Towards a multi-objective performance assessment and optimization model of a two-echelon supply chain using SCOR metrics

Abstract

This paper aims at multi-objective performance assessment and optimization of a multi-period two-echelon supply chain consisting of a supplier and a manufacturer. On the basis of the assessment system of the supply-chain operations reference model, the supply chain’s performance is investigated with respect to costs, assets, agility, reliability and responsiveness. First, methods to quantify these five performance attributes are put forward. Then a multi-objective mathematical programming model is developed for production decision making of components and products so that the supply chain’s performance frontier formed with Pareto efficient performance values can be achieved. Thereafter a simple augmented $\epsilon$-constraint method is proposed for searching for all Pareto efficient solutions of the multi-objective mathematical programming problem. Finally, efficiency of the method is demonstrated with a numerical example and a sensitivity analysis is implemented to reveal effects of capacity expansion on supply chains’ performance.

Appendices

Appendix A: Proof of Proposition 1

Proof

First, we observe that the set of feasible solutions of the three-dimensional model and that of the five-dimensional model are identical. Suppose $x^*$ is an efficient solution of the three-dimensional model. Then $x^*$ must be also a feasible solution of the five-dimensional model. Suppose it is not an efficient solution of the five-dimensional model, then there must exist an efficient solution of the five-dimensional model, e.g. $x^{**}$, whose corresponding non-dominated solution ($TC^{SC}(x^{**}),\, A^{SC}(x^{**}),\, F^{SC}(x^{**}),\, POF^{SC}(x^{**}),\, BO^{SC}(x^{**})$) dominates ($TC^{SC}(x^*), A^{SC}(x^*),\, F^{SC}(x^*),\, POF^{SC}(x^*),\, BO^{SC}(x^*)$). This means that constraints $TC^{SC}(x^{**}) \le TC^{SC}(x^*),\, A^{SC}(x^{**}) \le A^{SC}(x^*),\, F^{SC}(x^{**}) \ge F^{SC}(x^*),\, POF^{SC}(x^{**}) \ge POF^{SC}(x^*)$ and $BO^{SC}(x^{**}) \le BO^{SC}(x^*)$ hold with at least one strict inequality. As a result, ($a_1*TC^{SC}(x^{**}) + a_2*A^{SC}(x^{**}),\, F^{SC}(x^{**}),\, b_1*(\sum _{t=1}^Td_t - POF^{SC}(x^{**})) + b_2*BO^{SC}(x^{**})$) dominates ($a_1*TC^{SC}(x^*) + a_2*A^{SC}(x^*),\, F^{SC}(x^*)$, $b_1*(\sum _{t=1}^Td_t - POF^{SC}(x^*)) + b_2*BO^{SC}(x^*)$). That is to say, the solution $x^*$ does not belong to the set of efficient solutions of the three-dimensional model which contradicts our assumption. So $x^*$ must be an efficient solution of the five-dimensional model. On the other hand, when all non-dominated solutions of the five-dimensional model are aggregated a-posteriori, among the results obtained some solutions may be dominated. It means that some efficient solutions of the five-dimensional model are not efficient solutions of the three-dimensional model. So the set of efficient solutions of the three-dimensional model is a proper subset of the set of efficient solutions of the five-dimensional model. $\square$

Appendix B: Proof of Theorem 1

Since it is easy to observe the efficiency and effectiveness of the extension to algorithm acceleration with early exit, here we mainly concentrate on the proof of the efficiency and effectiveness of the acceleration algorithm with bouncing steps. Without loss of generality, let us take $e_{POF}$ as an example to discuss the problem of bouncing loop-control variables forward. Suppose current grid values of $e_A,\, e_F$ and $e_{POF}$ are $e_A^0,\, e_F^0$ and $e_{POF}^0$, respectively. The minimum of $POF^{SC}(q^s, q^m)$, the relatively worst value, is $RWV_{POF}^0$ after the following series of optimization models are solved when $e_{BO}$ moves from $BO^{nad}$ to $BO^{opt}$.

$$\begin{aligned} \min \; TC^{S\!C}\!(q^s\!,\! q^m) +\! \delta \left( \frac{A^{S\!C}\!(q^s\!,\! q^m)}{r_A} \!-\! \frac{F^{S\!C}\!(q^s\!,\! q^m)}{r_F} \!-\! \frac{POF^{S\!C}\!(q^s\!,\! q^m)}{r_{POF}} \!+\! \frac{BO^{S\!C}\!(q^s\!,\! q^m)}{r_{BO}}\right) \end{aligned}$$

$$\begin{aligned} st \qquad A^{SC}(q^s, q^m)&\le e_A^0 \nonumber \\ F^{SC}(q^s, q^m)&\ge e_F^0 \\ POF^{SC}(q^s, q^m)&\ge e_{POF}^0 \nonumber \\ BO^{SC}(q^s, q^m)&\le e_{BO}^\alpha \qquad e_{BO}^\alpha \in [BO^{opt}, BO^{nad}] \nonumber \\ (q^s, q^m)&\in S \nonumber \\ e_A^0, e_F^0, e_{POF}^0&\quad \text{ and } \quad e_{BO}^\alpha \text{ are } \text{ integers } \nonumber \end{aligned}$$

(40)

In the acceleration algorithm with bouncing steps, $e_{POF}$ is set to be $RWV_{POF}^0 + 1$ immediately after all above models are solved if $RWV_{POF}^0 < POF_{max}$. Obviously, if $RWV_{POF}^0 = e_{POF}^0,\, e_{POF}$ is $e_{POF}^0+1$ which exactly is what other $\epsilon$-constraint based methods do. So in the following discussion we mainly consider $RWV_{POF}^0 > e_{POF}^0$. In this situation the following series of models are not solved with the SAUGMECON method when the acceleration algorithm with bouncing steps is utilized.

$$\begin{aligned}&\min \; TC^{S\!C}\!(q^s\!,\! q^m) + \delta \left( \frac{A^{S\!C}\!(q^s\!,\! q^m)}{r_A} \!-\! \frac{F^{S\!C}\!(q^s\!,\! q^m)}{r_F} \!-\! \frac{POF^{S\!C}\!(q^s\!,\! q^m)}{r_{POF}} \!+\! \frac{BO^{S\!C}\!(q^s\!,\! q^m)}{r_{BO}}\right) \end{aligned}$$

$$\begin{aligned} st \qquad A^{SC}(q^s, q^m)&\le e_A^0 \nonumber \\ F^{SC}(q^s, q^m)&\ge e_F^0 \\ POF^{SC}(q^s, q^m)&\ge e_{POF}^{^{\prime }} \qquad e_{POF}^{^{\prime }} \in [e_{POF}^0+1, RWV_{POF}^0] \nonumber \\ BO^{SC}(q^s, q^m)&\le e_{BO}^\alpha \qquad e_{BO}^\alpha \in [BO^{opt}, BO^{nad}] \nonumber \\ (q^s, q^m)&\in S \nonumber \\ e_A^0, e_F^0, e_{POF}^{^{\prime }}&\quad \text{ and } \quad e_{BO}^\alpha \text{ are } \text{ integers } \nonumber \end{aligned}$$

(41)

We will prove that the series of models in (41) don’t need to be solved because of equivalence of the model in (41) and its counterpart in (40). Let us consider a concrete model (42) in (40) and a concrete model (43) in (41).

$$\begin{aligned}&\min \; TC^{S\!C}\!(q^s\!,\! q^m) +\! \delta \left( \frac{A^{S\!C}\!(q^s\!,\! q^m)}{r_A} \!-\! \frac{F^{S\!C}\!(q^s\!,\! q^m)}{r_F} \!-\! \frac{POF^{S\!C}\!(q^s\!,\! q^m)}{r_{POF}} \!+\! \frac{BO^{S\!C}\!(q^s\!,\! q^m)}{r_{BO}}\right) \end{aligned}$$

$$\begin{aligned} st \qquad A^{SC}(q^s, q^m)&\le e_A^0 \nonumber \\ F^{SC}(q^s, q^m)&\ge e_F^0 \\ POF^{SC}(q^s, q^m)&\ge e_{POF}^0 \nonumber \\ BO^{SC}(q^s, q^m)&\le e_{BO}^\alpha \nonumber \\ (q^s, q^m)&\in S \nonumber \end{aligned}$$

(42)

$$\begin{aligned}&\min \; TC^{S\!C}\!(q^s\!,\! q^m) +\! \delta \left( \frac{A^{S\!C}\!(q^s\!,\! q^m)}{r_A} \!-\! \frac{F^{S\!C}\!(q^s\!,\! q^m)}{r_F} \!-\! \frac{POF^{S\!C}\!(q^s\!,\! q^m)}{r_{POF}} \!+\! \frac{BO^{S\!C}\!(q^s\!,\! q^m)}{r_{BO}}\right) \end{aligned}$$

$$\begin{aligned} st \qquad A^{SC}(q^s, q^m)&\le e_A^0 \nonumber \\ F^{SC}(q^s, q^m)&\ge e_F^0 \\ POF^{SC}(q^s, q^m)&\ge e_{POF}^{^{\prime }} \nonumber \\ BO^{SC}(q^s, q^m)&\le e_{BO}^\alpha \nonumber \\ (q^s, q^m)&\in S \nonumber \end{aligned}$$

(43)

The following lemma holds:

Lemma 1

An optimal solution of the model (43) must be an optimal solution of the model (42).

Proof

Suppose that the optimal solution of the model (42) is {$TC_1^*,\, A_1^*,\, F_1^*,\, POF_1^*, BO_1^*$} and the optimal solution of the model (43) is {$TC_2^*,\, A_2^*,\, F_2^*,\, POF_2^*,\, BO_2^*$}. It is obvious that {$TC_2^*,\, A_2^*,\, F_2^*,\, POF_2^*,\, BO_2^*$} is a solution of the model (42) since $POF_2^* \ge e_{POF}^{^{\prime }} > e_{POF}^0$. If it is not an optimal solution of the model (42), we will have the following inequality:

$$\begin{aligned}&TC_1^* + \delta \left( \frac{A_1^*}{r_A} - \frac{F_1^*}{r_F} - \frac{POF_1^*}{r_{POF}} + \frac{BO_1^*}{r_{BO}}\right) \nonumber \\&\quad < TC_2^* + \delta \left( \frac{A_2^*}{r_A} - \frac{F_2^*}{r_F} - \frac{POF_2^*}{r_{POF}} + \frac{BO_2^*}{r_{BO}}\right) \end{aligned}$$

(44)

On the other hand, $POF_1^* \ge RWV_{POF}^0$ since $RWV_{POF}^0$ is the minimum of $POF^{SC}(q^s, q^m)$ obtained by solving the series of models in (40). Given $e_{POF}^{^{\prime }} \le RWV_{POF}^0$, we have $POF_1^* \ge e_{POF}^{^{\prime }}$. So {$TC_1^*,\, A_1^*,\, F_1^*,\, POF_1^*,\, BO_1^*$} is also a solution of the model (43). Since the optimal solution of the model (43) is {$TC_2^*,\, A_2^*,\, F_2^*,\, POF_2^*,\, BO_2^*$}, the following inequality holds:

$$\begin{aligned}&TC_1^* + \delta \left( \frac{A_1^*}{r_A} - \frac{F_1^*}{r_F} - \frac{POF_1^*}{r_{POF}} + \frac{BO_1^*}{r_{BO}}\right) \nonumber \\&\quad \ge TC_2^* + \delta \left( \frac{A_2^*}{r_A} - \frac{F_2^*}{r_F} - \frac{POF_2^*}{r_{POF}} + \frac{BO_2^*}{r_{BO}}\right) \end{aligned}$$

(45)

Apparently, inequalities (44) and (45) contradict with each other. So the optimal solution of the model (43) must be the optimal solution of the model (42). $\square$

Similarly, we can obtain the following lemma:

Lemma 2

An optimal solution of the model (42) must be an optimal solution of the model (43).

Based on the above two lemmas, a conclusion can be drawn that the optimal solution of the model (42) and the optimal solution of the model (43) are totally the same. Since the model (42) has been solved, there is no need to solve the model (43). For the same reason, the series of models in (41) don’t need to be solved. As a result, the number of single objective optimization models that need to be solved with the SAUGMECON method is reduced greatly. $\square$