1 INTRODUCTION The iron & steel industry is the pillar industry of the national economy with the huge energy consumption and the serious environmental pollution in the production process. Andersen[1], Larsson[2, 3], Sakamoto[4], Dutta[5], et al, in the study of energy distribution model, replace different fuel by equivalent heat to analyze and evaluate the energy consumption situation and level of the large iron & steel enterprise by the unified of perspective and standard, and then the improved measures were taken to solve it. For the continuous and discontinuous characteristics of the processes in the large iron & steel complex, Tang, et al [6, 7] found the key theoretical issues of the planning and scheduling optimization from the actual production management, and conducted an in-depth basic study. The above study results will help us understand and further study the energy saving and emission reduction issues in the large iron & steel complex. In order to give full play to the production ability of every process, we will study the resource scheduling problems for the purpose of the reduction-manufacturing. The establishment of appropriate mathematical models and algorithms are used to simulate and determine the less inputs and more outputs resource optimal allocation scheme, which can help the concerned sector to make decision.

This work is supported by National Nature Science Foundation under

Grant 61174183.

2 MATHEMATICAL DESCRIPTION OF THE LOGISTICS IN THE IRON & STEEL PRODUCTION

Figure 1 describes the five processes of coking, sintering, iron-making, steel-making and rolling and the material flows among them in a large iron & steel enterprises.

Fig 1. General situation of material flow in the production processes of the

large iron & steel enterprise Assume that (1) There are n supplies of inputs and outputs in the large iron & steel enterprise, numbered from 1 to n . (2) )(iP denotes the i th process, where 5,4,3,2,1=i denote the five processes of coking, sintering, iron-making, steel-making and rolling respectively.

(3) )(iP

jx and )(iP

jy denote the input or output number of

the j th material in the process of )(iP , where nj ...,,2,1= .

Mathematical Model and Algorithm of Optimal Resource Allocation in the Large Iron & Steel Complex

Deng Pan, Yingping Zheng School of Electronic & Information Engineering, Tongji University, Shanghai 201804, China

E-mail: pandengreal@126.com

Abstract: The reduction-producing is an important means of energy saving and emission reduction. For this purpose, the augmented vector is used to describe the inputs/outputs of every process in the large iron & steel complex, and then the optimal resource allocation is studied under the conditions of current equipment, technology and the operational level, the mathematical model is constructed to find an optimal resource allocation scheme for every process. The genetic algorithm is used to simulate the resource allocation of the large iron & steel complex, and determine the optimal resource allocation scheme. The taken measures include the reduction of the purchased materials, the full utilization of the recycled materials and the reasonable allocation of the input and output materials of every process. The simulation shows the validity of the mathematical mode and its algorithm. The purpose of energy saving and emission reduction can be achieved to some extent. Key Words: Large Iron & Steel Complex, Reduction-producing; Optimal Resource Allocation; Mathematical Model; Algorithm

3074978-1-4673-5534-6/13/$31.00 c©2013 IEEE

We set

[ ]TiPn

iPj

iPiPiP xxxxx )()()(2

)(1

)( ......= (1)

[ ]TiP

niP

jiPiPiP yyyyy )()()(

2)(

1)( ......= (2)

and have )()()( iPiPiP xX ∗= α (3) )()()( iPiPiP yY ∗= β (4)

where

=

)(

)(

)(2

)(1

)(

0

0

iPn

iPj

iP

iP

iP

α

α

αα

α ,

(5)

=

)(

)(

)(2

)(1

)(

0

0

iPn

iPj

iP

iP

iP

β

β

ββ

β (6)

)(ipjα and )(iP

jβ are equal to 0 or 1 ( nj ,...,2,1= , and can be determined by the practical situation of every process. The “0” denotes that the process )(iP need not put into or cannot produce the jth material in the process of the normal producing. The “1” denotes that need put into or can produce the jth material in the process of the normal producing.

3 MATHEMATICAL MODEL OF OPTIMAL RESOURCE ALLOCATION

3.1 Modeling Steps

Every process or every sub-process is not isolated, need to be coordinated with each other to achieve the maximum matching of its production capacity in the process cycle time. The so-called "matching" is not only the matching of the input materials with the output one in number but also in the quality requirements. The steps of building mathematical model of optimal resource configuration can be described as follows: (1) The input of process )(iP is divided into the materials from itself or other processes and the purchased materials, as shown in Figure2.

Fig 2. Matching model of production between the series process The materials from itself or other processes can be expressed as follows:

[ ]TiPn

iPj

iPiPiP xxxxx )()()(2

)(1

)( ......= (7) The purchased materials can be expressed as follows:

[ ]TiPn

iPj

iPiPiP xxxxx )()()(2

)(1

)( ...... ΔΔΔΔ=Δ (8) The input data of specific processes are to be augmented by the above vectors, the actual input data, which is the vectors of the material from itself or other processes and the

2013 25th Chinese Control and Decision Conference (CCDC) 3075

purchased materials (see (7) and (8)), can be respectively expressed below:

)()()( iPiPiP xX ∗= α (9) )()()( iPiPiP xX Δ∗=Δ α (10)

The value of )(ipjα is same with (5).

(2) As shown in Figure 2, based on the requirements of product quality, the output products of the process )(iP can be divided into the qualified products and the

unqualified products according to the main direction of the processes. The qualified products can be divided into the supplies to the next process and the unified scheduling supplies again. The qualified supplies to the next process can be expressed as follows:

[ ]TiPiPn

iPiPj

iPiPiPiPiPiP yyyyy )1()()1()()1()(2

)1()(1

)1()( ...... +→+→+→+→+→ = (11)

When i =5, )1()( +→ iPiPy does not denote the input of next process in the main direction but the end of production process. The unified scheduling qualified supplies, including the recycling waste such as scrap steel, miscellaneous iron, can be expressed as (12).

[ ]TiPn

iPj

iPiPiP yyyyy )()()(2

)(1

)( ˆ...ˆ...ˆˆˆ = (12) The unqualified products, including the emission of waste, can be expressed as (13).

[ ]TiPn

iPj

iPiPiP yyyyy )()()(2

)(1

)( .....= (13) The actual output products include the qualified one to next process, the unified scheduling qualified one and the unqualified one, as shown in (14), (15) and (16) respectively.

)1()()()1()( +→+→ ∗= iPiPiPiPiP yY β (14) )()()( ˆˆ iPiPiP yY ∗= β (15) )()()( iPiPiP yY ∗= β (16)

The value of )(iPjβ is same with (6).

(3) Calculation of the supplies )1( +iPX from every process to the process )1( +iP

The calculation of the supplies )1( +iPX from every process to the process )1( +iP is to calculate the input materials of the process )1( +iP from itself and other process, which is a part of the unified scheduling qualified supplies

)(ˆ lPY ( 5,4,3,2,1=l ) from every process, i.e. )1()(ˆ +→ iPallPY and )1()( +→ iPiPY .

We have

=

++→ ∗=5

1l

)l(P)1i(P),l(P)1i(P)all(P YY γ (17)

and )1()()1()()1( ˆ +→+→+ += iPallPiPiPiP YYX (18)

where

=

+

+

+

+

+

)1i(P),l(Pn

)1i(P),l(Pj

)1i(P),l(P2

)1i(P),l(P1

)1i(P),l(P

0

0

γ

γ

γγ

γ , (19)

10 )1i(P),( ≤≤ +lPjγ and 10

4

0i

5

1

)1i(P),( ≤≤= =

+

l

lPjγ ,

5,...,2,1=l , nj ,...,2,1= . (4) Optimization of resource allocation The optimizing of resource allocation is to give full play to the production ability of every process. Assume that

)1(max

+iPX denotes the maximum production ability of the

process )1( +iP . Under the optimization of resource

allocation, the purchased materials )1( +Δ iPX can be got by the below equation.

)1()1()1(max

)1( ++++ ∗−=Δ iPiPiPiP XXX α (20)

By (20), we can see that the minimum purchased materials can be realized by making full use of recycling materials.

3.2 Mathematical Model

By (7) to (20), we can get the objective function of optimizing resource allocation in the large Iron & steel Complex, as shown in (21).

)ˆ

(5

1

)()()(

)1()()()1()1(max

)1()1()1(max

)1(

=

+→++

++++

∗∗

+∗∗−=

∗−=Δ

l

lPlPlP

iPiPiPiPiP

iPiPiPiP

y

yX

XXX

βγ

βαα

(21)

(1) Function of optimal objective

3076 2013 25th Chinese Control and Decision Conference (CCDC)

=

+

=

Δ=Δ4

0

)1(5

1

)(

i

iP

i

iP XX

=

+→++

∗∗

+∗∗−=5

1

)()()(

)1()()()1()1(max

|ˆ

(|

l

lPlPlP

iPiPiPiPiP

y

yX

βγ

βα 22

(2) Constraint condition i) )(iP is the i th process, where 5,...,2,1=i denotes the five processes of coking, sintering, iron-making, steel-making and rolling respectively.

ii) =

)(

)(

)(2

)(1

)(

0

0

iPn

iPj

iP

iP

iP

α

α

αα

α ,

)(ipjα is equal to 0 or 1.

iii) =

)(

)(

)(2

)(1

)(

0

0

iPn

iPj

iP

iP

iP

β

β

ββ

β

O

O,

)(iPjβ is equal to 0 or 1.

iv)

=

+

+

+

+

+

)1i(P),l(Pn

)1i(P),l(Pj

)1i(P),l(P2

)1i(P),l(P1

)1i(P),l(P

0

0

γ

γ

γγ

γ

,

10 )1i(P),( ≤≤ +lPjγ and 10

4

0i

5

1

)1i(P),( ≤≤= =

+

l

lPjγ

5,...,2,1=l nj ,...,2,1= .

v) The purchased materials 0)( ≥Δ iPX , and

0)( ≥Δ iPjx , 5,...,2,1=i .

vi) )()(max

iPiP XX ≥ , 5,...,2,1=i .

vii) According to the main direction of production process, when i =5, )1()( +→ iPiPy does not denote the input of next process in the main direction but the end of production process.

viii) Under the adequate purchased materials, the maximum production ability of every process will use to maximize its output.

)(max

)()()1()()( )ˆ( iPiPiPiPiPiP Yyyy =++∗ +→β (23) where 5,...,2,1=i and the qualified rate

=

)(

)(

)(2

)(1

)(

0

0

iPn

iPj

iP

iP

iP

μ

μ

μμ

μ

O

O (24)

has been known. )(iPjμ can be express as follows:

%100)ˆ(

)ˆ()()()1()()(

)()1()()()( ∗

++∗+∗

= +→

+→

iPj

iPj

iPiPj

iPj

iPj

iPiPj

iPjiP

j yyyyy

ββ

μ (25)

where 5,...,2,1=i , nj ,...,2,1= and

%10000)( ==iP

jμ.

By (23) (25), we can get )ˆ( )()1()()( iP

jiPiP

jiP

j yy +∗ +→β , which is the number of every product when every process give full play to its maximum production ability.

(3) The set of the initial value: )(iPα and )(iPβ have been known. Under the adequate purchased materials, the maximum production ability of every process will use to maximize its output, i.e. )(

maxiPX and )(

maxiPY have also been

known.

4 ALGORITHM OF OPTIMIZING RESOURCE ALLOCATION

Step1: we set 1−=cycle and 10000)1( =−z .

Step2: we set 1+= cyclecycle and the other initial values are determined by Section 2.2.

Step3: After the coding of )1()( +→ iPiPy and )(ˆ iPjy , the

selection, crossover and mutation operators are used to calculate the vectors of )1()( +→ iPiPy and )(ˆ iP

jy according to the probability. Simultaneously, the determination of

)1()( +→ iPiPy and )(ˆ iPjy must satisfied the constraint

conditions listed in viii) of Section 3.2.

Step4: After the coding of )(iPγ , the selection, crossover and mutation operators are used to calculate the vectors of

)(iPγ according to the probability. Simultaneously, the

determination of )(iPγ must satisfied the constraint conditions listed in iv) of Section 2.2.

2013 25th Chinese Control and Decision Conference (CCDC) 3077

Step5: By substituting the initial values and the vector values of )(

maxiPX , )1()( +→ iPiPy , )(ˆ iP

jy , )(iPγ ( 5,...,2,1=i ) into the function of optimal objective

(22) to get =

Δ5

1

)(

i

iPX.

Step6: Assume=

Δ=5

1

)()(i

iPXcyclez ,

)1()( −≤ cyclezcyclez ? Then go to Step2, else set

)1()( −= cyclezcyclez , go to Step2. This step is used to determine the scheme of optimal resource allocation. Step7: Assume that ξ is an arbitrarily small positive data, when 200≥cycle , we calculate

|100/|))(|(|)(||99−=

−cycle

cyclem

mzqz , where

cycleqcycle ≤≤− 99 and Iq∈ . Go to Step8 if

ξ≤−−=

|100/|))(|(|)(||99

cycle

cyclem

mzqz is true to every q ,

else go to Step2.

Step8: We get )(iPXΔ ( 5,...,2,1=i ) of ever process

while =

Δ=5

1

)()(i

iPXcyclez , which are the optimal

values of the purchased materials of every process. The vector of

)ˆ(5

1

)()()()1()()()1(

=

+→+ ∗∗+∗∗l

lPlPlPiPiPiPiP yy βγβα

in the cycle th cycle are the optimal allocation scheme of every process, including the flow volume and flow directions of every material. The optimization configuration is complete.

5 EXAMPLE AND SIMULATION Taking the production of a large iron & steel complex in 2012 in southern China for an example, we use the genetic algorithm for the verification of our mathematical model and algorithm. We take 60=n and 5.0=ξ . Figure 3 describe the simulation of the genetic algorithm for our mathematical model.

Fig 3. Simulation of optimal resource allocation in the large iron & steel complex

With the increase of the parameter cycle , |)(| cyclez drops rapidly and shows a gradual convergence property so that we can get the optimal allocation scheme of every process. Comparison with the situation of this enterprise in 2011, the energy consumption of one ton steel reduced ranged from 3.675%~3.892%.

Fig 4. Decline rate of energy consumption of one ton steel

The simulation shows the effectiveness of our mathematical model and algorithm, which can be put into practice for further justification, improvement and enhancement.

6 CONCLUSION Under the condition of making full use of the purchased material to give fully play to the maximum production ability of every processes, the recycling of every materials, the reasonable distribution of the flow volume, and the control of the flow direction are used to achieve the optimal resource allocation of the large iron & steel complex so that the purchased materials can be reduced to minimum. The production processes of the large iron & steel enterprise are very complex, with the producing characters of continuous and discontinuous and different from the general assembly line. We study the scheme of the optimal resource allocation to make full use of the maximum production ability of every process in order to help every process complete the production target within the least time in its production cycle time. The optimization of the purchased materials realize the reduction-producing by the reasonable allocation and the full use of the recyclable materials, as a

3078 2013 25th Chinese Control and Decision Conference (CCDC)

result, play a role in energy conservation and emission reduction to some extent.

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models for the US steel industry, Energy, Vol.26, No.2, 137-159, 2001.

[2] M. Larsson and J. Dahl, Reduction of the specific energy use in an integrated steel plant-the effect of an optimization model, Iron Steel Inst Jpn, Vol.43, No.10, 1664-1673, 2003.

[3] M. Larsson , C. Wang and J. Dahl, Development of a method for analyzing energy, environmental and economic efficiency for an integrated steel plant, Applied Thermal Engineering, Vol.26, No.13, 1353-1361, 2006.

[4] Y. Sakamoto, Y. Tonooka and Y. Yanagisawa, Estimation of energy consumption for each process in the Japanese steel industry: a process analysis, Energy Conversion & Management, Vol.40, No.11, 1129-1140, 1999.

[5] G. Dutta and R. Fourer, A survey of mathematical programming applications in integrated steel plants, Manufacturing & Service Operations Management, Vol.3, No.4, 387-400, 2001.

[6] L. Tang, J. Liu, A. Rong and Z. Yang, A multiple traveling salesman problem model for hot rolling scheduling in Shanghai Baoshan Iron & Steel Complex, European Journal of Operational Research, Vol.124, No.2, 267-282, 2000.

[7] L. Tang, Intelligent Optimization-based Production Planning and Scheduling in Iron & steel Industry, Chinese Journal of Management, Vol.2, No.3, 263-267, 2005. (in Chinese)

2013 25th Chinese Control and Decision Conference (CCDC) 3079