Multi-criteria evaluation model: an application of MRP parameterization

This paper presents a multi-criteria evaluation model applied to the parameterization of the MRP method. Existing optimization approaches that address this problem tend to adopt a means of simulation. A simulated solution is characterized by a pair (parameters, performance indicators). In the context of the evaluation of solutions, the work of Barth, Damand et al. (2003) propose a heuristic approach to extracting knowledge from a solution set. The approach is based on the definition of a multi-criteria solution comparison function. The objective of this paper is to present the detailed modeling of this comparison function. Ultimately, this result contributes to the formalization of a multicriteria optimization problem. A problem solving strategy is proposed.


Introduction
Parameterization of the MRP method consists of dynamically specifying all the values of the method's input parameters (AP), taking into account the performance objectives, the parameter constraints and any new aspects of the situation under consideration. In some cases, it is possible to respond to the evolutions rapidly through programmed decisions. In the case of this paper, the decision-maker, hereinafter called "the planner ", does not have the prerequisites for the action, but is in a problemsolving situation. Conducting research implies having a means of simulation that can answer 'what if' type scenarios. The output values of the method enable the system's performance to be measured, generally expressed in performance indicators (PI). A solution is thus described as a relationship between an AP configuration and a PI configuration (hereinafter called an AP-PI relation). The evaluation of the set of solutions allows the properties characterizing the APs, the PIs and the AP-PI relations in the situation under consideration to be identified. The properties are inferred from an interaction between the planner's assessment questions and his representation of the overall structure of a set of solutions. In the performance space achieved, example generic evaluation questions include [2,6,8,15]: -What is the effect of APs on PIs? -Which performance areas are attainable? -Are there any equivalent solutions? -How does an AP-PI relationship stand with regard to other relations? -How does an AP-PI relationship stand in the absolute (rejection, waiting, acceptance)?
The type of questions presented above deal with issues concerning the multi-criteria decision-making support (sorting (), ranking () describing () selecting ()) introduced by [18]. In this case, the methods must, first, include the decider's preferences and second, propose an intelligible global representation depending on the issues under consideration. In the context of the parameterization of the MRP method, in [2], the authors put forward a model called the plan of preferences, which combines both the sorting and the ranking issues. As a research perspective, the authors motivate a performance exploration problem in the plan of preferences. The problem described is a combinatorial optimization problem. Two optimization problems are formulated as follows: (A) What are the best least good solutions with respect to the models of preference (ranking and sorting)? (B) Are there any solutions in this or that area of performance of the plan of preferences? The multi-criteria comparison function of the solutions is defined. However, the authors did not formalize the method required to construct the proposed model. The aim of this paper is therefore to formalize this method.
The paper is divided into 5 sections. Section 2 describes the theoretical background. Section 3 presents the plan of preferences concept. Section 4 describes the formalization of the plan of preferences construction method. Finally, section 5 concludes with the research perspectives generated by this work.
One recent state of the art [10], based on an analysis of 87 papers from 1976 to 2017, listed studies involving 31 APs, 29 PIs and 247 AP-PI relations. Not all the relations have been studied, however. The most widely studied APs are: demand variations, lot-sizing rule, planned lead time, freezing proportion, planning horizon, product structure, schedule rule, and replanning periodicity. Studies are mainly distinguished by the operating conditions and the number of AP-PI relations studied. Operating conditions are characterized in four ways: number of manufacturing levels (single-level or multilevel), type of demand (deterministic or stochastic), capacity (uncapacitated systems or capacitated systems), and item numbers (multiple or single). In [10], the authors highlight the contextual nature of the findings above. In effect, results may vary depending on changes to the operating conditions. For example, the relations 2 and 3 above are contradictory. The experiments by [21] and [1] respectively differ according to the operating conditions linked to the 'capacity' modality.
It is unlikely that the results of these studies could be used as a decision-making support for the parameterization of the MRP method in general cases. However, it would be feasible to use the 'simulation-experimental design' methodology linked to a set of AP-PI relations and interpreted by the planner of specific systems.
In such a case, one difficulty would lie in the multicriteria evaluation of the AP-PI relations. A parameterization study may include several hundred multicriteria relations; the study by [16] for instance is described as follows: 6 PI, 4 AP, 448 simulated AP-PI relations.

The plan of preferences concept
To facilitate multicriteria evaluation and comparison of AP-PI relationships in a specific environment, the paper titled "How Can We Ascertain, Understand and Interpret the Performance Level of A Production System? A Visual Method: The Plan of Preferences" [2] propose a way of considering support for the evaluation from a set of AP-PI relations thanks to the description of the properties related to ranking (rank of a AP-PI relationship) and sorting (class of a AP-PI relationship). The issues pertaining to ranking and sorting are complementary. The drawback in the case of ranking is that the evaluation of the AP-PI relationship is made with respect to the other relations. Thus, a situation can very well emerge as the best of a set of relations, even though it is bad in the absolute. Sorting, on the other hand, involves steering the investigation towards highlighting an assignment of relationships to predefined classes. These predefined classes specify the absolute value of relations. "Relative and absolute" evaluations correspond to two value judgements identified in cognitive psychology [3].
The AP-PI relations are represented on a plan ( Figure  1). The X and Y axes respectively support the two relations of preference (sorting and ranking). The coordinates of a solution s1 (a specific AP-PI relation) are: X assigned class C1, Y rank R1. This involves dialogue with the representation to extract properties. In [2], the authors suggest beginning the dialogue between the decision-planner and the plan of preferences with the following intuitive question: "How does the initial solution fit in with the new situation?" Subsequent questions are formed and linked together according to the interpretations of the properties extracted from the plan of preferences. This method draws on gradually acquired information. The decision planner intervenes throughout the multicriteria analysis to give information on his or her true preferences. Using the preference plan, the multicriteria method helps decisionmakers construct their overall preferences (called preorder of the decision-maker [18]).
This outcome is not only based on the logic of the preference plan, but also on the logic of the decision planner as it is revealed from information obtained at different stages of the evaluation process. These revelations, however, remain within the logical framework set by the model. This takes us on to the following section.

Formalisation of the plan of preferences construction method
This section presents respectively the formalization of AP-PI relations, scales of preferences supported by the XYaxes and the evaluation rules of the plan of preferences. The set of AP-PI relations derived from simulation are summarized in a table called the AP-PI table (Table 1).  Let's assume that the system is controlled by t AP i parameters. Each parameter is associated with several levels. If the parameter is qualitative, the number of levels will correspond to the number of possible parameter conditions. If the parameter is represented by a quantitative variable, only a finite number of possible values for the parameter will be retained. If we note Ni as the number of levels respectively associated with the APi parameters, the potential number of lines on the table will be: ∏ . The analysis space is the space whose dimensions are represented by performance indicators. In a multicriteria approach, solutions are estimated by analyzing the values they take for each PIj. Moreover, the PIj are real variables. The PIj space is thus always mathematically represented.

AP-PI table
To analyze the structure of the performances achieved, we need to prepare the information contained in the AP-PI table for this purpose.

Construction of the ranking relationship
Ranking consists of differentiating the solutions according to their relative interest in a decreasing order of preference. In the study context, PIs are highly heterogeneous and difficult to compare. Thus, an outranking method has been adopted. This approach is based on an elementary mechanism, namely, the pairwise comparison of solutions, PI by PI. It involves a set of conditions leading to the acceptance or rejection of an outranking at global level.
The family of outranking methods is dominated by the ELECTRE method [18] and the PROMETHEE method [4]. The outranking method applied is the PROMETHEE II method [4,5]. This is based on concepts with a physical interpretation that are easy for the decision-planner to understand. In addition, the nature of the outranking hypothesis introduced in the PROMETHEE method allows for the graduation of outranking credibility. This is the "... better than ..." relationship richer within the framework of the study of this paper than the "… at least as good as…" relationship of the ELECTRE method. When a si solution outranks a sk solution, it is indeed important to know the degree of credibility that can be attached to the statement "s i outranks s k ".
The principles of building and exploiting the ranking relations adapted to the problem are developed below.

Exploitation of the ranking relationship
The PROMETHEE method translates the outranking relationship through the information pertaining to the flow. The si solutions are positioned on a scale of preference whose states represent the solutions qualification ϕ (si). To visualise the ranking, the scale of preference is supported by a vertical and restricted axe, called Y-axis (

Construction of the sorting relationship
The sorting relationship is developed within the framework of the study of this paper.
Sorting consists of examining the values of each solution's PIj in order to put forward a recommendation from a set of possible recommendations, specified in advance. By linking a class to each recommendation, the problem boils down to the allocation of solutions to predefined classes. These classes are ordered and correspond respectively to the following judgements: "accepted or good", "pending or average" and "rejected or bad". The "good, average and bad" judgements are generally accepted by the decision-makers. The concepts relative to the definition of classes are defined below.
To express the users' preferences, each PIj is segmented  The segmentation of each PIj indicator makes it possible to characterize each si solution by a triplet (a(si), b(si), c(si)), where a(si), b(si) and c(si) are respectively the number of indicators whose value is judged as good for the si solution, the number of indicators whose value is judged as average for the si solution and the number of indicators whose value is judged as bad for the si solution.
The solutions characterized by the same triplet belong to the same class leaf. The potential number of classes is (with m: number of PI). These classes form a partition of all the performances. An si solution belongs to just one class and one only. For any given system, some classes may be empty. A total order is defined for all class leaves. Let there be two X and Y classes defined respectively by the triplets (ax, bx, cx) and (ay, by, cy). The relation of preference (>) between the classes is defined as follows: The conditions of preference define a total pre-order of solutions as several solutions may be assigned to a same class leaf. In this case, the solutions are equivalent. The relation of preference (³) between the solutions is defined as follows: Lastly, the class leaves are partitioned into three aggregate classes: 1 class defined by b = 0 and c = 0 (called "Good"); m classes defined by c = 0 and b  0 (called "Average"); ( +1) 2 classes defined by c  0 (called "Bad").

Exploitation of the sorting relationship
The successive definitions allow us to organize these in a hierarchy of partitions (Hata! Başvuru kaynağı bulunamadı.).
The class leaves are positioned on a scale of preference whose states represent the 1+m+

XY-plane, the rules of evaluation of the plan of preferences
The plan of preferences ( Figure 4) is obtained by crossing the Y-axis and the X-axis. The point of intersection of the coordinate axes is (Cmean, smean). Each si solution is represented by a coordinate point (Class leaf (si), ϕ (si)). The rules of evaluating preferences are illustrated and expressed as follows:  (17) With, "=": "is equivalent to"; ">": "is preferred to"; "<": "is not preferred to" The two axes discussed (via case 1, 2, 3, 4, 5 below) in the XY plane are complementary. The comparison between solutions according to the two axes of preference allows us to separate strong or robust preferences (case 1, 2, 3, 4) from the other non-robust preferences (case 5). A nonrobust preference corresponds to a contradictory ranking in accordance with the sorting and ranking axes. The contradictory rankings enable several preferred and incomparable solutions ({s1, s2}, case 5) to be identified in accordance with the two axes of preference.

Conclusion and perspectives
The actions described in this paper may be summed up as follows: -Motivate the theoretical background: there are no general AP-PI relation, -describe the concept of the plan of preferences, -formalize the two preference comparison relations (ranking and sorting) and the rules for reading a solution si (Class leaf (si), ϕ(si)). The ranking method applied is the PROMETHEE II method. The sorting method was constructed in the study of this paper.
In view of the work undertaken, the results of this paper, especially the detailed modelling of the two relations of preference (ranking and sorting), contribute to the formalization of the issues of combinatorial optimization identified in [2]. The resolution of problems (A) and (B) (see Introduction and reformulated below), is now both conceivable, via optimization methods.
(A) What are the best least good solutions with respect to the models of preference (ranking and sorting)?
(B) Are there any solutions in this or that area of performance of the plan of preferences?
To conclude, it is proposed to continue the work already conducted in the framework of this paper, as a strategy of problem-solving for general cases using the solutions to problems A and B: (1) Finding a solution to problem A with optimization methods and an initial definition of the performance zones accessible and non-accessible by the systems.
(2) Drawing up experimental designs which are as broad-sweeping as possible in view of the calculation potential.
(3) Applying the approach put forward in this paper to the dots calculated and deducing hypotheses on the structure of the performance levels.
(4) Fine-tuning the investigation and validating or invalidating the hypotheses by the detailed solving of multiple problems of type problem B.