1. INTRODUCTION
Manufacturing enterprises should adapt themselves to new business environment with diverse customer requirements (Peng et al., 2014). The manufacturing enterprises should retain various manufacturing resources to increase the adaptability. The manufacturing resources are divided into hard resources (e.g., manufacturing equipment, computational resources, material, storage) and soft resources (e.g., software, knowledge, personnel) (Wang and Xu, 2013). Since these resources are expensive for small and medium sized enterprises to retain, they cannot maintain sufficient resources and, therefore, often cannot meet the diversified requirements. As one of the solutions for this insufficient resource problem, innovative manufacturing models based on collaboration between the enterprises have recently been developed (Leng and Jiang, 2018;Tseng and Huang, 2009). Enterprises participating in these models produce customized manufacturing services collaboratively, which cannot be completed by a single company. The collaboration activities include information sharing (Ming et al., 2014), incentive sharing (Ding et al., 2016) and capacity sharing (Min et al., 2005).
Cloud manufacturing (CM), a collaborationbased manufacturing model, has received much attention from both industry and academia. CM was first introduced by Li et al. (2010), where they define it as “a computing and serviceoriented manufacturing model developed from existing advanced manufacturing models and enterprise information technologies under the support of cloud computing, the internet of things (IoT), virtualization and service oriented technologies, and advanced computing technologies.” These information and communication technologies enable efficient collaborations in CM. For example, cloud computing supports customers as they enroll their request documents (i.e., ondemand access), and also allows manufacturing resources that enterprises retain to be interconnected in the cloud environment (Ren et al., 2017). Namely, cloud computing enables enrollment of the customers and enterprises, and connects them in the cloud environment. As another example, IoT not only informs the customers of the manufacturing states regarding their requests, but also notifies the manufacturing process states that they are collaboration partners for communication efficiency (Guo et al., 2015).
Operation processes of the CM are as follows (Huang et al., 2013). First, a customer enrolls his or her request. Depending on the request content and statuses of the enterprises in the CM (e.g., available resources, capacity), a proper enterprise is selected to complete the request. If the request is too large to be completed by a single enterprise, then two or more enterprises are selected to complete the request through collaboration. Note that most requests are carried out by two or more enterprises. Once the enterprises are selected and they undertake the project to produce the manufacturing service, they freely share their resources and communicate with each other based on information and communication technologies. After the manufacturing project is completed, the customer evaluates the manufacturing service he or she obtained (Lu and Xu, 2017). For example, customers in MFG.com, which is one of the most promising CM companies, evaluate manufacturing services provided based on four criteria: project rating, quality, responsiveness, and delivery (Moore, 2016).
Since the requests in the CM are large, collaboration potential between enterprises participating in the CM highly impacts customer satisfaction (Ahn et al., 2016;Ahn et al., 2017). Collaboration potential is a term that denotes the expected effect of collaboration widely used in many fields such as scientific collaboration (Giuliani et al., 2010), collaboration in software projects (Magdaleno et al., 2015), and so on. For example, Giuliani et al. (2010) defines collaboration potential in scientific collaboration as the possibility that two researchers can collaborate effectively with each other. In this paper, the collaboration potential between enterprises in the CM is defined as the expected customer satisfaction for the manufacturing service provided through collaboration. In other words, an enterprise group that has higher collaboration potential will receive a higher rating from a customer than another group that has a lower collaboration potential. It is obvious that collaboration potential is one of the main factors affecting customer satisfaction, but there is no previous research developing a collaboration potential estimation model in CM.
In the CM, the task allocation problem is to allocate a customer request to a set of enterprises, which is one of the main operational issues in the CM. The task allocation considering collaboration potential in CM is to group the set of enterprises whose collaboration potentials are relatively high (i.e., the group will satisfy a customer with high probability), and then allocate a task to a proper set of enterprises. Although there is no research that solves the task allocation problem by considering the collaboration potential, some researchers attempted to solve it by considering geographical location and provided service type, which may impact the collaboration potential as summarized in Table 1.
For example, Liu et al. (2016) solved a manufacturing task scheduling problem considering task attributes, enterprise resources, and geographical distances between enterprises. The task attributes include urgency, workload and latency time, while the enterprise resources are turning, boring and welding. Liu et al. (2017) employed the GaleShapley algorithm to match each request and a set of enterprises in CM, considering the enterprise preference and geographical distances between enterprises. However, these researches did not consider collaboration potential directly.
This paper addresses dynamic estimation of collaboration potential problem, which is to estimate collaboration potentials of all enterprise pairs at next time period given collaboration potentials at current time period. Collaboration potential between enterprises can be estimated based on their rating histories in the CM, since the collaboration potential between enterprises highly impacts customer satisfaction. A Markov random field (MRF) model is developed to solve the problem because MRF can represent complicated relationships among many random variables as probabilistic joint distribution. That is, the considered problem has many random variables and there are complicated relationships among the variables. To be more concrete, the number of variables is ${(}_{2}^{n})\times t$, where n and t denote the number of enterprises and time periods, respectively. Therefore, MRF is employed to analyze the complicated relationship in this paper.
The rest of this paper is organized as follows. Section 2 describes the collaboration potential estimation problem including assumptions and notations is described in Section 2. Section 3 proposes the estimation model and explain how to construct the model and use it. Section 4 provides an example to illustrate the application of the proposed model. Finally, Section 5 conclude this research.
2. PROBLEM DESCRIPTION
The problem considered in this paper is to estimate the collaboration potentials among enterprises participating in the CM by utilizing the expected customer satisfaction rating. Such a rating in CM depends on the potential of the enterprise groups that complete the customer request. Customer satisfaction also depends on time, since the collaboration potentials may change with time. Time periods are considered to reflect that collaboration potentials are time varying. Accordingly, the problem is to calculate the collaboration potentials in time period t, given the collaboration potentials in time periods t − 1, t − 2, …, and t−τ , where τ denotes the number of time periods used as inputs.
Assumptions for the problem considered in this paper are as follows.

(Assumption 1) All the combinations of two enterprises collaborate with each other to produce manufacturing service at least once in each time period. In other words, all the combinations receive one or more customer satisfaction ratings for each time period.

(Assumption 2) The customer rating scale is a Likert type scale ranging from 1 to 5, where 1 represents “very dissatisfied” while 5 is “very satisfied.”

(Assumption 3) If three or more enterprises collaborated with each other and received m (m=1, 2, 3, 4, 5) points, then we assume every combination of two enterprises participating in the collaboration received m points. For instance, if enterprise 1, 2 and 3 collaborated with each other and received m points, then all three combinations (enterprise 1, enterprise 2), (enterprise 2, enterprise 3), and (enterprise 3, enterprise 1) receive m points.

(Assumption 4) Averages of customer ratings for two enterprises in time periods t and t−1 are positively correlated.
Notation used in this paper is presented in Table 2.
3. PROPOSED MODEL
Figure 1 shows the flow of the proposed model.
As seen in this figure, the process consists of the five steps: (1) defining random variables, (2) defining potential functions, (3) designing model, (4) training model, and (5) calculating collaboration potential. In the first step, random variables that express average of customer satisfaction ratings of two enterprises in each time are defined. In the second step, three types of potential functions that reflect relationships among the defined random variables are defined. In the third step, probability model to calculate collaboration potential is designed using the potential functions. The designed model includes unknown weights, which are estimated in the fourth step. In the final step, collaboration potential for all possible combinations of enterprises are calculated using the designed model with estimated weights.
Let ${S}_{i,j}^{t}$ be an average of customer satisfaction ratings for manufacturing services that any two enterprises E_{i} and E_{j} provided in time period t. For simplicity, we introduce a new variable ${X}_{i,j}^{t}$ that takes a rounded value of ${S}_{i,j}^{t}$. Namely, ${X}_{i,j}^{t}$ is defined for all i<j as follows:
Note that ${X}_{i,j}^{t}={X}_{j,i}^{t}$, and ${X}_{i,i}^{t}$ is not defined. Accordingly, ${X}_{i,j}^{t}$ is defined only for i<j≤n, where n denotes the number of companies in CM.
Figure 2 is a diagram of relationships that exist among ${X}_{i,j}^{t}$ for i< j ≤ n. As shown in Figure 2, there exist relationships between ${X}_{i,j}^{t}$ and ${X}_{i,k}^{t}$ for i < j ≠ k, and between ${X}_{i,j}^{t}$ and ${X}_{k,j}^{t}$ for k ≠ i < j.
According to the fourth assumption addressed in Section 2, ${X}_{i,j}^{t1}$ and ${X}_{i,j}^{t}$ are positively correlated for all i, j, t. Taking the correlations into consideration, we define three potential functions, which comprise an MRF as follows:
The potential function φ_{1} in Equation (2) is introduced by Assumption 4. The potential functions φ_{2} and φ_{3} in Equations (3) and (4) are constructed by the fourth assumption in Section 2.
An MRF efficiently yields the joint probability distribution of all variables, as follows (Murphy, 2012):
where c_{k} is the clique k in the set of all cliques C, φ_{l} denotes the l^{th} potential function, w_{l,k} is the weight assigned to the φ_{l} for c_{k}, and Z is a normalization function that ensures P(X = x) is a valid probability distribution, where Z is calculated as follows:
Then, the joint probability distribution of ${X}_{i,j}^{t1}$ and ${X}_{i,j}^{t}$ for all i and j can be expressed as follows:
X^{t} is a vector whose elements are ${X}_{i,j}^{t}$ for all i and j, and ${\alpha}_{i,j}^{t},{\beta}_{i,j,k}^{t},\hspace{0.17em}\text{and}\hspace{0.17em}{\gamma}_{i,j,k}^{t}$ denote the weights of ${\varphi}_{1}\left({X}_{i,j}^{t1},\hspace{0.17em}{X}_{i,j}^{t}\right),\hspace{0.17em}\text{and}\hspace{0.17em}{\varphi}_{3}\left({X}_{i,j}^{t},\hspace{0.17em}{X}_{k,j}^{t}\right)$, respectively. This probability is a basis used to calculate the expected customer rating for all enterprise groups in time period t. The normalization function at time period t, Z(t), is given by:
The log likelihood regarding the weights in Equation (7) is as follows (Murphy, 2012):
where ${\alpha}^{t}={({\alpha}_{1,2}^{t},\hspace{0.17em}{\alpha}_{1,3}^{t},\hspace{0.17em}\dots ,\hspace{0.17em}{\alpha}_{i,j}^{t},\hspace{0.17em}\dots ,\hspace{0.17em}{\alpha}_{n1,n}^{t})}^{\text{T}}$, ${\beta}^{t}={({\beta}_{1,2,3}^{t},{\beta}_{1,2,4}^{t},\hspace{0.17em}\dots ,\hspace{0.17em}{\beta}_{i,j,k}^{t},\hspace{0.17em}\dots ,\hspace{0.17em}{\beta}_{n2,n1,n}^{t})}^{\text{T}}$ and ${\gamma}^{t}={({\gamma}_{1,2,3}^{t},\hspace{0.17em}{\gamma}_{1,2,4}^{t},\hspace{0.17em}\dots ,{\gamma}_{i,j,k}^{t},\hspace{0.17em}\dots ,\hspace{0.17em}{\gamma}_{n2,n1,n}^{t})}^{\text{T}}$, and $\left({}_{2}^{n}\right)$ represent all possible combinations that consist of two enterprises.
Since the problem is to estimate the collaboration potentials in time period t given collaboration potentials in time period t−k(k = 1, 2,…, τ) , the weights ${\widehat{\mathit{\alpha}}}^{t},\hspace{0.17em}{\widehat{\mathit{\beta}}}^{t},\hspace{0.17em}{\widehat{\mathit{\gamma}}}^{t}$ can be calculated as follows:
A special approach such as metaheuristic methods should be employed to find the optimal weights ${\widehat{\mathit{\alpha}}}^{t},\hspace{0.17em}{\widehat{\mathit{\beta}}}^{t},\hspace{0.17em}{\widehat{\mathit{\gamma}}}^{t}$. As a genetic algorithm is usually employed to estimate MRF weights (Tohka et al., 2007;Ghosh et al., 2014;Lung, 2003;Mishra et al., 2012), this paper also adopt a genetic algorithm to estimate the weights.
The probability $\text{P(}{X}_{i,j}^{t}={x}_{i,j}^{t}{X}^{t1}={x}^{t1}\text{)}$ for ${x}_{i,j}^{t}=$ = 1, 2, 3, 4, 5 is obtained by marginalizing $\text{P(}{X}^{t}={x}^{t}{X}^{t1}={x}^{t1})$ as follows:
The nice Markovian property of the MRF enables us to avoid tedious calculations to obtain the probability in (11), which is presented in the following Equation (12) (Koller and Friedman, 2009):
where X_{(−i)} denotes the vector that includes all components except for X_{i}, and ${X}_{{N}_{i}}$ denotes the set of variables whose elements are all neighbors of X_{i}. Therefore, $\text{P(}{X}_{i,j}^{t}={x}_{i,j}^{t}{X}^{t1}={x}^{t1}\text{)}$ is given by:
where the normalization factor Z(t) is:
As mentioned before, the collaboration potential between two enterprises E_{i} and E_{j} is the expected value of ${X}_{i,j}^{t}$:
4. ILLUSTRATIVE EXAMPLE
In this section, we provide an example to illustrate the application of the proposed model. The application is to allocate each task to the proper set of enterprises in CM, called the task allocation problem. Let E_{i} (i=1,2,…, 20) be an enterprise i in CM, and E_{1}, E_{2}, …, E_{10} retain type 1 resources while E_{11}, E_{12}, …, E_{20} retain type 2 resources. Let T_{k} (k =1, 2,…,10) be task k to be handled, and each task requires both resources. Namely, two enterprises whose resource is different from each other are required to complete each task. The goal is to form a set of ten groups, each of which consists of two enterprises, that maximizes the total sum of collaboration potentials. The collaboration potentials at time t are calculated based on ${X}^{t1},{X}^{t2},\hspace{0.17em}\text{and}\hspace{0.17em}{X}^{t3}$. In other words, $({\widehat{\alpha}}^{t},\hspace{0.17em}{\widehat{\beta}}^{t},\hspace{0.17em}{\widehat{\gamma}}^{t})$ are estimated by maximizing ${\sum}_{k=1}^{3}\mathrm{log}l\left({\alpha}^{tk},\hspace{0.17em}{\beta}^{tk},\hspace{0.17em}{\gamma}^{tk}\right)$.
Table 3 shows conditional probabilities for finding optimal weights of $({\widehat{\alpha}}^{t},\hspace{0.17em}{\widehat{\beta}}^{t},\hspace{0.17em}{\widehat{\gamma}}^{t})$. ${X}_{i,j}^{t3}$ for i = 1, 2,…, 10 and j =11,12,…, 20 are assumed to be multinomially distributed with p=(p_{1}, p_{2}, p_{3}, p_{4}, p_{5})=(0.1, 0.25, 0.3, 0.25, 0.1) , where p_{m}(m=1, 2, 3, 4, 5) denotes the probability that ${X}_{i,j}^{t3}$ is m. Conditional probabilities $\text{P(}{X}_{i,j}^{tk}={n}_{2}\text{}{X}_{i,j}^{tk1}={n}_{1}\text{)}$ for k =1,2 are provided in Table 3.
We obtain $({\widehat{\alpha}}^{t},\hspace{0.17em}{\widehat{\beta}}^{t},\hspace{0.17em}{\widehat{\gamma}}^{t})$ by means of a genetic algorithm and then, using Equations (13) and (14), we obtain $\text{P(}{X}_{i,j}^{t}={x}_{i,j}^{t}{X}^{t1}={x}^{t1}\text{)}$ for =1, 2,…,10, j=11,12,…, 20, and ${x}_{i,j}^{t}=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4,\hspace{0.17em}5$. ${\text{CP}}^{t}({E}_{i},{E}_{j})$ for i =1, 2,…, 10 and j=11,12,…, 20 is calculated from $\text{P(}{X}_{i,j}^{t}={x}_{i,j}^{t}{X}^{t1}={x}^{t1}\text{)}$, and finally, Table 4 shows the calculation results.
We can allocate T_{k} (k = 1, 2,…,10) to a group of two enterprises using the results presented in Table 4. It can also be considered as an enterprise clustering problem, because every task is identical. As an example of the allocations (or enterprise clustering), if we allocate the tasks to ${({E}_{i},\hspace{0.17em}{E}_{10+i})}_{i=1}^{10}$, then the sum of the collaboration potential is 2.86+3.56+2.38+…+2.85= 29.60 . By searching all possible clusters, we find the best allocation C_{best} as {(E_{1}, E_{16}), (E_{2}, E_{12}), (E_{3}, E_{17}), (E_{4}, E_{20}), (E_{5}, E_{11}), (E_{6}, E_{14}), (E_{7}, E_{13}), (E_{8}, E_{19}), (E_{9}, E_{18}), (E_{10}, E_{15})} and Cworst as {(E_{1}, E_{19}), (E_{2}, E_{15}), (E_{3}, E_{13}), (E_{4}, E_{17}), (E_{5}, E_{20}), (E_{6}, E_{11}), (E_{7}, E_{12}), (E_{8}, E_{14}), (E_{9}, E_{18}), (E_{10}, E_{16})}. The sum of the collaboration potentials is 31.61 and 26.65 when allocating tasks to every cluster in C_{best} and C_{worst}, respectively. The difference in the sum of the collaboration potentials between the best case and the worst case is not big for the relatively small artificial example used in this paper. For more enterprises, resource types, and tasks, however, the difference would be much larger.
5. CONCLUSION
CM is an emerging manufacturing model based on cloud computing that enables efficient collaboration among diverse manufacturing enterprises. The collaboration is a main aspect of CM, but there is no previous research that estimates the collaboration potential. It is defined as the degree of customer satisfaction when two or more manufacturing enterprises provide manufacturing services that the customer wants through collaboration between them. Namely, an enterprise group that has a higher collaboration potential will receive a higher rating from a customer than a group that has a lower collaboration potential. The collaboration potential can be applied to solve various operational issues in CM. For instance, the collaboration potential from the model can be used as a base to allocate tasks in CM. Namely, a task will be allocated to a group that has high collaboration potential.
In this paper, we propose an MRF based estimation model to estimate the collaboration potential between enterprises participating in CM. The collaboration potential of two enterprises is measured by the expected customer rating of the manufacturing services they produced in the model. The variables used in the model are the rounded values of average customer satisfaction ratings for manufacturing services that two enterprises have provided in a time period. The model includes three kinds of potential functions to reflect the relationships among the variables and can yield the expected customer rating of two enterprises based on their collaboration histories. In practice, the proposed model can be employed to manage enterprise relationship participating in CM platform. That is, enterprises with high collaboration potential are grouped and CM manager can choose enterprise group and then two or more enterprises from the group to process a customer request.
In future research, we will apply the proposed approach to the realworld task allocation problems in CM, considering the collaboration potentials, and also develop more efficient algorithms to search the optimal weights or clustering methods to assign tasks to a more suitable group of enterprises.