Multi-criteria decision-making of manufacturing resources allocation for complex product system based on intuitionistic fuzzy information entropy and TOPSIS

Complex product system (CoPS) is a kind of technological system with long processing cycle, high complexity and great value of manufacturing resources, which are produced as customized, one-off or small batched capital goods items, such as aircraft engines, offshore oil equipment and cement equipment, distinct from mass-produced commodity products [1]. The manufacturing resources allocation (MRA) problem occurs in many respects for CoPS project, such as project approval and deliver, project schedule, project profitability and product service life, which relates to various types of manufacturing resources, such as equipment resources, human resources, design resources, simulation resources and data resources. The single or small batch, complicated structures and long processing cycle are main characteristics of CoPS which leads to a more complex MRA in resource selection, sequencing and evaluation. Further, MRA occupied a core position in the lifecycle of CoPS, so as to guarantee the reliability and service life of CoPS with lower cost.

In traditional MRA activities, most decision-makers usually make their choice from an intuitive point of view or partial information of CoPS lifecycle. However, for a CoPS, the problems are often not so easy and it is necessary to analyze the information in more detail with considering product lifecycle. Moreover, most of the modeling efforts in this area ensured MRA schemes smoothly only in manufacturing resources optimal stage (MROS) and manufacturing resources execution stage (MRES), while without considering the influence of product maintenance stage (PMS) in lifecycle [2]. For CoPS in a real life, the complexity of manufacturing resources set drives a higher demand for MRA with less expensive cost and reliability requirements.

MROS-MRES-based strategy may lead to the increasing of operating cost of CoPS which has generated customer’s unnecessary expenses and has reduced the reputation of CoPS manufacturing enterprise. Currently, MRA with considering PMS in lifecycle is far more concerned.

In terms of the problems mentioned above, motivations of the research can be concluded as follows:

To our best knowledge, there has no research proposed MRA model for CoPS from the view of product lifecycle. This paper contributed a richer MRA&CoPS multi-criteria decision-making (MCDM) model, which considering among MROS, MRES and PMS in product lifecycle. The model mainly focuses on MRA which simultaneously minimum MROS cost, MRES cost and PMS cost when CoPS delivers to customer. An 8-tuple evaluation criteria with respect to MROS, MRES and PMS are proposed to evaluate MRA for CoPS. Meanwhile, a novel fuzzy technique based on intuitionistic fuzzy sets (IFSs) is developed to express the expert preference and the objectivity of criteria in intuitionistic fuzzy environment. What is more, for such multi-criteria decision-making problem, it is desirable to employ information entropy (IE) to obtain criteria weights in intuitionistic fuzzy environment, rather than assign weights by expert experience. Drawing upon this, hybrid intuitionistic fuzzy information entropy (IFIE) and TOPSIS were often applied to cope with customer-involved MRA process of CoPS.

The remainder of this paper is organized as follows. “Literature review” presents related works. An overview of the background is summarized in “Background”. In “Three-layer criteria cumulative model for MRA&CoPS”, a new variant of MRA multi-criteria decision-making model of CoPS (MRA&CoPS) is proposed to evaluate MRA schemes from CoPS’s lifecycle perspective with respect to MROS, MRES and PMS. The model integrated a three-layer criteria cumulative strategy for process layer, component layer and product layer of CoPS. “The proposed approach” details the proposed approach of IFIE-TOPSIS. An illustrative example is validated in “Case illustration” and conclusions are given in “Conclusion”.

MRA is one of the most important strategic decisions in manufacturing system that has received the growing attention by researchers [2]. Traditionally, the evaluation criteria of MRA problem focus on MROS and MRES, such as time (makespan, tardiness), stability, cost, and reliability. Thekinen and Panchal [3] proposed a MRA evaluation model for matching service seekers and service providers, such as designers and machine owners, in cloud-based design and manufacturing. Chu [4] introduced a MRA method with knowledge-based fuzzy comprehensive evaluation for aircraft structural parts. The capability of planners, complexity of structural parts, reliability of machine tools and reliability of cutting tools are evaluated in the fuzzy comprehensive evaluation method. Wang [5] considered the credit of resource provider and constructed an resource allocation model to improve the efficiency in the process of trading. Liang [6] considered MRES in remanufacturing systems where the stochastic multi-product disassembly line balancing problem with maximal disassembly profit were studied and a chance-constrained programming model was formulated. Xu [7] proposed a bi-level manufacturing resources allocation model under fuzzy environment to satisfy customers’ expectation and to maximize suppliers’ profit for short lifecycles product. Fast non-dominated sorting genetic algorithm is employed to solve their model. Considering the ontology-based static manufacturing resource capabilities and the statistical nature of the manufacturing supply chain. Wu [8] proposed a novel Bayesian approach to produce the optimal and robust manufacturing resource allocation plan. Lee [9] presented an intelligent data management-induced resource allocation system with fuzzy logic. Saidi-Mehrabad [10] developed a liner programming for dynamic manufacturing resources allocation; the objective is to minimize machine maintenance and overhead, system reconfiguration, backorder and inventory holding, training and salary of worker costs. Lin and Chiu [11] applied particle swarm optimization algorithm to solve stochastic resource allocation problem for raising productivity. Bi [12] employed DEA method to generate resource allocation and target setting plan for each production unit. Ayyildiz [13] used the hybrid Best Worst Method (BWM) and Pythagorean fuzzy AHP method to determine the weight of metrics, in order to perform evaluation of the supply chain in the globalizing world.

To show the differences of this paper from literatures related to MRA problem, a systematic state-of-the-art reviews the existing works on the MRA problem corresponding to Table 1 in terms of methods, attributes, stage of lifecycle and problem environment.

It can be seen that the current research mainly focuses on manufacturing resources evaluation and optimization for shorter product lifecycle and time-to-market product, which concerns the MRA influence of manufacturing resources optimal stage (MROS) or manufacturing resources execution stage (MRES) without considering the product maintenance stage (PMS). However, CoPS has the characteristic of long lifecycle. The neglection of PMS for CoPS, such as ignoring the operating cost, may increase customer’s unnecessary cost and reduce the competitiveness of CoPS manufacturing enterprise. Thus, it is necessary to solve MRA problem from the view of MROS, MRES and PMS for MRA&CoPS. There are little previous studies related to MRA considering MROS, MRES and PMS, for example, saidi-Mehrabad [10] proposed a MRA model that considered the influence of the whole product lifecycle. They only focused on solving MRA problem under certain environment, while information for real-world MRA&CoPS are usually imprecise and vague.

Furthermore, decision of selecting the optimal manufacturing resources options are regarded as a multi-objective optimization problem in most studies. However, MRA&CoPS relates to long lifecycle with various influence factors which commonly include qualitative criteria in addition to quantitative criteria. Hence, MCDM is preferred for proposed MRA&CoPS. Indeed, MRA decision-making has been extensively studied in the community of expert and intelligent systems. In contrast to optimization model in previous studies with intelligent algorithms, qualitative or quantitative criteria are used to make the most suitable decision among a large number of alternatives for the MRA&CoPS problem. Compared to the linearized model for the assignment of available resources in dynamic manufacturing systems under certain environment [10], the proposed model is integrated with CoPS’s complex structure and uncertainty of decision information.

Recently, MCDM has received more attention for evaluation and selection problem. Considering uncertain environment, intuitionistic fuzzy sets and TOPSIS are widely used to solve MCDM problem with uncertain evaluation criteria. Fu [14] proposed a new mechanism to re-construct the published composite indicators, and an interval-valued TOPSIS procedure in conjunction with Shannon entropy objective weights was applied for modifying the value measure of health systems. Dwivedi [15] used a combination of subjective weights and entropy weights for each criterion. Then, added weights of PIS and NIS in the final step of TOPSIS to rank candidates. Garg [16] defined a novel algorithm to solve the MCDM process and illustrated numerical examples related to watershed’s hydrological geographical areas, global recruitments problem and so on. Niu [17] introduced two mentality parameters to address the risk attributes and mentality of decision-makers in MCDM process, which was regarded as interval-valued intuitionistic fuzzy numbers. Ulucay [18] considered the occurrences were more than one with the possibility of the same of the different membership and non-membership functions, and defined intuitionistic trapezoidal fuzzy-numbers (ITFM-numbers) in MCDM problem. Hussain [19] investigated the concept of generalized q-rung orthopair fuzzy sets and group generalized q-rung orthopair fuzzy sets to reduce uncertain errors in the original information to ensure the expert’s level of trust and improve the accuracy of final decision in MCDM method technique. Pourmehdi [20] used Analytic Network Process (ANP) and fuzzy TOPSIS to address the performance level of collection centers in reverse logistics, from perspective of sustainability dimensions in supply selection. Roszkowska [21] modified Fuzzy TOPSIS to scoring the negotiation offers in ill-structured negotiation problems. In their research, PIS and NIS are derived from a negotiation theory concept which called BATNA. Joshi [22] defined interval-valued intuitionistic hesitant fuzzy Choquet integral operator to aggregate interval-valued intuitionistic hesitant fuzzy sets (IVIHFS) considering the inter-dependency among the decision criteria, and then used TOPSIS method to rank alternatives in IVIHFS environment. Zhang [23] proposed a rural logistics center location model based on the theory of intuitionistic fuzzy TOPSIS. They calculated weight of decision-makers by rating fuzzy numbers and then determine the weight of the evaluation criteria according to weight of decision-makers. Aikhuele [24] proposed an intuitionistic fuzzy multi-criteria decision-making method to solve failure detection problem. They employed a type of entropy method to calculate weights of criteria, which only consider the membership degree and non-membership degree. Then, they got the relative closeness coefficient based on TOPSIS in intuitionistic fuzzy environment. Awasthi [25] presented a TOPSIS-based MCDM approach for evaluating green supplier under fuzzy environment. The employed linguistic terms to rate criteria and alternatives. Kacprzak [26] used TOPSIS method twice for MCDM problem. The first time it is used to determine the weights of decision-makers and the second time is used to rank alternatives. Chen [27] proposed a MCDM and similarity method between intuitionistic values. They ranked alternatives based on the degree of indeterminacy of each alternative with an extended TOPSIS method. Both the triangular fuzzy number (TFN) and TOPSIS were applied to rank the best alternatives of wind power potential in Weibull distribution model coupled with power law [28].

The research listed above all employed IF-TOPSIS which is mainly for solving selection problems with ranking alternatives by one-layer decision-making. For such problems, the decision-making information can be obtained from the evaluation object comprehensively. However, the structure of CoPS is highly complicated, which is manufactured by many processes with a large number of manufacturing resource alternatives. Assuming that a CoPS is composed of n components {C₁, C₂, …,C_n}, and component C_i (i ≤ n) is manufactured with m_i process \(\left\{ {P_{i}^{1} ,P_{i}^{2} , \ldots ,P_{i}^{{m_{i} }} } \right\}\), process \(P_{i}^{{m_{j} }}\) (\(m_{j} \le m_{i}\)) can be manufactured with \(S_{i}^{{m_{j} }}\) kinds of manufacturing resources \(\left\{ {R_{i1}^{mj} ,R_{i2}^{mj} , \ldots ,R_{{is_{i}^{{m_{j} }} }}^{mj} } \right\}\) and each manufacturing resource has \(A_{i}^{{m_{j} }}\) alternatives (\(A_{i}^{{m_{j} }} \ge S_{i}^{{m_{j} }}\)). The number of MRA schemes alternatives for a CoPS can be calculated as \(N{ = }C_{{A_{i}^{{m_{j} }} }}^{{{S_{i}{m_{j} }} }} \times P_{i}^{{m_{j} }} \times n\), which is a large quantity of alternative manufacturing resources combination schemes for a CoPS manufacturing task. Thus, it is difficult for experts to obtain comprehensive evaluation information of CoPS directly, and leads to more complex and time-consuming calculation and evaluation process of MRA&CoPS when employing one-layer method. To make the evaluation process more suitable for MRA&CoPS problem, this paper proposed a three-layer decision-making model, in which MRA schemes are determined through process dimension, component dimension and product dimension.

CoPS is produced as customized, one-off, or small batched capital goods items with higher complexity and value [41]. CoPS industries is identified as involving technology-intensive capital goods, systems integration, embedded and largely tacit knowledge and skills, project-based manufacturing, low-volume production (batch produced or individually tailored for specific customers), etc. [42]. Based on these studies of CoPS, reference [43] summarized typical manufacturing process of CoPS, i.e., designing activities oriented by customer, technical preparing, procuring raw materials, manufacturing, delivering and maintenance. The multi-level BOM structure is highly complex with many processes, materials and components requirements. In addition, it needs more manufacturing resources to support for the processes, materials and components in demand, while the cost, quality and time, etc. criteria of provided manufacturing resources are various. Moreover, the CoPS’s manufacturing task is decomposed into different sub-tasks and manufactured by the internal or outsourcing providers. It is difficult to evaluate manufacturing resources allocation schema from the overall perspective of CoPS lifecycle. Considering the multi-level BOM structure of CoPS, the mathematical models have been proposed to process layer, component layer and product layer.

Although a number of research have proposed MRA evaluation criteria, the currently studies mainly considers criteria in MROS and MRES and neglect criteria in PMS, such as operating cost. However, as mentioned in “Introduction”, PMS is also important for MRA&CoPS problem. Meanwhile, the current research only consider product layer for MRA problem. In real-life MRA&CoPS problem, data related to the problem are usually provided by resource suppliers. However, it is very difficult for resource suppliers providing data related to MRA&CoPS problem in product layer directly after the complex manufacturing of CoPS. Hence, the three-layer criteria cumulative model is proposed from process layer, component layer and product layer to evaluate manufacturing resources allocation of CoPS.

The CoPS’s lifecycle involves MROS, MRES and PMS, so the proposed 8-tuple criteria are summarized from three stages in this section. Further, the proposed evaluation criteria are successively accumulated in process layer, component layer and product layer.

An improved MCDM approach for MRA&CoPS is proposed, and the approach is mainly divided into the following three stages: (1) preparation stage: DMs determine alternatives (process layer, component layer and product layer), and define evaluation criteria and collect information of criterion for three layers of CoPS. (2) Calculation stage: DMs calculate fuzzy value by fuzzy method proposed in “Fuzzy process” to construct normalized fuzzy decision matrix. Then IE is extended to intuitionistic fuzzy environment as a weighting method to calculate weight of each criterion. Finally, DMs calculate \({\mathrm{D}}_{\mathrm{j}}^{*}\)\(D_{j}^{*}\) by TOPSIS in intuitionistic fuzzy environment. (3) Output stage: DMs sort the chosen alternatives by value \({\mathrm{D}}_{\mathrm{j}}^{*}\)\(D_{j}^{*}\) in decreasing order and judge the dimension. If the evaluation dimension is product layer, the minimum ranking order is the optimal MRA schema for CoPS. Otherwise, re-evaluate MRA process in next layer. Figure 2 illustrates the conceptual framework of the proposed method.

This section illustrates a case example to demonstrate the effectiveness of the proposed method to optimize MRA&CoPS. In this study, the case product cantilever rotary stacker produced by the case manufacturing enterprise in Tianjin of China is presented to test and to show the applicability and effectiveness of proposed model. The main manufacturing product of the case manufacturing enterprise is complicated and costly equipment, such as cement equipment, metallurgy equipment, and coal equipment. In addition, the customer may refuse to pay the warranty if the product broke down during one to two warranty periods. As Fig. 4 shows, the case product is mainly composed of stacker belt conveyor, hydraulic lift and belt speed control device, each of which is mainly composed by three key processes. In addition, each process can be completed by several kinds of different resources. To simplify the explanation and make this section easier to read, “Process layer” and “Component layer” will only illustrate the resource combination optimization process of stacker belt in process layer and component layer. “Product layer” will illustrate the resource allocation process of the case product.

MRA is essential in manufacturing resources decision-making of CoPS. However, the complex structure, long lifecycle and combined explosion, as well as incompleteness and fluctuation of decision information often resulted in the manufacturing resources decision-making a challenging task. It is highly desired to obtain MRA schemes, in terms of trade-off among CoPS’s manufacturing, operation and maintenance from the viewpoint of CoPS’s lifecycle. In this investigation, a MCDM model for MRA&CoPS is proposed with considering lifecycle-oriented 8-tuple evaluation criteria via three-layer accumulation. In addition, a hybrid IFIE-TOPSIS method by combining information entropy and TOPSIS has been extended in intuitionistic fuzzy environment to deal with the proposed MRA&CoPS model. The model is demonstrated with an example in a complex cement equipment manufacture company. Simulation results and sensitivity analysis have shown the effectiveness and superiority of proposed IFIE-TOPSIS method over classical TOPSIS.

MRA decision-making is increasingly becoming an integral component of the fully functioning expert and intelligent systems with applications across various manufacturing systems, especially for CoPS. Among various MRA models, the proposed MRA&CoPS model pays attention to MRA of CoPS with the viewpoint of trade-off among manufacturing, operation and maintenance of CoPS’s lifecycle. The proposed mathematical model can be categorized as a straight fuzzy MCDM (FMCDM) strategy according to the manufacturing resources evaluation framework. As a result, various alternatives of manufacturing resources should be identified to support different processes, components and products for a CoPS. In practice, medium-/long-term contracts should be made with the selected groups of manufacturing resources suppliers for the stable cooperation.

In the aspect concerning the limitations of this paper, it is important to notice that the 8-tuple evaluation criteria performance is possibly changed dynamically. Further, the proposed method stands in the view of manufacturers, whereas the CoPS’s lifecycle involves other parts, such as supplier, designer and costumer. Mover, the selected optimal MRA is a constant result for CoPS, while some manufacturing resources may change over time, such as the workers’ and production facilities’ productivity. Concerning the limitations of this method proposed in previous, the suggestions for further research are listed as follows.