Optimization analysis of supply chain scheduling in mass customization

ARTICLE IN PRESS Int. J. Production Economics 117 (2009) 197–211

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Optimization analysis of supply chain scheduling in mass customization Jianming Yao a,, Liwen Liu b a b

School of Business, Renmin University of China, Beijing 100872, China School of Economics and Management, Tsinghua University, Beijing 100084, China

a r t i c l e in fo

abstract

Article history: Received 12 June 2006 Accepted 20 October 2008 Available online 5 November 2008

How to deal with the contradiction between scale production effect and customized demand is the key problem on studying mass customization (MC). When MC is operating in supply chain environment, on one hand, the excellent operating character of the supply chain will give conditions for solving this problem. On the other hand, it will bring out several complicated contradictions and increase the difficulties of the analysis and research on the supply chain operating and scheduling, so it is important to settle the contradictions. Based on our earlier work, the dominant contradictions of the supply chain scheduling in MC and the ways to relieve them are briefly summarized in this paper. A dynamic and multi-objective optimization mathematical model and the appropriate solving algorithm are set up by introducing these relieving methods into the operating process. It is pointed out that the characteristics of the model and algorithm cannot only reflect the unique operating requirements for this special production mode, but also merge with the thought of relieving the dominant contradictions. The feasibility of the model and algorithm in practical application to improve the scheduling efficiency and to settle the key problem mentioned above ultimately gets validated through the analysis of an application case we followed and through the algorithm simulation of a numerical scheduling case. & 2008 Elsevier B.V. All rights reserved.

Keywords: Mass customization Supply chain scheduling Dominant contradictions Optimization

1. Introduction ‘Mass customization (MC) relates to the ability to provide customized products or services through flexible processes in high volumes and at reasonably low costs’ (Silveira et al., 2001). How to deal with the contradictions between scale production effect and the satisfied level of customized demand is the key problem on studying MC (e.g. Pine II, 1993; Fogliattoa and da Silveirab, 2008). In this aspect, the postponement has been widely used by many scholars (e.g. Brun and Zorzini, 2008; Li et al., 2007; Jiao et al., 2003; Ma et al., 2002; Ernst and Kamrad, 2000). The result of this tactics reflects mainly on

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the reduction and optimization of the customized quantity, and its core is to delay the customer order decoupling points (CODP) (e.g. Gu et al., 2002). However, the optimizing result may lead to the lower level of the customized diversification, which will finally affect the promotion of the customer satisfaction level. On the other hand, with the unceasing change of people’s consumption pattern and the increasing promotion of the customized level, the modes of approximately full customization and urgent customization will occur. These two modes are the concentrative reflections of the high degree of the customized demand. Then, there will be some contrary relations between the high customized demand and the traditional optimizing mechanism of the postponement to a certain extent. Therefore, it is required to look for a new effective method for this problem.

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When MC is operating in supply chain environment with the high flexibility, on one hand, the excellent character of the supply chain system will create the conditions for settling this key problem; on the other hand, the random information of the customer orders and the relation complexity of the supply chain cooperators will cause many complicated contradictions in operating process and increase the difficulty of the supply chain management (SCM). In this aspect, the main problem we should be concerned with is the supply chain scheduling optimization in MC. As a new production mode, the key problem of MC is to simultaneously obtain the satisfied customer response, cost benefit and production scale (Tu et al., 2001), and the operating efficiency will be embodied in the optimization of production management. So, the rational scheduling optimization will be a key idea to settle the problem in the former paragraph. Bose and Pekny (2000) pointed out that ‘the supply chain scheduling is a process to carry out effectively the flow of materials in the supply chain system’. This is the general description of the supply chain scheduling. However, the supply chain scheduling in MC is a typical stochastic and dynamic process. The dynamic character comes from the stochastic demand and the stochastic production capacity of different cooperators in supply chain environment (Yao and Zhou, 2003). It is well known that stochastic and dynamic problems are hard to solve (e.g. Gerard, 1999; Charles, 2001; Tormos and Lova, 2001; Cochran and Uribe, 2005). So to improve the efficiency of the supply chain scheduling, we should reasonably handle the problems in the stochastic and dynamic process. For these complicated stochastic and dynamic characters will probably induce several restriction factors to the scheduling process and then initiate the complicated contradictions and scheduling bottleneck problems, to settle or relieve the contradictions is the key in our paper. Therefore, the aim of our paper is to unearth the dominant contradictions of the supply chain scheduling in MC and to put forward the methods to relieve them. Then we will imbed these methods into the dynamic scheduling optimizing process and the settling process, and apply them to improve the customized service level and the comprehensive benefit level of the supply chain cooperators.

2. Related work 2.1. Problems in present study The supply chain scheduling optimization in MC belongs to a forefront research field, and the complete analysis and settling process of it are not found in the related works. So, we should understand the present researches from several different aspects and make them the background of our work. According to the main idea of this paper, we will briefly analyze the optimization of MC, the optimization of the supply chain and the combination of these two modes. By analyses, we are mainly to find out the ideas to settle the contradictions between the scale production effect and the customized level in MC, the

stochastic and dynamic characters in supply chain operation and the multi-objective problem of the production optimization. Since MC was put forward as a mainstream production mode in 21st century by Pine II (1993), it has been intensively studied in recent years (e.g. MacCarthy et al., 2003; Duray et al., 2000; Smirnov et al., 2003). To the optimization idea of it, most authors focused on the object dimension (the make-up and design of products) in twodimensional production system. Jiao et al. (2003) pointed out that the basis of MC is the ability of corporations to forecast and capture the potential demand information and the ability to develop according to the various demand. However, it is very important to deal with the inherent contradictions among quick responsiveness, customization and economy of scale in the implementation of MC. In other words, a successful MC should have the equilibrium of three factors including character, cost and scheduling. Then he put forward an idea to optimize MC by maximally using common components. As can be seen from the introduction, in the aspect of dealing with the contradictions between the scale production effect and the customized demand in MC, the postponement is widely used by many scholars (e.g. Jiao et al., 2003; Ernst and Kamrad, 2000). Gu et al. (2002) made a relatively comprehensive analysis of the operation optimization in MC in his paper, named ‘Research of the optimization methods for mass customization’. Firstly, he illustrate that the implementation of MC is a progressive procedure by an efficient optimization model; secondly, he put forward two optimization models to minimize the customization. It core idea is to expand the optimizing range of product class and push CODP to downstream at the same time. These former works laid the foundation of optimizing MC to a certain extent that is to decrease the customization and mostly utilize the common components and technological process to reduce the customization composition. Then, it can reduce the class variety induced by the customized demand and finally realize the unity of the scale production and the customized production. The essential of this idea is to give full scope to the postponement and then to promote the scale effect, which is obviously reasonable. The key of this optimization is to improve the flexibility of MC and the realization way of it lies in the improvement of the product developing and production technology. However, the promotion of the customized level reflected by the diversification of the product character and the delivery date will be probably influenced by reducing and optimizing the customization. There is obviously a contradiction in it and how to settle the contradiction is the key in present study. To settle this key problem, we integrate the mode of MC into the special operational environment of the supply chain, for the excellent flexibility of the supply chain system is advantageous to relieve this contradiction. The early idea of integrating MC with the supply chain can be seen in Pine II (1993). With the theoretical and practical development of SCM, this idea is increasingly unfolded before our eyes by many authors (e.g. Jack, 2001; Jonah et al., 2003; Akkermans et al., 2003). They point out

ARTICLE IN PRESS J. Yao, L. Liu / Int. J. Production Economics 117 (2009) 197–211

the importance and the urgency of this integration and also point out that MC is a better way to improve the customer service level. This improvement is inevitable for enterprises to guide their future development. This idea is a necessarily theoretical basis for our research on the integration of MC-SCM, the multi-angle analysis and the orientation of optimization objective. To deeply and systemically study the theory of SCM in MC and that of MC in SCM, Ghiassi and Spera (2003) proposed a typical and necessary canonical form of the supply chain system operating in MC. The most contribution of his work lies in summing up a new operational mode which is necessary for pull supply chain in MC and is the core that the more traditional studies of static push supply chain should be improved. The most prominent difference of the supply chain system in MC from the static supply chain system is that the former is a dynamic system and should be described by a non-linear program model. Furthermore, Klaus (2002) and Martin and Jiao, 2002 pointed out that the realization of the supply chain in MC must take the technical supporting system into consideration, and this is also the prerequisite to our work, for the relevant technical supporting methods are essential to realize the optimization. Here, how to reasonably implement the supply chain scheduling optimization in MC is the problem we should be most concerned about. For when the customized quantity of MC in the supply chain is determined, the production optimization process will tend to be translated into the problem of the pull supply chain scheduling optimization. Whether it is rational or not and whether its efficiency is high or not will decide directly the benefit level of the production scale and the customized services level. Different from the general supply chain scheduling, there are complicated particularities of the scheduling process in MC. Generally, in the optimization of the supply chain operation, the present work usually use the methods extended from the early ways of optimizing production operation (e.g. Lee et al., 2002; Moon et al., 2002), the methods of MILP (e.g. Edgar et al., 2003; Vasilios, 1996), the methods of decision analysis (e.g. Sahin et al., 2008; Biswas and Narahari, 2004; Sabri and Beamon, 2000; Bose and Pekny, 2000 and the methods of pure mathematical theory (e.g. Mokashi and Kokossis, 2003; Lancioni et al., 2000; Lakhal et al., 2001) to make concentrated analyses on setting up and solving the supply chain scheduling model. In the supply chain optimization, no matter what method is adopted, the ways to settle the dynamic characters are mainly to transform the dynamic information to the static one (e.g. Edgar et al., 2003; Vasilios, 1996; Lee et al., 2002; Moon et al., 2002). Edgar et al. (2003) and Vasilios (1996) analyzed the settlement of the stochastic and dynamic problems about the supply chain system. The method they use to transform the dynamic characters is the ‘differential analysis’ that make differentiation to time proceeding by making every slight time corresponding with the moment ‘t’ as a relatively static unit and then make the static optimization.

199

This method is effective to settle the complicated dynamic characters of the supply chain and is an effective idea to settle the general complicated dynamic problems. Here, it is most important to rationally determine the differential moment. Generally speaking, many scholars are accustomed to select the differential moment with the equal period (e.g. Edgar et al., 2003; Vasilios, 1996). Edgar et al. (2003) adopt one day as a scheduling period (differential period) when he study the scheduling problem of the supply chain with a distributed multifactory. The reason he adopt the equal differential period is that it can be handled and realized easily. But in MC, we should rationally consider the equilibrium relations between the decrease of the customization brought about by the postponement and the satisfaction level of the customized services. The substance of improving the scale production effect through postponement is realized by maximally using the group operation scale of the common procedures and decreasing the operation scale of the special procedures. Obviously, the group production scale of the common procedures has direct ratio to the number of the customer orders (this ratio relation is not a linear one) during a definite period. So, if we prolong the concentrated time of handling the orders every batch, the group production scale of common procedures will expand to a certain extent, the effects of the postponement will be enhanced and the efficiency of the production will be improved. On the other hand, if we prolong that time every batch, the customized service level and the production continuity of the different cooperators in the supply chain will be affected. For this reason, it is critical to decide the differential moment and period (scheduling moment and period) to reasonably transform the dynamic problem into a static one. In addition, the complexity of the relations among the supply chain cooperators is also an important factor influencing the scheduling level. The characters of this factor are seldom found in non-supply chain environment. Therefore, to solve all the problems mentioned above we must analyze the feasibility and practicability of the methods from a new angle. 2.2. Earlier work relative to this paper In a word, to settle the discussed complicated problems in MC and in supply chain, there are many works (e.g. Akkermans et al., 2003, Jack, 2001) pointed out the ways to probe the settlements for various problems. All these are usefulness we can learn from. In our early work (e.g. Yao and Zhou, 2003; Yao et al., 2004, 2005a, b), we analyzed several restrictive factors which greatly influence the supply chain scheduling. These factors contain the remained problems put forward in the introduction. To grasp the key idea and probe the method to solve these problems, we conclude two decisive dominant contradictions which are different from but correlate with each other in MC and the supply chain operation process. The first is the contradiction between the satisfaction level of the customized services and the comprehensive profits of the cooperators in the supply chain system; the second is the contradiction between the improved

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3.1. Ways of relieving dominant contradictions

qualities of products, which can be reflected by the physical characters of products and be corresponding to the classification of GO and SO and also contains the customized delivery date of products which can be corresponding to the classification of RO; on the other hand, it contains completely the realization of the production efficiency in the supply chain scheduling in MC which can be reflected by the complexity of the production process, be corresponding to the classification of GO and SO and can be reflected by the same time production lot corresponding to the definition of the time threshold. So, to plan and optimize the supply chain scheduling based on the order classification by the time threshold is a good idea to satisfy the different demands of the customer with the higher production efficiency. This is the key way to relieve the first dominant contradiction. The existence of the ROs and SOs may make it more complicated and difficult to handle the supply chain scheduling. But as a real meaning customized system, to greatly satisfy the customer demands is necessary. And only on this base can the excellent service brand be set up gradually and can the supply chain cooperative relationships develop and last greatly and long. At the same time, to reduce the ROs and SOs gradually will be the objective for the enterprises to get by reform and process reorganization. In the following paragraphs, we introduce the idea of the time threshold and the order classification into the dynamic scheduling process to determine the scheduling adjustment moment.

3.1.1. Thought of order classification based on time threshold The key problem of MC to solve is to satisfy the different customized demands with the scale production efficiency. As for a manufacturing supply chain, it is to make the system’s comprehensive profits maximum on the premise of rationally realizing the production efficiency to different orders. Obviously, these constitute a contradictory body. To relieve these dominant contradictions in the dynamic scheduling and to show the effects of the scope economy in MC, we put forward the idea of the time threshold based on the elementary classification of the stochastic orders. The main idea of the time threshold is that the core enterprise should put forward a time horizon to wait dynamically for other orders’ coming by weighing all factors in the supply chain system when it has received any order in the scheduling. At the same time, to promote the customized service level maximally and realize the approximately full customization, we also given an idea of the secondary classification of orders based on the time threshold. The main contents of the secondary classification are as follows: Definition 1, the orders received by the enterprise during the time threshold whose parts will be made by the unique designing and machining technology are named Special Order (SO); Definition 2, the orders besides SO are named General Order (GO); Definition 3, the orders that should be given the highest priority to operate to fit the urgent delivery date of customer are named Rush Order (RO) which are not restricted by the time threshold. We can see that, on one hand, the idea of the secondary order classification contains completely the customized

3.1.2. Determination of dynamic scheduling moment In early work, we point out the stochastic demand of the customer and the stochastic production capacity of every node are the key factors which will result in the dynamic characters of the supply chain scheduling in MC. These two factors are also the key reasons for the contradiction between the customer service level and the cooperators’ profits in the system. To know more about the relations among the production time, production cost and available production capacity in quantity, we selected a typical fabrication node in a supply chain system and took sample data every day during one month. The result is shown in Fig. 1 and the data of it has been normalized. From Fig. 1 we can learn that to satisfy the different demands of the customers, the information both coming from the stochastic customer demand and from the dynamic changes of the cooperator resources can make a large wave to the curve of the node’s available production capacity in the supply chain network. The change of the available production capacity will certainly lead to the difference of the production time and cost to the same operation in different times. But the relation between production time and available production capacity is not a regularly inverse one and is determined by the complexity of the MC production. At the same time, the pre-production preparation, the working procedure and the post-production treatment of the different customized products may not be identical with each other which will cause the wave to the curve of the production cost, though its trend is smooth on the whole. It does not like the

available production capacity of the cooperators resulted from optimizing their operational time of each stage during production and the increase of the extra inventory cost at some production stages. These two dominant contradictions are the basic causes of many restrictive factors. At the same time, the dominant contradictions will certainly cause the bottlenecks of the supply chain scheduling in MC. So, it is very important to probe the methods to relieve these dominant contradictions. 2.3. Main work of this paper Based on the earlier achievements, this paper gives a brief summary of the methods to relieve the dominant contradictions. On this foundation, the paper lay stress on imbedding the relief methods into the operating rule of the dynamic scheduling and establish a mathematical optimization model of the supply chain scheduling in MC. In order to solve the complicated scheduling problems in MC, the paper also puts forward an appropriate algorithm and makes an application case to illustrate the implementation of the scheduling methods and a simulation case to verify the convergence character of the algorithm. 3. Optimization model based on relieving dominant contradictions

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201

Unit Production Time Available Production Capability Unit Production Cost

1 0.8 0.6 0.4 0.2 0 3

1

5

7

9

11

13

15

17

19

21

23

25

27

29

Fig. 1. Sample data of a typical fabrication node every day during one month.

The nth batch of orders … The second batch of orders

The moment of receiving the rush order (TR); The moment of determining the dynamic time threshold by the core enterprise (T0)

End of production

Rush orders

Rush orders The first batch of orders

0 TR1TR2

T01 TR3TR4

T01+T02

A,T,C

…

t

A T C

0 TC1 TC2

TC3 TC 4TC5

TC6

…

0 TD1 TD2

TD3TD4TD5

TD6

…

t

Providing dynamically the production parameters information by the cooperators; Distribution of the dynamic sampling information by the core enterprise (TC) The moment of implementing the dynamic scheduling (TD)

Fig. 2. Determination and adjustment in supply chain scheduling in MC.

general problem of the workshop scheduling such as FSS and JSS (e.g. Lee et al., 1997) in which the production cost is invariant. Because these characters will influence the supply chain scheduling process, we should embody the solving ideas in building the scheduling model. The solution ideas of the dynamic scheduling are always the research focuses in the academe. The key one of them is how to make rational description of the dynamic character and how to transform it into a static question by the available methods. Edgar et al. (2003) once described and solved the problems by using the differential time slice when he built up a dynamic optimization model of MILP to study the multi-factory distributed supply chain scheduling, and the crux of the methods is to organize the production based on the orders received every day as a handling unit. But it is hard to avoid the inflexibility of them and is not suit for solving the dynamic problems of the supply chain scheduling in MC, because the production cost and the production time will vary with the production course in this special scheduling mode. In this paper, we will handle this stochastic problem by dynamically sampling with time t to determine the scheduling moment and operating cycle. The determination of the scheduling moment is not in random, but has it regularity based on the ideas of the time threshold and the order classification mentioned above as shown in Fig. 2 where A, T and C are, respectively, the available production capacity, the production time and the production cost of every cooperator.

From Fig. 2 we can know that the dynamic scheduling may be carried out at the end of the time threshold calculated after every batch of orders coming or at the arrival of every RO. Because the customer should pay more for his rush demand (Jiao et al., 2003), the supply chain must consider the system’s comprehensive profits when adjusting the scheduling for it. The adjustment process of the supply chain dynamic scheduling in MC is shown to Fig. 3. Because the scheduling adjustment at every moment is based on the constraint of the available production capacity in the supply chain, there are no conflicts among the scheduling commands. Even though, for there are probably a lot of ROs, the handling cost and quantity of their information will increase and the process of the scheduling will be more complicated. 3.1.3. Dynamic profit preference of supply chain cooperators We also put forward the idea of the cooperators’ dynamic profit preference to handle the complex cooperative and competitive relations in our early papers. It is the key to solve the contradictions between the production time and cost and to solve the restrictive factors induced by the extra inventory in the supply chain. The profit preference can reflect the cooperators’ partial action to the comprehensive profits with the production time and cost which are the basic operating parameters. It can also realize the identical handling on these two parameters to output the optimization result with solo

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Start Ensure the supply chain system and the cooperators realize their actual and expected profit maximum.

Receive a order Receive a rush order

No

i =i +1

If adjust the scheduling, does the system can get the overall profit?

No

Receive a non rush order i If operate order i, does the system can get the overall profit? Yes Time: t =T0

Yes

Return

Implement scheduling

The production implementation should base the orders classification method to solve the contradiction between the scale production effect and the customized level.

Fig. 3. Adjustment process of dynamic supply chain scheduling in MC.

objective. All these help us to reduce the complexity, promote the efficiency of the scheduling and realize the building of the scheduling optimization algorithm. When the relations about the profit preference are introduced into the supply chain scheduling, they are mainly embodied in the restrictive relations on the satisfaction level of this preference, in the constraints of the scheduling model and in the maximization of the system’s income. To make the system income maximum is a practical optimization objective on the premise of guaranteeing every cooperator’s profits above the minimum satisfaction degree. The ideas of the order classification based on the time threshold and of the determination on the scheduling adjustment moment are the core for relieving the first dominant contradiction. The idea of the dynamic profit preference is mainly to relieve the second dominant contradiction. In substance, the latter idea is used by cooperators to determine dynamically their profit preferences to balance the available production capacity they can improve by optimizing the production time of each stage during the production process against the increase of the extra inventory cost at these corresponding stages. So, to introduce the idea of the profit preference can make cooperators rationally weigh the profit expectations from these two aspects and the win-to-win between the supply chain system and the customers can be attained. 3.2. Optimizing mathematical model 3.2.1. Description of dynamic character and handling of irregularity Based on the relieving ideas mentioned above, we set up a dynamic and multi-objective optimizing model which has two aims. On one hand, it should show the unique character of the supply chain scheduling in MC. That character is reflecting the requirements for the satisfaction level of the customized service and of the supply chain cooperators’ comprehensive profits. It should also realize the operating objective of the supply chain system at the same time. On the other hand, the thought of relieving the two dominant contradictions should be

reflected fully in the building of this model, which is the base for bringing its practical value into playing. We will also meet the problem of the production proceeding irregularity in the supply chain scheduling. The production irregularity comes from the objective reality of the special one and the common one in the combination style of the same batches of the customized products. Here, the irregularity is that the operating process of one or more parts of a customized product may span some special sequence in the production proceeding. This irregularity will have great influence on the supply chain scheduling, not only on the implementation of the scheduling optimizing, but also on the complexity of the description of the parameters about the dynamic scheduling characters. This is more different from the simple partition in the general dynamic programming. To reflect the dynamic scheduling characters, match the above idea of the scheduling adjustment moment and describe the production proceeding irregularity rationally, the parameter t is introduced into the model to describe the dynamic character. The dependent variables in the model will change with the parameter t. At this time, when the core enterprise in the supply chain makes sampling analysis on the basic parameters of the cooperators at moment t, it must analyze the operating parameters of the related cooperators at moment t+Dt determined by the normal and irregular production proceedings in advance. In the model, these complicated problems will be described by the united operating parameters in order to contain the proceedings of the relative fictitious stages so as to realize the scheduling optimization and algorithm implementation. 3.2.2. Building of model To explain conveniently, the analysis of the model will be done after it has been built. In the model, the core enterprise can join in one or more production stages. That may not affect the optimizing process but only change the implications of some variables. The assumptions of the model, the enactments and explanations of the parameters and indexes are shown in Table 1.

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Table 1 Explanations of model assumption, indexes and parameters. t K

The starting moment of the supply chain scheduling The total number of the stages in the supply chain for MC orders. The stages may include material supply, fabrication, assembly, distribution, inventory, transportation, delivery and so on The index of the production stages, k ¼ 1,2,y,K The starting moment of the k stage The total number of the cooperators at every stage or the number of the production/work groups at the stage in which the core enterprise takes part The index of the nodes at every production stages, r ¼ 1,2,y,Nk The index of every node The number of the orders received during the time threshold besides the rush orders At moment tk, the number of the starting stages for the different orders divided by the core enterprise The index of every class in G The total number of the orders contained in G by the basic order classification method, g ¼ 1,2,y,G The index of the customer order class, i ¼ 1,2,y,M The number of the orders in every class, m ¼ 1,2,y,Mg The index of the customer orders in every customer order class, j ¼ 1,2,y,Nm The index of every order The delivery date of order (h,i,j) The production time for node (k,r) operating order (h,i,j)

k tk Nk r (k,r) NT 0 G H Mg i Nm j (h,i,j) T Dhij T kr hij ðt k Þ C kr Thij ðt k Þ

The production cost for node (k,r) operating order (h,i,j) except the extra inventory cost

T kr exp hij ðt k Þ

T kr exthij ðt k Þ

The expected production time for node (k,r) operating order (h,i,j) decided by the analysis of the subjective and objective influence factors The expected production cost for node (k,r) operating order (h,i,j) decided by the analysis of the subjective and objective influence factors The extra inventory time for node (k,r) operating order (h,i,j)

C kr exthij ðt k Þ

The extra inventory cost for node (k,r) operating order (h,i,j)

C kr exp hij ðt k Þ

ext

The unit extra inventory cost for node (k,r) operating order (h,i,j), C ext ðt k Þ ¼ T ext ðt k ÞC ext Dt ðt k Þ The absolute maximum time limit for the difference between the actual production time and the expected production time for node (k,r) operating order (h,i,j) when (h,i,j) is produced at (k+1,r) The demanded production capacity for stage k operating order (h,i,j) at moment t

C Dt ðt k Þ T kþ1;r Dhij ðt k Þ AkDemhij ðt k Þ AkSupprhij ðt k Þ

The available production capacity of node (k,r) for operating the orders

dkr hij ðt k Þ b bmax

If order (h,i,j) is operated by node (k,r), then dhij ðt k Þ ¼ 1, otherwise dhij ðt k Þ ¼ 0

kr

kr

Q kr hij

The tolerant parameter of the due date The maximum of b and it should be defined by the core enterprise and other cooperators in the supply chain system. The financial compensation is needed when exceeding the due date considering the accidents in production such as the accident of traffic, collapse of storehouse, cut of water, power failure and so on The operating quality of order (h,i,j) at node (k,r), including transport, inventory, service and so on

Q kstahij

The standard quality demand of order (h,i,j) operated at stage k

kr hij ðt k Þ

The profit preference factors for node (k,r) to order (h,i,j) decided by the analysis of the subjective and objective influence factors

Ukr(tk) Umin  kr(tk) USC(tk) fkr(tk)

The profit preference satisfaction degree of node (k,r) The minimum profit preference satisfaction degree of node (k,r) The comprehensive profit preference satisfaction degree of the supply chain system at moment tk The contribution factor of (k,r) to the supply chain system comprehensive profits, (0pfkr(tk)p1). The factor of the node (k,r) is different from each other. Obviously, when all the factors reach the maximum and all fkr(tk) ¼ 1, the comprehensive profits of the supply chain will reach the maximum of their ideal final value

The complete formulation of the model is as follows: Objective function: min Z 1 max SCprofits

Mg X Nk X Nm K X G X X ext kr ¼ f½ðC kr T:hij ðt k Þ þ T ext:hij ðt k Þ  C Dt ðt k ÞÞ k¼1 r¼1 h¼1 i¼1 j¼1

Subject to: Mg X Nm G X X

AkDemhij ðt k Þp

Nk X

AkSupprhij ðt k Þ

(5)

r¼1

h¼1 i¼1 j¼1

kr

kr þ kr hij ðt k Þ  T hij ðt k Þ  dhij ðt k Þg

(1) min Z 2 max CUSTprofits

kr kr ¼ jT kr exp hij ðt k Þ  ½T hij ðt k Þ þ T exthij ðt k Þj þ b

  X   Nm kr   dhij ðtk Þ  N m  min Z 3 ¼  max MCprofits  j¼1  max U SC ðtÞ ¼

Nk K X X k¼1 r¼1

fkr ðtk ÞU kr ðtk Þ

(2)

(3)

(4)

T Dhij ðtÞp

Nk K X X kr kr ½T kr hij ðt k Þ þ T exthij ðt k Þ  dhij ðt k Þ k¼1 r¼1

pð1 þ bÞT Dhij ðtÞ kr kr kþ1 jT kr exp hij ðt k Þ  ½T hij ðt k Þ þ T exthij ðt k ÞjpmaxT Dhij ðt k Þ

Mg X Nk X Nm G X X r¼1 h¼1 i¼1 j¼1

dkr hij ðt k Þ ¼ N T 0

(6)

(7)

(8)

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dkr hij ðt k Þ ¼ 1

(9)

U kr ðt k ÞXU minkr ðt k Þ

(10)

k Q kr hij XQ Stahij

(11)

r¼1

Here: 0pbpbmaxo1; k ¼ 1,2,y,K; r ¼ 1,2,y,Nk; h ¼ 1,2, y,G; i ¼ 1,2,y,Mg; j ¼ 1,2,y,Nm. In the model, Eq. (1) is a multi-objective function, its operating main line lies in optimizing the customized cost. Its outstanding character is the introduction of the cooperators’ profit preference factor. The profit preference can reflect the partial action of the cooperators’ comprehensive profits by the production time and cost which are the basic operating parameters. It can also realize the identical handling on these two parameters and realize the output in single objective. All these can decrease the complexity of the supply chain scheduling, promote the scheduling efficiency and realize the building of the optimization algorithm. The meaning of Eq. (1) is that at moment t, every cooperator (k,r) should determine the value of their profit preference factor e(t) by the preference decision process based on order (h,i,j)’s planning states and the subjective and objective circumstances and regard them as the basic data on the supply chain scheduling. At the same time, each cooperator should provide their production parameters to the core enterprise at moment tk in order to implement the scheduling. Based on the rational calculation of these data, we regard the minimum comprehensive production cost of order (k,r) in the supply chain system as the optimizing objective and draw up the scheduling plan. The determination of tk is shown as Eq. (12) where kA(1,K). tk ¼ t þ

k X

T xexp ðtÞ

(12)

x¼1

Eq. (2) is an optimization function of the delivery date satisfaction and its operating main line lies in optimizing the punctual delivery of the customized products. From Eq. (2) we can learn that as a supply chain system, the core enterprise has its corresponding expected production time (the expected delivery date) for every task at the corresponding stage. The closer the actual production time to its expected time is, the better the product delivery to customer is guaranteed, and the best the stabilization of the supply chain system is strengthened. b in Eq. (2) is a tolerant parameter of due date. The smaller the value of b is, the higher the service level of the delivery date provided to customer is. As a result, it is important to promote the service level of the delivery date to customer. Here we list it in Eq. (2) because its meaning is identical with that of Eq. (2). Eq. (3) is an optimization function of the scale production effect and its operating main line lies in optimizing the scale operation of the customized products. We can learn from Eq. (3) that the smaller the value of Z3 is, the greater the level of the scale production effect is. The division of Nm is based on the above order classification, and the direct aim of the order classification

is to relieve the contradiction between the operation of the postponement and the promotion of the customer service level, that is also the aim of Eq. (3). Eq. (3) can be developed for two scenarios. 1. When the available production capacity of (k,r) is greater than Nm needed at this stage: Nm X

AkDemhij ðtÞpAkSupprhi ðtÞ

(13)

j¼1

At this time, the production of Nm can be completed by one cooperator, or part of its task can be completed by other cooperators at the same stage. This decision process is similar to the profit preference decision. Obviously, the scale production effect of the former is higher than latter. But, as mentioned above, the supply chain system is a systemic and organic cooperative system and to realize the maximum comprehensive profits is its terminal goal, so, a practical way we should adopt is to weigh rationally the relation between the system’s overall and local interests. 2. When the available production capacity of (k,r) is smaller than Nm needed at this stage: AkSupprhi ðtÞo

Nm X j¼1

AkDemhij ðtÞp

Nk X

AkSupprhij ðtÞ

(14)

r¼1

At this time, because the constraint of the available production capacity will give its full scope, the production of Nm cannot be completed by one cooperator obviously. This is an inevitable selection to avoid the problem of production congestion. So, Eq. (3) can show the idea of optimizing the scale production effect, no matter under the conditions of Eq. (13) or Eq. (14). Eq. (4) is an optimization function of the supply chain’s comprehensive profits. Because the supply chain system which operates the customized action is an organic and systematic system, it is a practical optimization objective to maximize the supply chain system’s comprehensive profits on the premise that every cooperator can achieve its satisfaction level. Eq. (5)–(11) are the model’s constraints. Eq. (5) ensures that the available production capacity of all nodes in the supply chain exceed the related demand of all the orders during the same operating period. Eq. (6) is the constraint of the punctual delivery date. Considering the accidents in production such as traffic accident, storehouse collapse, water cut, power failure and so on, we introduce b as a tolerant parameter of due date. At the same time, b is also an optimization objective embodied in Eq. (2). Eq. (7) is the constraint of continuous production. It ensures that the same product should have a continuous order at every stage. Eq. (7) also indicates that in the supply chain system, every cooperator must consider the profits of others when it makes its own profit decision in order to realize the reasonable relationship, to complete the production tasks smoothly and to make all cooperators obtain their own satisfaction level. Eq. (8) is the constraint of production stages. It ensures that all orders received in T0 should go through all the production stages (but some orders’ tasks can only go through the fictitious stages). Eq. (9) is the constraint of the unique production. It

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ensures that there should be no phenomenon of repeat production. Eq. (10) is the constraint of the satisfaction degree of the cooperators’ profit preference. It ensures that the cooperative relations should be built on an acceptable level which every cooperator can achieve their own satisfaction degree of profits. Eq. (11) is the constraint of production quality.

3.2.3. Hidden meaning of model to relieve dominant contradictions It should be further point out that the former four optimization functions are not isolated but influence and restrict each other. At the same time, these functions must act together when realizing the optimization objectives. In order to indicate the hidden meaning of this optimization model, we unify these four objective functions of the model to the form min Z ¼ K 1 min Z 1 þ K 2 min Z 2 þ K 3 min Z 3 þ K 4 ð1  K 5 max Z 4 Þ

(15)

In Eq. (15), K1–K4 are unitized coefficients. On one hand, they can realize the unification of the different dimensions; on the other hand, they can embody the different weights of the optimization objectives in the supply chain scheduling. K5(0pK5 max Z4p1) is a coefficient of transformation. If we summarize and abstract the characters from Eq. (15), we can get the visible meaning and hidden meaning of the model’s objective functions as shown in Fig. 4. From Fig. 4 we can learn that the model’s objective functions can implicate the core ideas of relieving the dominant contradictions. So, the analysis of this visible and hidden meaning can embody the meaning of the model and verify the main idea and the feasibility of the model.

Optimizing the = supply chain scheduling in MC

Realization

Promoting the level of customization

Optimizing the system’s production cost

=

Promoting the profit of the suppliers and demanders

4. Optimization algorithm for dynamic scheduling 4.1. Optimization mechanism and description of algorithm In this paper, we use the improved ant algorithm to solve the scheduling problem, for ant algorithm has more advanced characters (e.g. Bonabeau et al., 2000). However, as shown in the former text, there are much more new operating characters and complexities in the supply chain scheduling in MC compared with the general scheduling problems such as JSS and FSS. In order to make the algorithm rational and suitable, we must develop the general ant algorithm. First, we regard every cooperator in the supply chain as an independent unit and it has relatively definite operating parameters during every production period. Here we are concerned with the main parameters such as the available production capacity A(t), production time T(t) and production cost C(t) which can directly influence the supply chain scheduling character. The analog relations between the actions of the supply chain scheduling in MC and of the ants’ optimization of looking for food are shown in Table 2. The second issue we discuss is the change of allowable field. Now we take the adjustment of scheduling aroused from the coming of the RO as an example. Suppose the supply chain system is composed of set M ¼ {m1,m2,y,mn} of cooperators. At a certain moment tD aimed at one RO, the cooperators in M which can provide production ability to some production stage of RO consist the set MRO, but at tD, the cooperators in M which can provide available production capacity to RO consist set M RO , so there is M RO AMROAM(t ¼ tD). If at the moment, one cooperator in MRO wants to transform into a member of MRO by adjusting its own operating plan such as improving the utilization of equipments, extra working hours and so on, M RO will makes expansion. When this

Optimizing the Optimizing the Optimizing the + system’s punctual + system’s scale + system’s overall profit delivery production effect Visibal meanings Hidden meanings

Reducing the cost of customization

+

Promoting the punctual level of + customization

Promoting the = comprehensive effect of the + supply chain system

Promoting the levels of relieving or solving the dominant contradictions

205

=

Promoting the Promoting the variety level of + relationship level of customization cooperation

Promoting the service level of the customers

Relieving or solving the 1st dominant contradiction

+

+

Promoting the profit of the cooperators

Relieving or solving the 2nd dominant contradiction

Fig. 4. Visible and hidden meanings of model.

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course gets stable, the production capacity constraint of the supply chain to RO will be smooth. So, the optimization at this time will become a searching process for the suitable cooperators in the allowable field at every production stage and realizing the linkage of the different production stages to make the comprehensive incomes of the supply chain maximum. The next issue is the problem of the customized delivery on time. To satisfy the customized service level maximally, the delivery on schedule is necessary. In other words, delivery on schedule is better than early or late. For there are expected delivery dates for every stage of every MC product, in scheduling adjustment process we can treat it as an optimization objective that searching for the cooperators whose practical delivery date is near to the expected one at every stage. This problem must be solved when setting up the algorithm. We also need to consider the problem of production congestion in the supply chain network. We can learn from the former text that the different stages of MC can be divided into the common production components and the special components to realize the efficiency. Because of the cooperative character of the supply chain, some stages in MC can span several stages such as the class B shown in Fig. 5. So, when there are several orders being produced at the same time, the congestion problem will probably come into being at some nodes, such as node (k  r) in Fig. 5, for there are production class B and C being operated at the same time. This is also the problem we should pay more attention to when setting up the algorithm.

Table 2 Similar relations between supply chain scheduling in MC and ants looking for food. Actions of supply chain scheduling in MC

Actions of ants looking for food

Beginning of dynamic scheduling End of dynamic scheduling Different customized production tasks at each stage Each cooperator (node) in supply chain network Difference of cooperators’ production parameters Optimization scheduling

Starting from nest Arrival at food Corresponding ants of different categories Each travel of looking for food Different attraction and exclusion probability of travels to ants Optimization of looking for food and of physical strength expenditure

(1.1)

Finally, in order to ensure that the supply chain system can get its maximum comprehensive profits, it is necessary that the cooperators should not make arbitrary changes of their partial delivery dates when the scheduling adjustment has been done. Of course, the cooperators can make internal adjustments according to their cooperative relationships with the other supply chain. 4.2. Mathematical description of algorithm Suppose that a supply chain network at the moment of dynamic scheduling adjustment is composed of the source node, object node and cooperator nodes as shown in Fig. 5. The stage division of the network will be determined dynamically by the practical production circumstance of the several orders at the moment tD. In the running of algorithm, ants will move from the source node to the object node through the network and die then. As the ants cannot return, the pheromone (e.g. Bonabeau et al., 2000) of different roads will be determined intelligently by the different cooperators’ production parameters. The structure of the algorithm is as follows. First, we consider the structure of the ants and the forbidden nodes. To realize the algorithm we adopt the special method to make ants. That is: we make two-step division to the ant classification. The first step is to classify the ants by the orders’ production classification. Each order class is corresponding with an ant classification; the second is to classify them by the beginning stage of the production in the same order classification, and the different beginning stages are corresponding with the different ant classes. Suppose at moment tD, the number of the order classes that should be made the scheduling adjustment is n and the index of the different beginning stages in every classification is mi (i ¼ 1,2,y,n). Then, the P number of the ant class at moment tD is ni¼1 mi and every ant class is signified as Aij(i ¼ 1,2,y,n; j ¼ 1,2,y,mi). For each ant class, we set the forbidden nodes according to its allowable field. Then, we set the path selection probability to solve the problems of the third and fourth issues in Chapter 4.1, the probability to select the different paths should be determined as follows. 1. The attraction probability of the paths to ants. Given M ij is the allowable field of Aij at tD. The set of cooperator r(r ¼ 1,2,y,R) at stage k(k ¼ 1,2,y,K) is signified by M ij:kr . As one of the optimization objectives is to minimize the production cost, the quantity (signified by pð1Þ ) of the ij:kr

(2.1)

B

(n.1)

A (1.2)

(2.2)

C

… (1.r)

… (2.r)

C

(n.2)

A

Source Node

The First Stage Production Nodes

The Second Stage Production Nodes

B C …

… (k.r) The kth Stage Production Nodes

Fig. 5. Construct network of dynamic supply chain scheduling in MC.

Object Node

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remained pheromone by ant class Aij is in the inverse ratio of the production cost when they passing through M ij:kr . Then the attraction probability of class (1) for cooperator r to Aij at stage k can be described as , ð1Þ P ð1Þ A ¼ pijkr

R X

pð1Þ ijkr

(16)

r¼1

At the same time, to solve the problem of the third issue in Chapter 4.1, let the expected production time limit (expected production delivery date) of Aij to kr is TE. Considering the dynamic character of the supply chain’s cooperative relationship, some cooperator should make a competitive bidding according to its own production process because of its probable cooperative relations with the other supply chain networks at the same time. Then let its competitive bidding delivery date be TS and let T D ¼ jT E  T S j. We can learn that to realize the punctuality of the customized production and the connections of the different production stages, the smaller TD is, the better it is. Certainly, some times the quantity of TD has the relation with the production cost which should be weighed by the supply chain system. Let pð2Þ denote the ij:kr pheromone quantity which is inversely proportional to TD that ant class Aij left when they pass through node kr. So the attraction probability of class (2) for cooperator r to Aij at stage k can be described as , P ð2Þ A

¼p

ð2Þ ijkr

R X

pð2Þ ijkr

(17)

r¼1

2. The exclusion probability of the paths to ants. Scholar Navarro Varela and Inclair (1999) has given the idea that the pheromone of the same ant class will attract each other and that of the different class will exclude. So, in order to solve the production congestion discussed in Chapter 4.1, we set the exclusion probability of the paths to ants. Let cpq.kr denote the quantity of the remained pheromone by non-Aij ant class at kr, then the exclusion probability of it to Aij can be described as , P R ¼ cpqkr

R X

cpqkr

r¼1

ðp ¼ i; qaj; pai; q ¼ j; pai; qajÞ

(18)

3. The probability of the ants to select paths. Let the probability of Aij to select path kr be ð2Þ P ij:kr ¼ aPð1Þ A þ bP A þ gð1  P R Þ

(19)

In Eq. (19), a,b,g(0oa,b,go1;a+b+g ¼ 1) are the parameters for the algorithm and they can reflect the expected weight relations between the attraction and exclusion probabilities. We also need to determine the update rule of the pheromone. Different from the traditional algorithm, the ants in this paper only have one-way motion character, so the algorithm should update the pheromone at every production node automatically. Here we use F to denote all the former p(1), p(2) and c to simplify the indications

207

and the update rule is

Fðt þ 1Þ ¼ FðtÞ þ DFðt; t þ 1Þ  xFðtÞ ¼ ð1  xÞFðtÞ þ DFðt; t þ 1Þ

(20)

where F(t) and F(t+1) denote, respectively, the pheromone quantity remained at the node at the tth and the (t+1)th batch, DF(t, t+1) denotes the pheromone quantity left at the (t+1)th batch and x(0oxo1) denotes the volatile coefficient of the pheromone. 4.3. Description of algorithm steps In scheduling, when the scheduling adjustment moment comes, we operate the algorithm to adjust dynamically the assignments of the production tasks among the supply chain cooperators. Because there are many complex cooperative and competitive relations among the cooperators, it is difficult to find the exact optimization solution. In practice, we can weigh in several aspects (e.g. the production cost and production delivery date in this paper) and put forward an expected satisfaction level before scheduling. When the algorithm convergence makes every optimizing index reach its corresponding level, the algorithm can be over. The steps are: Step 1: The core enterprise determine the scheduling adjustment moment according to the procedure shown in Fig. 3. Step 2: When the algorithm begins, we judge the order classification according to the description (in the second issue in Chapter 4.1 and the first and second issues in Chapter 4.2) and set up the ant classification, then determine the allowable fields for them. Step 3: Set the expected production delivery date TE, set the production cost of all kinds of MC production class corresponding with the different kinds of ant class Aij at different production stages. Determine the relations between TD and the remained pheromone of all kinds of ants by the way of sampling and analyzing the present data. Step 4: Determine the expected satisfaction level of every optimization index such as the production cost and delivery date according to the historical experience and the data on-the-spot. Step 5: Set and adjust the value of a, b, g and x. Step 6: Generate ant batch t (in the beginning, t ¼ 1) at source node and there are kinds of ant class in every batch. Let the number of the ants in every class be x times as much as the number of the corresponding kinds of orders (3oxo10 is quietly suitable for the small supply chain network). Make the ants move to the object node and die when they arrive. Update the pheromone at every node according to the rule described in the fourth issue in Chapter 4.2 and then let t ¼ t+1. Step 7: Account the ant number passing every node in every batch. Make the judgment of whether the ant number of selecting every node has become stable. We can judge the stable state by comparing the number of the ants selecting one node in this batch with the former batch to see if there is no obvious change, or by observing the number of the ants that always changes in a small

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dictions. The series of the smart simulation toys are main products developed in this company. The product structure is shown in Fig. 6. The major product types (selecting the simulation dog as an example) are shown in Table 3. Customer can choose the existed types or submit orders with the custom types. The supply chain network structure (divided into different stages according to Fig. 6) is shown in Fig. 7. There is more than one cooperator in one ‘group’. There are also other core enterprises in outside supply chain having collaborations with the members of this supply chain during the specific period of time. By using our solutions of MC, the toy Inc. determines dynamically the scheduling adjustment moment according to the different order types and order arrival moments to realize the allocation of the production tasks among the companies denoted in Fig. 7 at different stages. Through six month tracking analysis of its production, the results show that the solutions discussed in this paper has effectiveness in solving the key problem between the ‘scale production effect’ and the ‘customized service level’ in MC and improving the cooperative production efficiency of the supply chain system simultaneously. Here, we give a brief scheduling case of the toys’ control circuits in the electronic apparatus plants (EAP) including EAP1, EAP2 and EAP3 and of the toys’ external simulation furs in the fabric processing factories (FPF) including FPF1, FPF2 and PFP3 in Fig. 7 as an example to verify the algorithm validity. In the verification, the production capacity demand of the tasks to the EAP individuals is 0.75 (the data in this chapter has been normalized and unified) and to the FPF individuals is 0.67. All the individual parameters are shown in Tables 4 and 5. In the algorithm, let the batch equal 100 and let a ¼ 0.50, b ¼ 0.20, g ¼ 0.30 and x ¼ 0.05. Let A denote the ant class choosing EAP individuals and B denote the ant

scope during the recent several batches. Then, we give the assignment of MC tasks to different cooperators according to the number of the ants distributed at the different nodes, calculate the optimizing level of all kinds of indexes and judge if they have reached the satisfaction level. If yes, we can stop the algorithm and turn to step 8, otherwise turn to step 6. Step 8: Output the algorithm result and make scheduling implementation according to it. Here, we should also pay attention to several issues. Firstly, if the move of the ants cannot reach the stable state even they have passed all batches, we should adjust the values of all kinds of parameters to a larger range, which means we should turn to step 5. Secondly, if every index cannot reach the satisfaction level even if the algorithm has run in a long time, we should revise the expected satisfaction level, which means we should turn to step 4. Finally, when the algorithm reaches the stable state, the problems of production congestion at some nodes are disappeared, and even if the quantities of the attraction pheromone at these nodes are larger, the ant current through these nodes will not increase. It is owed to the role of the exclusion probability. 5. Application and simulation The model and algorithm are applicable not only to the production enterprises of MC, but their essential principle and idea are also applicable to the service enterprises of MC. The difference between them lies in the re-classification of the service customization process and of the different order processing. In short, there are two steps for the applications of the model and algorithm to MC practice. Firstly, the scheduling center (core enterprise) should balance the profits of the supply chain system and of the cooperators involved in. Then, it should determine dynamically the reasonable scheduling adjustment moment according to the different order types and order arrival moments. Secondly, at every scheduling adjustment moment, by implementing the algorithm, the core enterprise can allocate rationally the tasks among the supply chain individuals to realize the smooth convergence of the production at different stages and realize the production punctuality. In our study, we select a smart toy Inc (core enterprise) and its supply chain as a tracking target to verify the solutions we put forward to relieve the dominant contra-

Table 3 Main product types of smart simulation dogs. Dog type

Size

Moving type

Smart control type

Delivery type

ZA ZB ZC Custom

L M S Custom

YA YB YC Custom

KA KB KC Custom

Rush

Smart Simulation toy Internal components

Moving Components Motors

Moving Connectors

Control Components

Control Circuits

Control Connectors

Toy Packaging

Metal framework

Finished Products External Simulation Fur

Simulation Fur Basic Additional Material Accessories

Control Software Various Materials

Fig. 6. Product structure division of smart simulation toy.

Semi-finished Products Parts

Components

Materials

Non rush

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Home Appliances Company (Core Enterprise 3)

Group of Machinery Group of Packaging Group of Fabric Companies Processing Factories Assembly Companies

Customers

Number of ants passing EAP individuals 800 700 600 EAP individual 2

Ant number

Clothing Company (Core Enterprise 2)

Group of Home Appliance Customers

Fourth Stage

Smart Toy Inc. (STI) (Core Enterprise 1)

Group of Clothing Customers

Third Stage

Group of Toy Customers

209

500 400 300 200 EAP individual 3

100 EAP individual 1

0 0

Group of Textile mills

Group of Jewelry Processing Centers

40

60

80

100

120

Operating batch

Second Stage

Group of Small Group of Assembly Plants Tanneries

20

Fig. 8. Simulation result about EAP individuals.

Number of ants passing FPF individuals

Group of Software Group of Development Electronic Companies Apparatus Plants

All types of raw material supplies: electronic components, mechanical parts, packing materials, textile materials, chemicals, leather materials, jewelry and other raw materials Fig. 7. Supply chain structure of smart toy Inc.

450 400

FPF individual 1

350 Ant number

Group of Machinery Plants

Materials

Group of Motor Factories

First Stage

500

300 250 FPF individual 3

200 150 100 50

Table 4 Operational parameters of EAP individuals in supply chain.

FPF individual 2

0 10

Operational parameter

EAP 1

EAP 2

EAP 3

Unit production cost C Delivery date tolerance value y Available production capacity

0.44 0.46 0.66

0.63 0.31 0.90

0.15 0.70 0.83

Table 5 Operational parameters of FPF individuals in supply chain. Operational parameter

FPF 1

FPF 2

FPF 3

Unit production cost C Delivery date tolerance value y Available production capacity

0.33 0.45 0.30

0.43 0.22 0.74

0.25 0.36 0.40

class choosing FPF individuals. By computing, the simulation results are shown in Figs. 8 and 9. From Fig. 8 we can know that to ant class A, because its feasible domain (EAP 1, 2 and 3) has no other tasks simultaneously, the production congestion does not exist at these nodes. After several batches of operating, the ant current reaches the stable state and all of the ants select EAP 2 to complete the task. This is because although the

20

30

40

50

60

70

80

90

100

Operating batch Fig. 9. Simulation result about FPF individuals.

unit production cost of EAP 2 is high, its production time is accordance with the basic requirement of MC activity. In Fig. 8, the ant number through EAP 3 ascends first then drops. It is because the initial ascendant unit production cost of EAP 3 attracts a lot of ants. With the algorithm running, the ant number is oriented to the control of the tolerance value of the delivery date then decreases. This reflects the operational flexibility of the ant algorithm in solving the complex multi-objective supply chain optimization in MC. For ant class B, from the view of the production capacity constraint, only FPF 2 can complete the tasks independently. However, due to its high unit production cost, FPF 1 and 3 are chosen by the ants ultimately. See from Fig. 9, the algorithm has greater flexibility in settling the capacity constraint of the production or service activities. This is the role of the exclusion probability in the algorithm and will reflect its ability to settle the congestion at some nodes in the supply chain.

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The simulation results also indicate that we can obtain the better convergence time and effect by the appropriate adjustment of the parameter (a, b, g and x) value according to the practical circumstances of the supply chain scheduling in MC and obtain the satisfactory results. 6. Conclusions The supply chain scheduling in MC has the complexities of the two aspects initiated, respectively, by MC and supply chain environment at the same time. Based on our earlier work, this paper probes the relevant solutions of them, mainly embody the follows. 1. We briefly sum up the ideas to relieve the dominant contradictions which reflect the relations between the scale production effect and customized satisfaction level and reflect the complicated cooperative and competitive relations among the cooperators in the supply chain. In relieving the first one, we put forward the thoughts of a developed order classification, the time threshold and the secondary order classification based on the time threshold. In relieving the second, we point out the rational profit preference decision of the supply chain cooperators is the key way. 2. We built a dynamic, multi-objective mathematical optimization model and put up an appropriate optimization algorithm to reflect the complicated process of the supply chain scheduling in MC and to solve the bottlenecks by introducing the ideas of relieving the dominant contradictions into the scheduling. The characters of the model and algorithm cannot only reflect the unique requirements for the operating characters of MC, but can also introduce the thought of relieving the dominant contradictions into the scheduling implementation. 3. We also point out that there are complicated problems when implementing the supply chain scheduling in MC, such as the problem of the production congestion at some nodes, the problem of changing the allowable field, the problem of the punctual delivery, the description of the dynamic characters and the handling of the irregularities and so on. Then, we consider the solutions of these problems when design the model and algorithm. In addition, as a study for this cutting-edge research topic in supply chain, this paper raises several relevant questions for further consideration. Several aspects that we think are worthy of additional work. For example, in the scheduling, how can core enterprises and other numbers capture the more accurate information of the production parameters; combining with the operational characters of MC, how to solve the dynamic and stochastic problems more efficiently; reflecting in this article, that is how to determine the time threshold more rationally and flexibly and how to establish a reasonable procedure mechanism from the ‘dynamic sampling’ to ‘local mathematical analysis’ to ‘dynamic control’; the extra inventory problem is also a special expression of the

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