i
_j} is designed to be learnt from RLS algorithm.

3.2.2 Recursive least square learning algorithm

The learning process, as shown in Fig. 2, also applies to SFS-SISOM. The main difference is step 1: the construction of input matrix, parameter vector and output vector. According to Equation (17), the input matrix of SFS-SISOM can be represented as: $I = [\begin{matrix} μ_{1}^{1} (X_{1}^{1}) μ_{2}^{1} (X_{1}^{1}) \dots μ_{i_{1}}^{1} (X_{1}^{1}) & μ_{1}^{2} (X_{1}^{2}) & \dots μ_{i_{2}}^{2} (X_{1}^{2}) \dots μ_{i_{n}}^{n} (X_{1}^{n}) \\ ⋮ \\ μ_{1}^{1} (X_{r}^{1}) μ_{2}^{1} (X_{r}^{1}) \dots μ_{i_{1}}^{1} (X_{r}^{1}) & μ_{1}^{2} (X_{r}^{2}) & \dots μ_{i_{2}}^{2} (X_{r}^{2}) \dots μ_{i_{n}}^{n} (X_{r}^{n}) \\ ⋮ \\ μ_{1}^{1} (X_{M}^{1}) μ_{2}^{1} (X_{M}^{1}) \dots μ_{i_{1}}^{1} (X_{M}^{1}) & μ_{1}^{2} (X_{M}^{2}) & \dots μ_{i_{2}}^{2} (X_{M}^{2}) \dots μ_{i_{n}}^{n} (X_{M}^{n}) \end{matrix}]$ (19)

and the input matrix I consists of M rows and S columns.

The parameter vector is written as: $θ = (C_{11}, C_{12}, \dots, C_{1 i_{1}}, C_{21}, \dots, C_{2 i_{2}}, \dots, C_{{ni}_{n}})^{T}$ (20) and the output vector is: $Y = ({\hat{y}}_{1}, \dots, {\hat{y}}_{r}, \dots, {\hat{y}}_{M})^{T}$ (21)

Similar to Equation (12), the SFS-SISOM can be represented as: $Y = I θ$ (22)

where Y is an M-by-1 matrix, I is a M-by-S matrix and θ is a S-by-1 matrix.

Once the input matrix, parameter vector and output vector are constructed according to Equations (19)–(21), the remaining steps of Fig. 2 are the same as described in Standard FS, they are omitted here.

4 Experimental results and discussions

4.1 Experimental data

Three datasets which vary from the number of records and dimensions are used in this work:

4.1.1 DS1 - MP3 player dataset

This dataset is collected by conducting a survey which aims to promote MP3 player sales by providing potential customers with customised discounts. The survey is electronically distributed via mailing list to students and academic staff in the University of Manchester, U.K. Eventually, 248 feedbacks are returned and collected for further analysis. Four influential attributes, usefulness (5 values), importance (5 values), budget (4 values) and knowledge about this product (5 values), are accounted in this dataset. To start, participants are invited to perform a self-assessment based on these factors. Then, if the participant finds that the current price is too expensive, he/she is required to provide a minimal discount (represented by numerical values) that would persuade him/her to purchase. To maximise the company’s profits, the specified discount is also the one that pricing managers most likely to provide.

4.1.2 DS2 - Kingston apartment dataset

This dataset is collected from an international real estate company, named Kingston Real Estate Co., Ltd (http://www.kingston-hotel.cn). The dataset captures the prices of 441 apartments, distributed in 6 different buildings, developed by Kingston in 2005. The apartment price varies from location, layout and size. Four input attributes are detailed in Table 2, in which Building No., Floor No. and Apartment No. reflect different locations and layouts, respectively. To better interpret the effects of these attributes in resulting negotiation price, the raw dataset is pre-processed by interviewing sales experts, with an aim of assigning evaluation scores to input attributes.

First, each building is evaluated based on various criteria (e.g. location, convenience and facing direction). For example, Building No.6 receives the highest score due to its convenience (close to sport centre, shopping mall and hospital), diverse apartment sizes and nice river views. Second, as pointed out by the experts, floor No. plays an important role in identifying the negotiation price. In this dataset, each building has 16 floors. Higher floors tend to be more desirable due to its views and air quality. For this reason, higher floors receive higher evaluation scores. Third, whilst floor No. indicates the vertical properties of an apartment, the apartment No. represents horizontal properties, such as layout and facing direction. The obtained evaluation scores are summarised in Table 1. Higher scores indicate better properties. At last, since different customers may have different requirements in terms of size, it is difficult to provide objective judgments at this point; thereby no extra process is required for area attribute.

4.1.3 DS3 - Boston house sataset

This dataset was collected by the U.S Census Service which concerns the house price for house sale in Boston Mass in 1976. It is publicly available at StatLib archive (http://lib.stat.cmu.edu/datasets/boston). The value of a residential property is determined by 11 influential factors (see Table 2 for details). Based on various property factors and customer’s features, the negotiation house price can be quite different. The experimental results obtained here can be used to demonstrate the applicability and utility of proposed methods by identifying the right negotiation price for these Boston house properties. In particular, different from previous two datasets, this dataset is selected to verify the capability of proposed method in dealing with high dimensional problems.

4.2 Experimental results

4.2.1 DS1 - MP3 player dataset

In this experiment, both standard FS and SFS-SISOM are employed for prediction in a wide variety of situations which vary in the number of training/testing samples and fuzzy attribute partitions. The obtained results reveal that when the divided sub-spaces are well covered by training samples, standard FS and SFS-SISOM perform well. Similar results are obtained in which standard FS performs better in terms of goodness of fitting (higher accuracy in training dataset), whereas SFS-SISOM achieves higher accuracy in testing dataset with less fuzzy rules. This is evident in the Exp ₁,Exp ₇ and Exp ₁₃ in Tables 3, 4, respectively.

When only 50 instances (Exp ₄ - Exp ₆ in Tables 3, 4) are used for modelling, SFS-SISOM achieve more accurate results (around 92.6%) in testing dataset than standard FS. When 248 instances are all employed in the experiment, with the increase of the number of training data samples, standard FS starts to get rid of over-fitting problem. For example, Exp ₇, Exp ₉, Exp ₁₁, and Exp ₁₃ in Tables 3, 4 obtain similar performances, however, the number of required fuzzy rules in SFS-SISOM (#10) is less than one third of that required in standard FS (#36).

In some cases, the user would like to partition the input variable into a specified number of sub-spaces, such that the negotiation behaviour in a certain sub-space could be observed. Unfortunately, since the number of required fuzzy rules increases exponentially with the number of divided sub-spaces in standard FS, standard FS easily suffers from over-fitting problem when the number of available training sample is not sufficient to cover the sub-spaces (see Exp ₈, Exp ₁₀ and Exp ₁₂ in Tables 3). Under the same circumstance, SFS-SISOM is still capable of accomplishing such tasks (see Exp ₈, Exp ₁₀ and Exp ₁₂ in Tables 4).

In short, when the number of partitions in each attribute is carefully selected, the results in this experiment demonstrate that both standard FS and SFS-SISOM are able to provide reliable and understandable negotiation pricing decision support and they can also be widely expanded to real-life applications.

4.2.2 DS2 - Kingston apartment dataset

The apartments data of individual building is used, in return, as testing dataset, whilst the remaining data is used for training in this experiment. As discussed in [24], important attributes are suggested to include more fuzzy sets. Hence, the floor No. attribute is divided more finely than others in this experiment. The obtained results by employing standard FS and SFS-SISOM are reported in Tables 5, 6, respectively.

The results reveal that both the standard FS and SFS-SISOM perform very well (92.64% –98.28% testing accuracy), when the input space is partitioned by using 1,2,1,1 partition parameter. In most cases, standard FS achieves slightly higher testing accuracy than SFS-SISOMs, but requires 15 more fuzzy rules. However, when the input space is divided into more sub-spaces (see Exp ₇ –Exp ₁₂ in Tables 5, 6), the performance of standard FS drops dramatically, whereas SFS-SISOM still produces reliable results. The main reason for causing this is that the number of required fuzzy rules is exponentially increased in standard FS approach. In addition, take a closer examination into the unacceptable results (see Exp ₁₀ and Exp ₁₂ in Tables 5 building No.4 and No.6 both receive unique building evaluation scores (see Tables 1). When being used as testing dataset, no similar samples are provided in training dataset, this may result in unacceptable performance in modelling.

4.2.3 DS3 - Boston house dataset

This dataset contains 11 input attributes and 499 historical records are available. Even for the simplest partition, suppose that each attribute only includes 2 fuzzy sets, thus 2¹¹ = 2048 fuzzy rules are required in standard FS approach. This makes it difficult, even impractical, to model such high dimensional problem by using standard FS. For this reason, only SFS-SISOM is considered in this case. The first 414 records are used in the training dataset and the remaining 85 records are left out for testing. The results of using SFS-SISOM approach and employed parameters are reported in Tables 7.

The experimental results show that SFS-SISOM performs well in this dataset and capable of providing reliable decision support. SFS-SISOM approach achieves 84% –87% accuracy in testing dataset. In addition, the number of required rules and parameters is small.

4.3 Discussions

4.3.1 Interpretability and transparency

Interpretability and transparency is one of the most significant advantages of using FS approaches in data mining and it is crucial to the negotiation pricing problem at hand. In this work, the acquired knowledge can be represented in the form of IF-THEN fuzzy rules. Such rules not only provide a neat formalism of representing knowledge in computer, but also simulate the cognitive behaviour of human beings. To demonstrate the utility of proposed methods regarding interpretability, an example is shown in Tables 8. For the Exp ₇ in Tables 3, the input attributes are represented by 2, 3, 2 and 3 fuzzy sets, respectively. The results show that the derived knowledge well reflects the reality. For instances, given the same conditions in other attributes, the better understanding of the product a customer has, the more discount he/she tends to receive, and vice visa (see r ₁ –r ₃). Other reasonable negotiation behaviour applies to importance, usefulness and budget attributes. The more useful and important the product to a customer, fewer discounts would be provided (see r ₁, r ₇, r ₈, r ₁₃ and r ₂₆). Similarly, the more budget a customer intends to spend, fewer discounts would be considered (see r ₃ and r ₆).

When it comes to the extreme cases, r ₃₁ suggests to provide only 3.2514% discount when a MP3 player is useful and important to a customer. Meanwhile, the customer has relatively limited knowledge about this product, but with sufficient budget. A contrary example is illustrated in r ₆, in which pricing manager tends to provide the largest discount (33.7592%) when the product is neither useful nor important to a customer, who has good understanding of this product, but with limited budget. The above examples demonstrate that such FS approach facilitates a consistent and intuitively understandable negotiation pricing decision support, thereby; it can be more trustworthy and transparent for pricing managers.

4.3.2 Accuracy

The accuracy of proposed methods is reported in both average percentage error (APE) and mean square error (MSE). The accuracy is relevant to the probability that the divided sub-spaces are well covered by training samples. To provide a fair basis for comparison, it is assumed that there are n input attributes and each attribute is divided into m sub-spaces (includes m + 1 fuzzy sets). The numbers of divided sub-spaces for standard FS and SFS-SISOM are m ⁿ and m × n, respectively. In most cases, m × n is smaller than m ⁿ. Thus, given the same training dataset and partitions, the divided sub-spaces in SFS-SISOM stand a better chance to be covered.

In addition, the selected learning algorithms contribute to the model performance. The recursive least square learning algorithm used in standard FS and SFS-SISOM is capable of finding the global optimal solutions.

4.3.3 Generality

Compared to SFS-SISOM, standard FS easier suffers from “the curse of dimensionality” problems, thus it become less generic in handling diverse problems. This is evident in the experiment on DS3 that standard FS fails to build predict model for the problem at hand, due to the large number of rules required. Similarly, standard FS suffers from the over-fitting problem in DS2 (as shown in Exp ₁₀ in Tables 5), whereas SFS-SISOM is still functioning properly under the same circumstance.

4.3.4 Applicability

The above experimental results reveal that standard FS approach would be advantaged in dealing with low dimensional problems with good data coverage (sufficient training samples). This is because low dimensional problems require relatively small number of sub-spaces in the input domain; training samples then stand a good chance of covering all sub-spaces. Thus, the prediction accuracy would be improved. In addition, compared to SFS-SISOM which only considers one attribute at a time, the rules generated from standard FS cover all combinations of input attributes by using fuzzy intersection operators. This makes standard FS become more applicable to simulate global negotiation pricing behaviour.

On the other hand, SFS-SISOM tends to better handle high dimensional problems with sparse data coverage (insufficient training samples). Since SFS-SISOM constructs linear fuzzy inference between individual input attribute and output attribute, when the number of input attributes is large, the number of required fuzzy rules is greatly smaller than standard FS approach. However, this would also results in that SFS-SISOM is only capable of modelling one aspect of negotiation pricing behaviour at a time.

5 Conclusion

This paper has proposed a systematic approach which employs fuzzy techniques, for the first time, to provide negotiation pricing decision support. Apart from the standard FS, a novel method, SFS-SISOM, has also been developed to overcome the curse of dimensionality. The effectiveness and applicability of proposed approaches has been demonstrated by three experimental datasets which vary from dimensionality and data coverage. In particular, the performance of two different approaches are compared and discussed with respect to some important properties: interpretability and transparency, accuracy, generality and applicability. The experimental results reveal that fuzzy approaches are capable of providing reliable and reasonable negotiation pricing decision support, if the appropriate method is chosen for the given problem.

Although the proposed method is promising, much may be done through the future work. The proposed method is currently for proof-of-concept purpose. The promising results can be used to encourage commercial companies start to collect more detailed negotiation pricing data. Particularly, e-business and e-commerce platform can be used to assist and accelerate this process. This would then contribute to the development of pricing decision support system. Also, due to the availability of the collected datasets, the customer demands are not covered in this work. More effort will be put to collect the demand side data in future work. Since the proposed approach is general, once the demand data is available, it can be taken into account as an influential factor in negotiation pricing, and the proposed model can be readily applied to suggest the price boundary. In addition, it would be interesting to apply other FS approach (e.g. hierarchical FS [24, 30]) to the datasets used in this work, and exploit its performance and applicability.

Acknowledgments

This work is supported by the National Nature Science Foundation of China (No. 71301133, 71371159) and Humanity and Social Science Youth foundation of Ministry of Education, China (Grant No. 13YJC630033). Thanks also go to all participators who helped to complete the MP3 dataset questionnaire, and Kingston Real Estate Co., Ltd for the donation of the Kingston dataset.

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