[?1]Det ? ???.. ?? ??? ???? ???

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

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

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??? ?????

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

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.