Drawing a Strategy Canvas Using the Fuzzy Best–Worst Method

Abstract

Blue ocean strategy is a new approach to strategy-making and innovation with the aim of sustained performance and growth of the company. In this approach, an analytical and decision-making tool known as Strategy Canvas is introduced in order to create blue oceans and to innovate value. Strategy Canvas provides a basic framework for taking into account all competitive and key factors in the current industry. In this study, the proposed fuzzy best–worst method is employed as a multi-criteria decision-making method to map Strategy Canvas of a manufacturing company in Iran. The proposed method is concerned with the proposition of a new approach to eliciting the weight vector from the fuzzy pairwise comparison matrices. For this purpose, a nonlinear optimization model was proposed. After solving the model, crisp weights were extracted from the fuzzy pairwise comparison matrices. The proposed method was an execution of the best–worst method in a fuzzy environment. This method is able to carry out fewer and more consistent comparisons. This method is also capable of substituting the fuzzy AHP method.

Appendix

Theorem 1

The priorities derived by the LFPP methodology from the upper triangular elements of a fuzzy pairwise comparison matrix are exactly the same as those derived from the lower triangular elements of the fuzzy pairwise comparison matrix (Wang and Chin 2011).

Theorem 2

The LFPP methodology produces the unique normalized optimal priority vector for any fuzzy pairwise comparison matrix (Wang and Chin 2011).

Theorem 3

FBW measures a unique normalized optimal priority vector.

Proof

The FBW’s objective function is a convex function, and the constraints of the model are linear. Hence, the feasible area is also convex. The optimal solution is often unique and the model presents unique weights based on the convex programming theory.

Theorem 4

The proposed FBW method ends in a more compromised and better solution compared with LFPP method for weighting the criteria or alternatives in a fuzzy pairwise comparison with lower calculations and comparisons. The optimal solution of FBW is, in other words, less than or equal to the optimal solution of LFPP.

Proof

To avoid complexity, the worst criterion is set in the first position (W = 1) and the best at the last position (B = n) of the pairwise comparison.

$\tilde{A}$	$C1 = C_{\text{w}}$	$C_{j}$	$Cn = C_{\text{B}}$
$C1$	$1$	$\_$	$\_$
$C_{i}$	The worst comparison:$\tilde{\varvec{a}}_{{\varvec{iw}}}$	The other comparison:$\left[ {\tilde{\varvec{a}}_{{\varvec{ij}}} } \right]$	$\_$
$Cn = C_{\text{B}}$	$\tilde{\varvec{a}}_{{\varvec{Bw}}}$	The best comparison:$\tilde{\varvec{a}}_{{\varvec{Bj}}}$	$1$

Applying Theorem 1, no variation is created whether the elements under the main diagonal of a fuzzy pairwise comparison matrix are considered or those above. The elements below are considered, and the LFPP model is rewritten by FWB notation:

$$\begin{aligned} & {\text{Minimize}}\;Z_{\text{LFPP}} = (1 - \lambda )^{2} + M \cdot \left[ {\sum\limits_{{i \in k^{{\prime }} }} {\left( {\delta_{i1}^{2} + \eta_{i1}^{2} } \right)} + \sum\limits_{{j \in k^{{\prime \prime }} ,j \ne w}} {\left( {\delta_{nj}^{2} + \eta_{nj}^{2} } \right) + \mathop \sum \limits_{{j \in k^{{{\prime \prime \prime }}} ,j \le n - 1}} \mathop \sum \limits_{{i \in k^{{{\prime \prime \prime }}} ,i > j}} \left( {\delta_{ji}^{2} + \eta_{ji}^{2} } \right)} } \right] \\ & \left\{ {\begin{array}{*{20}l} {{\text{The}}\;{\text{Worst}}\;{\text{Comparisons}}\left\{ {\begin{array}{*{20}l} {x_{i} - x_{w} - \lambda \ln \left( {\frac{{m_{iw} }}{{l_{iw} }}} \right) + \delta_{iw} \ge \ln l_{iw} ,i \in k^{'} } \hfill \\ { - x_{i} + x_{w} - \lambda \ln \left( {\frac{{u_{iw} }}{{m_{iw} }}} \right) + \eta_{iw} \ge - \ln u_{iw} ,i \in k^{'} } \hfill \\ \end{array} } \right.} \hfill \\ {{\text{The}}\;{\text{Best}}\;{\text{Comparisons}}\left\{ {\begin{array}{*{20}l} {x_{B} - x_{j} - \lambda \ln \left( {\frac{{m_{Bj} }}{{l_{Bj} }}} \right) + \delta_{Bj} \ge \ln l_{Bj} ,j \in k^{{\prime \prime }} ,j \ne w} \hfill \\ { - x_{B} + x_{j} - \lambda \ln \left( {\frac{{m_{Bj} }}{{l_{Bj} }}} \right) + \eta_{Bj} \ge - \ln l_{Bj} ,j \in k^{{\prime \prime }} ,j \ne w} \hfill \\ \end{array} } \right.} \hfill \\ {{\text{The}}\;{\text{Other}}\;{\text{Comparisons}}\left\{ {\begin{array}{*{20}l} {x_{j} - x_{i} - \lambda \ln \left( {\frac{{m_{ji} }}{{l_{ji} }}} \right) + \delta_{ji} \ge \ln l_{ji} ;\;i \& j \in k^{{{\prime \prime \prime }}} ,j \ne w,j = 1, \ldots , n - 1;\; i > j} \hfill \\ { - x_{j} + x_{i} - \lambda \ln \left( {\frac{{u_{ji} }}{{m_{ji} }}} \right) + \eta_{ji} \ge - \ln u_{ji} ;\;i \& j \in k^{{{\prime \prime \prime }}} ,j \ne w,j = 1, \ldots , n - 1;\; i > j} \hfill \\ \end{array} } \right.} \hfill \\ {k = \left\{ {1,2, \ldots ,n} \right\},k^{'} = \left\{ {1,2, \ldots ,w - 1,w + 1, \ldots ,n} \right\},k^{{\prime \prime }} = \left\{ {1,2, \ldots ,B - 1,B + 1, \ldots ,n} \right\},k^{{{\prime \prime \prime }}} = k - \left\{ {B,w} \right\}} \hfill \\ {\delta_{ij} ,\eta_{ij} ,x_{i} ,\lambda \ge 0,k^{'} \cup \left\{ w \right\} = k,k^{{\prime \prime }} \cup \left\{ B \right\} = k} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(17)

Based on Theorem 2, by solving Model (17), the optimal solution $x_{\text{LFPP}}^{ *}$ is computed. The FBW model has been rewritten again.

$$\begin{aligned} & {\text{Minimize}}\;Z_{\text{FBW}} = (1 - \lambda )^{2} + M \cdot \left[ {\sum\limits_{{i \in k^{\prime}}} {\left( {\delta_{i1}^{2} + \eta_{i1}^{2} } \right)} + \sum\limits_{{j \in k^{\prime\prime},j \ne w}} {\left( {\delta_{nj}^{2} + \eta_{nj}^{2} } \right)} } \right] \\ & \left\{ {\begin{array}{*{20}l} {{\text{The}}\;{\text{Worst}}\;{\text{Comparisons}}\left\{ {\begin{array}{*{20}l} {x_{i} - x_{w} - \lambda \ln \left( {\frac{{m_{iw} }}{{l_{iw} }}} \right) + \delta_{iw} \ge \ln l_{iw} ,i \in k^{'} } \hfill \\ { - x_{i} + x_{w} - \lambda \ln \left( {\frac{{u_{iw} }}{{m_{iw} }}} \right) + \eta_{iw} \ge - \ln u_{iw} ,i \in k^{'} } \hfill \\ \end{array} } \right.} \hfill \\ {{\text{The}}\;{\text{Best}}\;{\text{Comparisons}}\left\{ {\begin{array}{*{20}l} {x_{B} - x_{j} - \lambda \ln \left( {\frac{{m_{Bj} }}{{l_{Bj} }}} \right) + \delta_{Bj} \ge \ln l_{Bj} ,j \in k^{{\prime \prime }} ,j \ne w} \hfill \\ { - x_{B} + x_{j} - \lambda \ln \left( {\frac{{m_{Bj} }}{{l_{Bj} }}} \right) + \eta_{Bj} \ge - \ln l_{Bj} ,j \in k^{{\prime \prime }} ,j \ne w} \hfill \\ \end{array} } \right.} \hfill \\ {k = \left\{ {1,2, \ldots ,n} \right\},k^{'} = \left\{ {1,2, \ldots ,w - 1,w + 1, \ldots ,n} \right\},k^{{\prime \prime }} = \left\{ {1,2, \ldots ,B - 1,B + 1, \ldots ,n} \right\}} \hfill \\ {\delta_{ij} ,\eta_{ij} ,x_{i} ,\lambda \ge 0,k^{'} \cup \left\{ w \right\} = k,k^{{\prime \prime }} \cup \left\{ B \right\} = k} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(18)

Assuming the comparisons’ decrease in FBW model, the model constraints also declined. Solving Model (18), the only global optimal solution $x_{\text{FBW}}^{ *}$ is calculated considering Theorem 3. The possible areas of Models (17 and 18) are, respectively, called $S_{\text{LFPP}}$ and $S_{\text{FBW}}$.

The solution area is expanded with the decrease of model constraints. This expansion ends in two various results: The expansion does not affect the objective function value or decrease the objective function value (minimum OB). Thus, the FBW model value is equal to or lower when compared with the optimal solution of LFPP model expressed as follows:

$$\begin{aligned} S_{\text{LFPP}} \subset S_{\text{FBW}} \Rightarrow \forall x \in S_{\text{LFPP}} \Rightarrow x \in S_{\text{FBW}} \hfill \\ x_{\text{LFPP}}^{*} \in S_{\text{LFPP}} \;{\text{and}}\;x_{\text{FBW}}^{*} \in S_{\text{FBW}} \Rightarrow Z_{\text{FBW}} \left( {x_{\text{FBW}}^{*} } \right) \le Z_{\text{LFPP}} \left( {x_{\text{LFPP}}^{*} } \right) \hfill \\ \end{aligned}$$

The FBW’s value of the objective function is constantly lower in comparison with the LFPP’s. On the other hand, the positive term $\mathop \sum \limits_{{j \in k^{{{\prime \prime \prime }}} ,\;j \le n - 1}} \mathop \sum \limits_{{i \in k^{{{\prime \prime \prime }}} ,i > j}} \left( {\delta_{ji}^{2} + \eta_{ji}^{2} } \right)$ in the LFPP’s objective function is greater than the objective function of FBW method. Accordingly, given this value, LFPP model’s response is worse compared with FBW model.