?1Det ? ?? ?? ????? ?? ???? ????? ?? ?, t=10?? ???/

Ro? ?? 13??? ????

??? ?????

Det? ?? (????) t=17? ? ????.

Penalty cost? det?? t=11?? ??

Ro? t=14?? ?? ?????(??? ???)

Triangle? ?? Det? t=14? ?? ???????.

?? ????? det(t=11)/ro(t=14)? ????.

v2Deterministic ?? ????? ? ?? ???

Penalty cost ????? ??????

–

Deterministic? penalty cost? ??? ??? ? ????…

v3Other results? ? ?????

v4Table 4? 0.1? ?????

?? ????

???? ? ???? 0.1? ???? ?? ?? ?????

??? ?.. ?? ??, 0.01, 0.05, 0.1, 0.2 ???…

??? triangle distribution? ????

??, deterministic ?? ??? ? ???.. ??

??? ???? ??? ???.

?5?? 0.1? ??????? ? ????.

Deter /box(3???) /distri p=0.6 ?? ??? ??????.

<4??? distri p = 0.5 ? ????.>

Our model solves the extreme

conservatism problem by yielding stable results in any distribution.Robust optimization is added

to the proposed model which shows effective response to processing volatility

according to personalized demand. We found The Robust optimization is more

stable in various uncertainties. Furthermore, we proposed a

distributionally robust model to solve the conservancy of the existing robust

model and obtained better results. And the

We propose a model that considers

all factors to solve the problem of optimal solution guarantee of the

individual decision-making model. Decentralized systems, in which existing

companies share the process, do not guarantee the optimal solution, but they

can guarantee the optimal solution considering all factors such as time,

process, and factory.

5. Conclusion

Finally, various

are

considered to compare the performance of the solutions as in Table 5. The

results are also more stable in any uncertainty. And we can see that the volatility becomes smaller

considering the distributionally robust model with higher probability.

=0.4

=0.5

=0.6

=0.7

Mean

219,450

190,158

188,013

185,190

Standard deviation

3,162

904

698

0

Max

228,450

190,560

189,780

185,190

Table

5 Mean value compared with various

v4 in Triangle di?5 stribution.

Robust model, as mentioned above,

has higher objective function cost than deterministic model because it

constructs optimal network against the worst case. However, we can

see the results

v3 in 100 simulation

experiments with uncertain

. The consideration of the worst case increased the

cost of objective value, but guaranteed a more stable solution as Table 3 and Table 4. Also, distributional model is better than RC with box set in mean and less

than RC with

box set in variance. It is shown that the robust model guarantees an appropriate result in

the uncertain situation than the deterministic case, and distributionally robust

model outperforms the traditional RC with box uncertainty set. Next, we have

also experimented with uncertainty having a wide range z.

Deterministic

RC

with box set

Distributionally Robust

Mean

200,245?1

194,502

188,013v2

Standard deviation

105

965

698

Max

200,345

196,580

189,780

Table

4 Mean value compared with all models in Triangle distribution.

Deterministic

RC

with box set

Distributionally Robust

Mean

190,345

189,598

177,145

Standard deviation

2,456

Max

200,345

189,598

177,145

Table

3 Mean value compared with all models in Normal distribution.

Random

processing time data are generated from a normal distribution and a triangle

distribution, and simulation experiments are conducted to compare the

performance of the each solution (i.e. deterministic solution, robust solution

with box uncertainty set, distributionally robust solution) in the face of

uncertainty. The method of

simulation proceeds as follows.

After fixing the network consisting of the results of decision variables

related to supply network design as each of the three models, the performance of the network

with the decision variables related supply network operations is computed by

generating 100 uncertainties. And if there are productions that take

time-limit, treat them as penalty cost. So that only penalty cost is

calculated at variable cost. The results are summarized in Table 3 and Table 4.

4.2.

Simulation Experiments

Deterministic

RC with box set

Distributionally

robust

Objective

value

158,590

198,644

187,435

Lead

time

10

10

10

Table

2 Objective value table

Figure 2: Connected smart factory network in

process.

As

shown in Figure 2, each model

generated different network. This is because the deterministic

model can obtain the optimal value with all fixed parameter values, while RC

with box set and distributionally robust model have the solutions corresponding

to the worst-case value and feasible for the any value realized from the

partially known information. Table 2 shows that the objective function value and production

lead time from each model. It may look like that deterministic model is the

best solution with the least objection function value, but it is not true when

the production operations are executed as planned in the face of uncertainty

4.1.

Objective Function Values

Parameters

Value

D

100

TL

14

2098, 3750

0.81, 0.93

1060, 1870

38,65

164,000, 177,000

2500, 4500

Table

1: Input data

, where

and

are lower bound

and upper bound, respectively. 3) For a distributionally robust solution with uncertainty set,

processing time uncertainty is considered with any distribution. Next, processing time is

randomly generated and realized objective values are calculated by implementing

the deterministic solution and robust solution. The parameter values used for

the numerical experiments are summarized in Table 1. We used GAMS (The General Algebraic Modeling

System) version 24.85 to solve the deterministic model and robust counterpart,

and Cplex solver was used. The CPU

of the computer is Intel Core (i) Core i5-6400 @ 2.70GHz (4 CPUs), ~ 2.7GHz,

memory is 8GB, and operating system is Windows 10 Pro 64 bit operating system.

= {

,

}

In this section, we analyze the performance

of the proposed model. First, we obtain three

optimal solutions: 1) For a deterministic solution, it is assumed that a

decision maker do not consider uncertainty in Equation (10) and

2) For a robust

counterpart (RC) with box set, processing time uncertainty is considered with

, considered for the capacity

constraint. It is assumed that

processing time is uncertain and belongs to a specific uncertainty set

Uncertainty set is considered for the uncertain processing

time with

of

uncertain data.

4. Numerical experiments

,n,t (20)

In the equivalent robust

counterpart, Equation (18) is replaced by Equation (20).

We note that

can be any value in the uncertainty set

. Therefore, Equation (18) must be replaced by a equivalent deterministic constraint in

uncertain data .

Equation (11) is the objective function that consists of

factory operation cost, setup cost, production cost, and transportation cost. Equation

(12) is a constraint on the balance of raw material inputs and outputs at each operation

in each plant. The transportation takes place considering the lead time

from

factory i to factory j. Equation (13) is a constraint on the

balance of processed parts or finished goods in each process in each factory. Equations

(14), (15), and (16) use the Big M condition to determine whether

and

are selected

or not. Equation (17) is a demand fulfillment condition. Equation (18) is a capacity

constraint of each process, and uncertain processing time is considered with

the box set defined in Section 3.1. Finally, Equation (19) is a time limit

condition of the final process time for meeting due date.

,

,j,n,t

,

,n,t

t n=final (19)

,n,t

(18)

,t n=final

(17)

,t n=final (16)

,n,t (15)

,n,t (14)

,j,n,t (13)

,j,n,t (12)

(11)

Model

now present the distributionally robust model as follows:

3.2. Model

Therefore, the (2) is satisfied

Since the superior value of

is less than the minimum value of

, the minimum value must be found. If we differentiate by b, we can

easily find the minimum value of b. Since only the outside of Chebyshev

inequality has been considered, the

is

Thus

So (6) is reformulated by (8)

According to the external characteristic of Chebyshev inequality, (6)

follows the inequality.

(6) can be transformed as follows.

(5) is equivalent of (6). And b is a variable.

It is because One-Side-Chebyshev inequality. (4) develops as follows.

First, as we all know, the standard deviation is

(4)

Then, we can find superior probability with mean and variance 11, When

Robust

model finds the inferior value of the probability because it needs to find the

worst case. So model need to find

. It is equivalent to the convex second-order

constraint

(3)

For any distributionally robust model, we must find the appropriate minimum value

for the uncertain distribution

.Distributionally robust follows through this process 10.

so

and

(2)

where

.

A classical chance

constraint approach to the solution of under uncertainty is to

formulate Equation (1) with the concept of probability 9.

In a personalized production system, processing time is more uncertain than mass production system, in which the same product is produced repeatedly and processing time is almost constant when the process is stable. However, in a personalized production system, it is difficult to calculate or estimated accurate processing time because a variety of products are made in small quantities. Traditionally, the robust problem with box uncertainty set with given minimum and maximum values ??may lead to very conservative results. In this section, we show the process of solving a robust problem that is less conservative when knowing the mean and variance of uncertain parameters. Prior to this, we apply the following steps for the reformulation.

3.1. Distributionally

robust model

Figure 1: Diagram of the problem

Figure 1 shows a brief process of the proposed model. All process have

in-out buffer. Process n at factory j get raw materials{

} from previous process. And process

makes products with the raw materials {

}. The products are delivered to the next process along with

production stocks

, and the factories have their respective delivery lead times. Every

factory goes through this series of processes and provides personalized

products for customers.

n

TL Time limit

A

mixed-integer programming model is developed to design a smart supply chain and

determine production plan simultaneously. In other words, the proposed an optimization

model determines not only smart factories within CSFN and processes in the

selected smart factories, but also the time when customized products will be

made. Three indices including ‘factory(i)’,’process(n)’, and ‘time(t)’ are introduced to find an optimal solution of the

integrated decision structure with the following notations:

3. Mathematical Model

In this paper, we develop a mathematical

model for smart supply chain design and operations of CSFN in which smart

factories can share their processes and resources through ICT. Due to the

characteristics of the manufacturing sector, there is a statistical

distribution of the production system, and a smart supply chain is constructed

using the partially knwon statistical distribution.

The solution of the optimization under uncertainty is to select an X

that can feasible for all possible constraints of the problem with the

uncertainty set U. And it ensures stability. However, securing the

extreme stability assuming a worst case can be also very conservative. Many

studies have attempted to solve the extreme conservatism of the robust model

and have shown meaningful results 8,9. These studies are applicable to

various models and show good solutions. The robust model for solving extreme

conservatism starts from a few basic assumptions. For example, distributionally

robust optimization begins with the assumption that there is a distribution of

uncertainty and that we know the partial information on the distribution such

as mean and variance.

s.t.

In problem (1), A, B and C are given parameters, and

objective function is to find feasible X. Assume that parameter A

is uncertain, then we solve the following problem;

s.t.

(1)

The goal of robust model is to get an optimal solution which is stable

under uncertainty sets U in parameters. So all sets are met

under the condition that the Worst case in the box sets is met. For example,

consider a optimization

problem.

2.2. Robust

Optimization

In a series of manufacturing developments, consumers are demanding more and more customized demand, and to meet these needs, manufacturing enterprises are building collaborative processes through various technologies.From the 21st century, there is a new paradigm throughout the manufacturing industry, such as service-oriented architecture (SOA) 3 and IoT 4. Especially, with the development of communication technology, the rise of IoT technology has introduced a new concept of integration by facilitating information 4. This new concept means interconnected supply chains that consider multiple factories in a network at a plant through the new technologies 5. There are also many research for resource allocation for virtual manufacturing 6 and agile manufacturing 7. Traditionally, manufacturing has reduced costs and achieved economies of scale through mass production. The development of multiple technologies such as computer-aided-design(CAD) has led to more advanced manufacturing processes, which have improved personalized demand and product quality. In the meantime, the emergence of Additive Manufacturing (AM) presented a new concept of the integration concept of manufacturing 1. Based on the development of various technologies and the AM concept, manufacturers began to focus on providing consumers with better quality products. Therefore, many manufacturing companies have focused on mass customization to achieve the economy of scope 2.

2.1. Smart

Supply Chain Management

2. Literature review

Therefore, the concept of smart supply chain to be discussed in the future

should be considered together with uncertainty, and a robust model against

uncertainty must be considered together.

On the other hand, many

factories have individual and diverse uncertainties, such as weather, power

outages, and workers’ ability. Customization in the smart supply chain also implies uncertainty. Factory production

capacity is always irregular, and irregular supply chains, especially with

customized consumption, can increase the uncertainty of processing times. But existing optimization model

is very vulnerable to this uncertainty because it obtains a highly variable

optimal solution by parameters.

We refer

to the new supply chain with a concept of the CSFN as the smart supply chain.

The smart supply chain must be able to respond quickly to the small demands of

personalized products and have flexible connections between factories.

The CSFN

is an environment in which a supplier, factory or customer in the network are

connected through the cloud. With the CSFN, participants who are members of the

network can freely share information and each member can make better decisions.

Consumers can be more specific about their needs through the network. They want

to customize consumption in the network. Factories share the resources and

availability of each other to build an optimal supply chain. It also

establishes an inter-factory network to respond flexibly to the consumer’s

customized needs. This CSFN should be easy to connect and disassemble.

Suppliers can make cost-effective decisions by making optimal decisions based

on these inter-factory networks and customized consumer needs. Thus this

enables better planning and collaboration of the supply chain.

As we

enter the fourth industrial revolution, manufacturing is in a new phase. In

fact, some factories are trying to reduce costs by combining various advanced

technologies. In particular, Internet of Things (IoT) and Cloud technologies are the most popular in the manufacturing industry.

These two technologies are regarded as cutting-edge technologies that enable

free resource sharing between factories, and are being integrated through

various experiments. If these advanced technologies enable smooth sharing of

resources between factories, a new concept of smart factory that is different

from the existing one can be considered. We call these factories the Connected Smart

Factory Network (CSFN).

1. Introduction

Keywords: Smart Supply Chain, Personalized

Production, Uncertainty, Robust optimization.

Abstract. In the era of the fourth industrial

revolution, new technologies such as IoT, Cloud and 3D printers are integrated

into manufacturing system. In particular, connected smart factories are

expected to efficiently produce a variety of personalized products with small

lot size. Therefore, it is necessary to manage new supply chain based on

connected smart factories differently from the existing supply chain for mass

production. In the new environment, processing time may be not stable and

different depending on the factory environment because it produces a small

amount of product. In this paper, we propose a distributionally robust

optimization model to construct and operate a smart supply chain by sharing

resources of smart factories within a given lead time at a minimum cost in the

face of processing time uncertainty. It overcomes the conservativeness

issue of the traditional robust optimization model with box uncertainty set. Simulation experiments

demonstrate the outperformance of the proposed model compared to a deterministic

model and robust counterpart with box uncertainty set in terms of robustness

against uncertainty.