An integrated Taguchi loss function–fuzzy cognitive map–MCGP with utility function approach for supplier selection problem

Abstract

Due to the effects of supplier evaluation and selection problem on the quality of products and companies’ business activities, supplier selection is considered as a strategic issue in organizations’ development plans. The purpose of this study is to provide an integrated framework for supplier selection problem regarding to the loss of criteria deviation from specification limits, causal relationships between criteria and the preferences of decision makers (DMs) in the supplier selection problem. Thus, in the first step, the loss of each criterion is calculated using Taguchi loss function (TLF), then fuzzy cognitive map (FCM) and hybrid learning algorithm are applied to determine criteria weights. Finally, considering outputs of TLF and FCM methods, multi-choice goal programming with utility function (MCGP-U) is used to select an optimal supplier and to increase the DMs’ expected utility values, simultaneously. The results of implementation of proposed framework based on the extended MCGP-U model on an active company in paint and coating industry show that delivery time criterion has the most effect and priority on suppliers’ evaluations. Also among six qualified suppliers, a supplier with the least total loss value and the most utility values is selected as the optimal supplier for the under consideration company.

Appendix

$$\begin{aligned} {\text{Min}}\,Z & = 0.581\left( {d_{1}^{ + } + f_{1}^{ - } } \right) + 0.743\left( {d_{2}^{ + } + f_{2}^{ - } } \right) + 0.446\left( {d_{3}^{ + } + f_{3}^{ - } } \right) + 0.615\left( {d_{4}^{ + } + f_{4}^{ - } } \right) + 0.441\left( {d_{5}^{ + } + f_{5}^{ - } } \right) + 0.728\left( {d_{6}^{ + } + f_{6}^{ - } } \right) \\ & \quad + 0.631\left( {d_{7}^{ + } + f_{7}^{ - } } \right) + 0.590\left( {d_{8}^{ + } + f_{8}^{ - } } \right) + 0.552\left( {d_{9}^{ + } + f_{9}^{ - } } \right) + 0.711\left( {d_{10}^{ - } + f_{10}^{ - } } \right) + 0.695\left( {d_{11}^{ - } + f_{11}^{ - } } \right) \\ \end{aligned}$$

(10)

Equation (10) is the objective function of proposed model and minimizes the deviations of goals from aspiration levels to satisfy the utility values of DM. The calculated weights are multiplied to the goals indicate the priority of goals based on the causal relationships between goals.

$$3.59(S_{1} ) + 2.18(S_{3} ) + 0.71(S_{4} ) + 7.51(S_{5} ) + 4.45(S_{6} ) - d_{1}^{ + } + d_{1}^{ - } = y_{1}$$

(11)

Equation (11) represents the sales price criterion goal (the less is better goal). It should be noted that, the goal is to decrease the loss, so the less is better for such goals. Coefficients are the loss values of suppliers for the sales price criterion.

$$431.06 \le y_{1} \le 437.73$$

(12)

Equation (12) represents the y bound for the sales price criterion.

$$\lambda_{1} \le \frac{{437.73 - y_{1} }}{6.67}$$

(13)

Equation (13) represents LLUF (for the less is better).

$$\lambda_{1} + f_{1}^{ - } = 1$$

(14)

Equation (14) represents the high value of 1 for utility values. In cases that utility value is not perfect, it can take value in [0,1].

$$4.44(S_{1} ) + 9.99(S_{2} ) + 1.11(S_{3} ) + 1.11(S_{5} ) + 4.44(S_{6} ) - d_{2}^{ + } + d_{2}^{ - } = y_{2}$$

(15)

Equation (15) represents the delivery time goal function (the less is better goal).

$$1.05 \le y_{2} \le 1.08$$

(16)

Equation (16) represents the y bound for the delivery time criterion.

$$\lambda_{2} \le \frac{{1.08 - y_{2} }}{0.03}$$

(17)

Equation (17) represents LLUF (for the less is better).

$$\lambda_{2} + f_{2}^{ - } = 1$$

(18)

Equation (18) represents the high value of 1 for utility values.

$$9.62(S_{1} ) + 9.74(S_{2} ) + 9.78(S_{3} ) + 9.49(S_{4} ) + 9.57(S_{5} ) + 9.86(S_{6} ) - d_{3}^{ + } + d_{3}^{ - } = y_{3}$$

(19)

Equation (19) represents the quality goal function (the less is better goal).

$$8.84 \le y_{3} \le 8.93$$

(20)

Equation (20) represents the y bound for the quality criterion.

$$\lambda_{3} \le \frac{{8.93 - y_{3} }}{0.29}$$

(21)

Equation (21) represents LLUF (for the less is better).

$$\lambda_{3} + f_{3}^{ - } = 1$$

(22)

Equation (22) represents the high value of 1 for utility values.

$$8.85(S_{1} ) + 9.51(S_{2} ) + 8.26(S_{3} ) + 10.00(S_{4} ) + 7.73(S_{5} ) + 9.29(S_{6} ) - d_{4}^{ + } + d_{4}^{ - } = y_{4}$$

(23)

Equation (23) represents the customer satisfaction goal function (the less is better goal).

$$6.24 \le y_{4} \le 6.36$$

(24)

Equation (24) represents the y bound for the customer satisfaction criterion.

$$\lambda_{4} \le \frac{{6.36 - y_{4} }}{0.12}$$

(25)

Equation (25) represents LLUF (for the less is better).

$$\lambda_{4} + f_{4}^{ - } = 1$$

(26)

Equation (26) represents the high value of 1 for utility values.

$$8.32(S_{1} ) + 8.88(S_{2} ) + 9.29(S_{3} ) + 8.88(S_{4} ) + 7.97(S_{5} ) + 9.96(S_{6} ) - d_{5}^{ - } + d_{5}^{ - } = y_{5}$$

(27)

Equation (27) represents the environment management systems goal function (the less is better goal).

$$6.84 \le y_{5} \le 6.98$$

(28)

Equation (28) represents the y bound for the environment management systems criterion.

$$\lambda_{5} \le \frac{{6.98 - y_{5} }}{0.14}$$

(29)

Equation (29) represents LLUF (for the less is better).

$$\lambda_{5} + f_{5}^{ - } = 1$$

(30)

Equation (30) represents the high value of 1 for utility values.

$$7.06(S_{1} ) + 10.50(S_{2} ) + 8.75(S_{3} ) + 7.57(S_{4} ) + 6.91(S_{5} ) + 9.20(S_{6} ) - d_{6}^{ + } + d_{6}^{ - } = y_{6}$$

(31)

Equation (31) represents the research and development goal function (the less is better goal).

$$5.20 \le y_{6} \le 5.37$$

(32)

Equation (32) represents the y bound for the research and development criterion.

$$\lambda_{6} \le \frac{{5.37 - y_{6} }}{0.17}$$

(33)

Equation (33) represents LLUF (for the less is better).

$$\lambda_{6} + f_{6}^{ - } = 1$$

(34)

Equation (34) represents the high value of 1 for utility values.

$$9.19(S_{1} ) + 5.79(S_{2} ) + 8.26(S_{3} ) + 7.11(S_{4} ) + 6.18(S_{5} ) + 5.42(S_{6} ) - d_{7}^{ + } + d_{7}^{ - } = y_{7}$$

(35)

Equation (35) represents the technical capabilities and laboratory facilities goal function (the less is better goal).

$$4.63 \le y_{7} \le 4.77$$

(36)

Equation (36) represents the y bound for the technical capabilities and laboratory facilities criterion.

$$\lambda_{7} \le \frac{{4.77 - y_{7} }}{0.14}$$

(37)

Equation (37) represents LLUF (for the less is better).

$$\lambda_{7} + f_{7}^{ - } = 1$$

(38)

Equation (38) represents the high value of 1 for utility values.

$$4.45(S_{1} ) + 7.78(S_{2} ) + 6.40(S_{3} ) + 5.10(S_{4} ) + 4.25(S_{5} ) + 3.99(S_{6} ) - d_{8}^{ + } + d_{8}^{ - } = y_{8}$$

(39)

Equation (39) represents the services goal function (the less is better goal).

$$3.45 \le y_{8} \le 3.52$$

(40)

Equation (40) represents the y bound for the services criterion.

$$\lambda_{8} \le \frac{{3.52 - y_{8} }}{0.07}$$

(41)

Equation (41) represents LLUF (for the less is better).

$$\lambda_{8} + f_{8}^{ - } = 1$$

(42)

Equation (42) represents the high value of 1 for utility values.

$$5.62(S_{1} ) + 2.50(S_{2} ) + 5.62(S_{4} ) + 1.22(S_{5} ) + 3.02(S_{6} ) - d_{9}^{ + } + d_{9}^{ - } = y_{9}$$

(43)

Equation (43) represents the geographical location goal function (the less is better goal).

$$243.75 \le y_{9} \le 248.75$$

(44)

Equation (44) represents the y bound for the geographical location criterion.

$$\lambda_{9} \le \frac{{248.75 - y_{9} }}{5}$$

(45)

Equation (45) represents LLUF (for the less is better).

$$\lambda_{9} + f_{9}^{ - } = 1$$

(46)

Equation (46) represents the high value of 1 for utility values.

$$14(S_{1} ) + 7(S_{2} ) + 8(S_{3} ) + 11(S_{4} ) + 12(S_{5} ) + 9(S_{6} ) - d_{10}^{ + } + d_{10}^{ - } = y_{10}$$

(47)

Equation (47) represents the financial stability goal function (the more is better goal). Coefficients are the financial stability values of suppliers.

$$7 \le y_{10} \le 14$$

(48)

Equation (48) represents the y bound for the financial stability criterion.

$$\lambda_{10} \le \frac{{y_{10} - 7}}{7}$$

(49)

Equation (49) represents RLUF (for the more is better).

$$\lambda_{10} + f_{10}^{ - } = 1$$

(50)

Equation (50) represents the high value of 1 for utility values.

$$10(S_{1} ) + 6(S_{2} ) + 10(S_{3} ) + 9(S_{4} ) + 8(S_{5} ) + 11(S_{6} ) - d_{11}^{ + } + d_{11}^{ - } = y_{11}$$

(51)

Equation (51) represents the experience goal function (the more is better goal). Coefficients are the experience values of suppliers.

$$6 \le y_{11} \le 11$$

(52)

Equation (52) represents the y bound for the experience criterion.

$$\lambda_{11} \le \frac{{y_{11} - 6}}{5}$$

(53)

Equation (53) represents RLUF (for the more is better).

$$\lambda_{11} + f_{11}^{ - } = 1$$

(54)

Equation (54) represents the high value of 1 for utility values.

$$S_{1} + S_{2} + S_{3} + S_{4} + S_{5} + S_{6} = 1$$

(55)

Equation (55) shows that a supplier selection constraint.

$$\begin{aligned} & S_{i} \in \left\{ {0,\left. 1 \right\},} \right.\quad i = 1, \ldots ,6 \\ & \lambda_{j} ,y_{j} ,d_{j}^{ + } ,d_{j}^{ - } ,f_{j}^{ - } \ge 0,\quad j = 1, \ldots ,11 \\ \end{aligned}$$

(56)

Equation (56) specifies variable types of the model.