The Contribution concept for the control of a manufacturing multi-criteria performance improvement

Abstract

By dealing with an overall manufacturing performance improvement context, we introduce in this paper the “improvement contribution” concept. A framework that integrates such a concept to the quantification of a multi-criteria interacting performance is proposed. The improvement contribution is defined as a new intelligent functionality that quantifies the impact of the improvement of a single (or a set of) mono-criterion performance(s) on the improvement of an overall performance. When performances are interacting, the quantification of such a contribution cannot be direct. The proposed approach consists of an extension of a previously developed Performance Measurement System (PMS). The considered PMS integrates an aggregation operator—the Choquet Integral (CI)—for the expression of an overall performance by handling weights and interactions between the mono-criterion performances. The principles of the improvement contribution and its quantification are thus presented in addition to the way the improvement contribution can be used for helping decision-makers in their manufacturing improvement control. As an illustration, the use of these contributions within successive iterations of improvement actions is shown using a case study submitted by the SME Fournier Company.

Appendices

Appendix 1: Fundamentals of the choquet integral (CI)

The initial form of the CI operator

To deal with the involved performance interactions, the CI operators provide powerful aggregation operators. To be more precise, the CI operators can be seen as an extension of the WAM operators, and are able to consider not only weights for mono-criterion performances but in addition weights for any subset of performances (Grabisch 1996; Labreuche and Grabisch 2003). However, such a model requires ($2^{n}-2)$ parameters in order to take into account the weight of each possible subset of performances (from 1 to n corresponding mono-criterion performances). This represents a huge identification task in practice. Hence, more tractable models have been introduced with a sufficient information level, e.g., the 2-additive CI that only considers weights for pairs of performances (called in that case interactions between 2 (Grabisch 1997). These operators are a kind of compromise between complexity and richness of the model, since only $\left( {\frac{n(n+1)}{2}} \right)$ parameters are required. We propose hereafter notation in accordance with the PMS ones, where $p_i^A$ is a given mono-criterion performance (defined on [0,1]) at the state A and $p_j^A$ is another one mono-criterion performance at the same state A.

2 kinds of coefficients are involved in a 2-additive CI-based aggregation model:

• the weight of each mono-criterion performance in relation to the overall performance by the so-called Shapley coefficients $\phi _{\mathrm{i}}^{\prime }s$, that satisfy the condition$\sum _{{\mathrm{i}}={1}}^{\mathrm{n}} {\phi _{\mathrm{i}}} =1$ (which is a quasi-natural condition);

• the interaction coefficient of any pair of mono-criterion performances $p_i^A$ and $p_j^A$, denoted $I_{\mathrm{ij}}$. $I_{\mathrm{ij}}$ ranges within the interval $\left[ {-1;1} \right]$. The “−1“ value expresses a complete redundancy between 2 mono-criterion performances, whereas the “−1“ value means complete complementarity of these mono-criterion performances.

Thus, the overall performance $p_{Ov.}^A$ aggregated by a 2-additive CI is given by (Grabisch 1997):

$$\begin{aligned}&p_{Ov.}^A =CI(\vec {P}^{A})=\sum \nolimits _{{\mathrm{i}}={1}}^{\mathrm{n}} {(\phi _{i}} -\frac{1}{2}\sum _{j\ne i} {\left| {I_{ij} } \right| }) p_i^A\nonumber \\&\quad +\sum _{I_{ij} >0} {\left| {I_{ij} } \right| } \min (p_i^A ,p_j^A )+\sum _{I_{ij} <0} {\left| {I_{ij} } \right| } \max (p_i^A ,p_j^A ) \end{aligned}$$

(4)

with the property $\left( {\phi _{\mathrm{i}} -\frac{1}{2}\sum \limits _{j\ne i} {\left| {I_{\mathrm{ij}}} \right| } } \right) \ge \hbox { }0,\hbox { }\forall \hbox { }i\hbox { }\in [1,n]$ in order to ensure the monotonicity of the aggregation operator.

Note that the mechanism of determination of the Shapley and interaction coefficients is based on the expertise of the decision-maker. The handling of this expertise may be carried out with the MACBETH method in a coherent way according to the measurement theory (Krantz et al. 1971). Readers interested in the details of this method can find more information in (Clivillé et al. 2007, Bana e Costa et al. 2012).

Example 4

As an illustration of the CI performance aggregation model, let us consider the case study of § 5 which involves 4 mono-criterion performances. Thus, by applying (4) to this particular case, the overall performance associated to state $p_{Ov.}^A$ becomes:

$$\begin{aligned}&p_{Ov.}^A =0.05p_1^A +0.1p_2^A +0.075p_3^A +0.025p_4^A \\&\quad +0.3\min (p_1^A ,p_2^A )+0.2\min (p_1^A ,p_3^A )\\&\quad +0.25\min (p_3^A ,p_4^A ) \end{aligned}$$

The piecewise linear form of the CI operator

By examining (4), one can see that it is possible to eliminate the min and max operators by considering the ranking of any pair of performances$\left( {p_i^A ,p_j^A } \right)$:

• When $p_i^A >p_j^A$, the term $\sum \limits _{I_{ij} >0} {\left| {I_{ij} } \right| } \min (p_i^A ,p_j^A )+\sum \limits _{I_{ij} <0} {\left| {I_{ij} } \right| } \max (p_i^A ,p_j^A )$ of (4) becomes $\sum \limits _{I_{ij} >0} {\left| {I_{ij} } \right| } p_j^A +\sum \limits _{I_{ij} <0} {\left| {I_{ij} } \right| } p_i^A$; and we have in this case $\Delta p_{Ov.}^A =\sum \nolimits _{{\mathrm{i}}={1}}^{\mathrm{n}} {w_i^{\prime } } p_{i}^A$

with

$$\begin{aligned} w_i^{\prime }= & {} \phi _{i} -\frac{1}{2}\sum _{j\ne i} {\left| {I_{ij} } \right| } +\sum _{I_{ij}<0} {\left| {I_{ij} } \right| }\nonumber \\= & {} \phi _{i} -\frac{1}{2}\sum _{I_{ij} >0} {\left| {I_{ij} } \right| } +\frac{1}{2}\sum _{I_{ij} <0} {\left| {I_{ij} } \right| } \end{aligned}$$

(5)

• When $p_i^A <p_j^A$, the term $\sum \limits _{I_{ij} >0} {\left| {I_{ij} } \right| } \min (p_i^A ,p_j^A )+\sum \limits _{I_{ij} <0} {\left| {I_{ij} } \right| } \max (p_i^A ,p_j^A )$ of (4) becomes $\sum \limits _{I_{ij} >0} {\left| {I_{ij} } \right| } p_i^A +\sum \limits _{I_{ij} <0} {\left| {I_{ij} } \right| } p_j^A$; and we have $\Delta p_{Ov.}^A=\sum _{{\mathrm{i}}={1}}^{\mathrm{n}} {w_{i}''} p_i^A \quad \Delta p_{Ov.}^A =\sum _{{\mathrm{i}}={1}}^{\mathrm{n}} {w_{i}''} p_i^A$ with

  $$\begin{aligned} w_i''= & {} \phi _i -\frac{1}{2}\sum _{j\ne i} {\left| {I_{ij} } \right| } +\sum _{I_{ij} <0} {\left| {I_{ij} }\right| }\nonumber \\= & {} \phi _i -\frac{1}{2}\sum _{I_{ij}>0} {\left| {I_{ij} } \right| } +\frac{1}{2}\sum _{I_{ij} >0} {\left| {I_{ij} } \right| } \end{aligned}$$

  (6)

Example 5

Let us consider again the example 4 corresponding also to example 3 in § 3.3.2

When $p_1^A>p_2^A>p_4^A >p_3^A$ then $w_1 =0.05,w_2 =0.3+0.1=0.4,w_3 =0.075+0.2+0.25=0.525,w_4 =0.025$.

When $p_1^\alpha>p_2^\alpha>p_3^\alpha >p_4^\alpha$ then

$$\begin{aligned} w_1= & {} 0.05,w_2 =0.3+0.1=0.4,\\ w_3= & {} 0.075+0.2=0.275,w_4 =0.025+0.25=0.275 \end{aligned}$$

When $p_2^A>p_1^A>p_3^A >p_4^A$ then

$$\begin{aligned} w_1= & {} 0.05+0.3=0.35,w_2 =0.1,\\ w_3= & {} 0.075+0.2=0.275,w_4 =0.025+0.25=0.275 \end{aligned}$$

This result can be generalized to any Choquet integral beyond the 2-additive ones (Montmain et al. 2005).

Appendix 2: Improvement contribution quantification without knowledge of the intermediate states

In § 3.3, the improvement contribution quantification is based on the knowledge of the performance ranking changes. When it is not the case, i.e. the intermediate states between states A and B are not known, the preceding approach is not applicable. Nevertheless, it is possible to quantify the lower and upper values of the improvement contribution, by maximising, and respectively minimising, versus $p_i^A$, the non-linear part of the CI, i.e. $\sum \limits _{I_{ij} >0} {\left| {I_{ij} } \right| } \min (p_i^A ,p_j^A )+\sum \limits _{I_{ij} <0} {\left| {I_{ij} } \right| } \max (p_i^A ,p_j^A )$ leading thus to $C_i^{A\rightarrow Bup}$ and $C_i^{A\rightarrow B\hbox { }low}$.

Example 6

Let us consider again the preceding example where the mono-criterion performances improve from $\vec {P}^{A}=\left( {0.6,0.55,0.2,0.35} \right)$ to $\vec {P}^{B}=\left( {0.6,0.7,0.5,0.5} \right)$ . Therefore the upper and lower improvement contributions are quantified as follows:

$C_3^{A\rightarrow B\hbox { }low} =0.525\times 0.15+0.275\times 0.15=0.12$ and $C_3^{A\rightarrow B\hbox { }up} =0.525\times 0.3=0.1575$

$C_4^{A\rightarrow B\hbox { }low} =0.025\times 0.15=0.00375$ and $C_4^{A\rightarrow B\hbox { }up} =0.275\times 0.15=0.04125$

In this case, whatever the improvement way the contribution of improving $p_3^A$ notation is much higher than improving $p_4^A$ notation.