Piecewise Linear Approximation Based MILP Method for PVC Plant

Subscriber access provided by AUSTRALIAN NATIONAL UNIV

Article

Piecewise Linear Approximation based MILP Method for PVC Plant Planning Optimization Xiaoyong Gao, Zhenhui Feng, Yuhong Wang, Xiaolin Huang, Dexian Huang, Tao Chen, and Xue Lian Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b02130 • Publication Date (Web): 19 Dec 2017 Downloaded from http://pubs.acs.org on December 29, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Industrial & Engineering Chemistry Research is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Piecewise Linear Approximation based MILP Method for PVC Plant Planning Optimization Xiaoyong Gao1, 6, Zhenhui Feng2, Yuhong Wang2*, Xiaolin Huang3, Dexian Huang4*, Tao Chen5, Xue Lian2 1. Institute for Ocean Engineering, China University of Petroleum, Beijing 102249, China 2. College of Information and Control Engineering, China University of Petroleum, Qingdao, Shandong 266580, China 3. Institute of Image Processing and Pattern Recognition, Shanghai Jiao Tong University, Shanghai 200240, China 4. Department of Automation and Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China 5. Department of Process and Chemical Engineering, University of Surrey, Guildford GU2 7XH, U.K. 6. Department of Automation, China University of Petroleum, Beijing 102249, China ABSTRACT： This paper presents a new piecewise linear modelling method for the planning of polyvinyl chloride (PVC) plants. In our previous study (Ind. Eng. Chem. Res., 2016, 55 (48), 12430–12443), a multiperiod mixed-integer nonlinear programming (MINLP) model was developed to demonstrate the importance of integrating both the material processing and the utility systems. However, the optimization problem is really difficult to solve due to the process intrinsic nonlinearities, i.e. the operating cost or energy consuming characteristics of calcium carbide furnaces, electrolytic cells and CHP units. The present paper intends to address this challenge by using the piecewise linear modelling approach that provides good

ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

approximation of the global nonlinearity with locally linear models. Specifically, a hinging hyperplanes (HH) model is introduced to approximate the nonlinear items in the original MINLP model. HH model is a kind of continuous piecewise linear (CPWL) model, which is proven to be effective for any continuous linear functions with arbitrary dimensions on compact sets in any given precision, and is the basis for the linearization MINLP model. As a result, with the help of auxiliary variables, the original MINLP can be transformed into a mixed-integer linear program (MILP) model, which then can be solved by many established efficient and mature algorithms. Computational results show that the proposed model can reduce the solving time by up to 97% or more and the planning results are close to or even better than those obtained by the MINLP approach.

KEYWORDS: PVC, Multiperiod planning, MINLP, Piecewise linear approximation, HH, MILP.

1．． Introduction The PVC production by calcium carbide method accounts for over 80% market shares in China. 1 The substantial energy consumption associated with this production method has provided strong motivation for optimal operation of PVC plants, and plant-wide planning optimization has become an attractive option. In our previous work,2 multiperiod planning of a PVC plant was considered, integrating both the material processing and the utility systems, and an MINLP model was proposed. The results showed that the total cost saving amounts to 3.35% on average. However,


Page 2 of 37

Page 3 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


because of the highly nonlinear energy consumption characteristics and multiple parallel facilities to be considered, the resulted integrated model is too large to solve in reasonable and acceptable time when confronted with industrial scale problems. Accelerating the computation is an urgent but unmet need in this area. With respect to MINLP problems, there are many research reports and commercial solvers. Jaffal et al. presented a K-best branch and bound technique for MINLP problem,3 which decreases the computational complexity of the conventional branch and bound technique in the context of dynamic resource allocation for multiuser downlink Orthogonal Frequency Division Multiplexing systems. Aras et al. proposed a hybrid branch and bound (BNB) approach, and the results revealed that the hybridization of BNB with particle swarm optimization (PSO) and genetic algorithm (GA) was superior to other algorithms on optimal design of shell and tube heat exchangers.4 Méndez et al. replaced a very complex MINLP formulation by a sequential MILP approximation for the blending and scheduling of large-scale problems in oil-refinery applications.5 An improved PSO algorithm is proposed to solve NLP/MINLP problems with equality constraints by Luo in 2007.6 Besides, evolutionary

algorithms,7

Bernstein

global

optimization

algorithm8

and

outer-approximation algorithm9-10 have been widely used to solve the problem. Despite these valuable progresses on MINLP solving, due to the large scale of industrial PVC planning problem, its unsatisfying convergence speed is also the main obstacle for industrial use. Especially, it is easy to fall into a local optimum. The efficient solution for industrial planning problem remains challenging.



Compared with the hard-to-solve MINLP, there are well established and computationally efficient solvers, both open source and commercial, for MILP. To avoid the solving difficulty from the model nonlinearity, a natural idea is to approximate nonlinear characteristics by piecewise linear (PWL) models. And original MINLP can thus then be replaced by MILP. There are lots of fruitful results in PWL representations, such as hinging hyperplanes model,11 generalized piecewise linear functions,12 high level canonical piecewise linear functions,13 generalized hinging hyperplanes (GHH) model14 and adaptive hinging hyperplanes (AHH) model.15 PWL has an explicit property of global nonlinearity and local linearity, and thus, it provides an appropriate global approximating performance while maintaining the local linearity. More recently, Gao et al. presented a novel high level canonical piecewise linear model based on the simplicial partition16 and then applied it in oil refinery scheduling optimization problem.17 Besides, the piecewise linear approach has widely used in model predictive control and nonlinear process description.18-19 However, there are few applications in the industrial process optimization, particularly in planning and scheduling optimization. To the best of our knowledge, there is no any report to introduce piecewise linear model as nonlinearity approximation method for the large scale MINLP planning optimization. In the field of PVC process optimization, the existing results mainly focused on PVC polymerization process to delivery process.20 Tian et al. addressed the integrated scheduling problem, but only considered the material processing system.21-23 In the recent result, Wang et al. proposed a plantwide MINLP model to solve the planning


Page 4 of 37

Page 5 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


optimization considering both material processing and utility systems, but this model is very hard to solve because of the nonlinear characteristics.2 In this paper, the hinging hyperplane model proposed by Breiman 11 is introduced. Compared to other forms of PWL model, HH model has a simple model structure and its parameters are easy to train. Moreover, it is sufficient for industrial applications. We use the HH model to approximate all the nonlinear items in the previously proposed MINLP model. After that, the original MINLP model can be transformed into an identical MILP model by introducing piece-indicating binary variables. Then, we have several mature algorithms to solve the MILP. This method can avoid the nonlinear problems in modeling stage. Compared with the MINLP model, the efficiency of the MILP has a significant improvement, and it is easier to get the global optimum. The rest of the paper is organized as follows. Section 2 provides the process description and problem statement. The nonlinear items in MINLP model and its piecewise linear model is presented in Section 3. In Section 4, the detailed MILP model is described which is transformed from the original MINLP. After that, cases study showing the effectiveness of the MILP model is given in Section 5. Finally, some conclusions are drawn in section 6. 2．． Problem statement A typical system in Figure 1 is adapted from the flowchart of our previous paper,2 and the same PVC process is considered. The PVC process is described as two systems, i.e., utility system and material processing system. In utility system, the electricity supply comes from combined heat and power (CHP) units and the state grid. In material



processing system, PVC production is composed by two sequential parts: the continuous vinyl chloride monomer (VCM) processes and the batch-wise polymerization process. The detailed PVC process is described elsewhere.2

Figure 1. PVC production process.

Due to the intrinsic nonlinearity in energy consumption characteristics, the MINLP model was presented in our previous work.2 The proposed model can reflect start−stop operation of several energy-intensive processing units, and also has taken into consideration the energy consumption and coal consumption. Because of the nonlinearity of those equipment and large model scale, the resulted MINLP is hard to get the solution for this planning optimization problem in reasonable and acceptable time. Also, the result gap is relatively high. Moreover, there is very limited help for result gap decrease to lengthen the solution time, according to our experience. The major solving difficulty lies in the complex nonlinearity and excessive binary variables in the model. Moreover, there is no way to neglect these intrinsic


Page 6 of 37

Page 7 of 37

nonlinearities. Especially, there are highly nonlinear energy consuming characteristics about this equipment in real-world PVC plant, according to the data obtained from the plant, as shown in Figures 2 with blue cross points representing the collected data such as calcium carbide furnaces, electrolytic cells, and CHP units. The detailed curves for electricity consumption can be found in Supporting Information of the previous paper.2 In other words, nonlinear characteristics involved in the plantwide planning model makes it really difficult to solve. How to deal with the nonlinear characteristics is the key to accelerate the solving process. In this paper, to linearize the processes nonlinearities, a typical piecewise linear model, i.e. hinghing hyperplane model, is introduced to approximate the intrinsic nonlinear items of the original model. Moreover, the main nonlinear items lie in the operating cost or energy consuming characteristics of the process. The detailed procedures will be discussed in the next section.

400

350

300

250

200

150 80 (a)

100 120 140 160 180 Production rate of calcium carbide furnace /(t/day)

11

660

10

640

9

620

Coal consumption /(g/Kwh)

Energy consumption of unit production /(Kwh/Kg)

450 Energy consumption of unit production /Kwh/t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8 7 6 5

600 580 560 540

4

520

3

500

2 500

1000 1500 2000 2500 3000 3500 (b) Production rate of electrolytic cell /(Kg/h)

480 100

110 (c)

120 130 CHP unit load/MW

140

Figure 2. Collected data for Electricity consumption of (a) calcium carbide furnaces, (b) electrolytic cells and (c) Coal consumption of CHP units

3．． Nonlinearity in MINLP model and its piecewise linear model As mentioned before, due to the nonlinearity and large scale binary variables in our proposed MINLP model, 2 the model is difficult to solve. To accelerate the solving speed, the piecewise linear approximation method is introduced to approximate the



Page 8 of 37

model nonlinearities and then a corresponding MILP model is resulted. The whole procedure is depicted in this section, shown in Figure 3.

Figure 3. Procedure of piecewise linear approximation based method for MINLP 3.1 Nonlinear representation in the MINLP model For the electricity generation/consumption constraints, the consumed electricity of calcium carbide furnaces and electrolytic cells are the dominant energy consumption form. The nonlinear relationship between electricity consumption and production rate of Cl2 or CaC2 is given respectively in Figure 2. ,, and ,, are the nonlinear functions, respectively. Then, the total electricity consumption is formulated in eqs. 3. 1~3. 3.

,, ∗ ,, ,, ∗ ,, ∀ ∈ ∈

(3. 1)

∈

,, , ∗ ,, ∗ ∀ ∈ , ∈


(3.2)

Page 9 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


,, , ∗ ,, ∗ ∀ ∈ , ∈

(3.3)

The total electricity is related to the production output and the nonlinear functions . The output {,},, can be expressed by the product of working state , , production rate {,},, and production time {,} . In real processes, production rate of Cl2 and CaC2 should be restricted in a reasonable range shown in eqs 3.4~3.5. !"# !% , ∗ ,, ≤ ,, ≤ ,, ∗ , ∀ ∈ , ∈

(3.4)

!"# !% , ∗ ,, ≤ ,, ≤ ,, ∗ , ∀ ∈ , ∈

(3.5)

In this model, the electricity elsp is supplied by CHP units and the state grid, as shown in eq3.6. Then, to satisfy the demand of the electricity consumption, electricity supply should be restricted in eq3.7. Finally, as shown in eqs3.8~3.9, , is the electricity supply load of CHP units, which is determined by working state , , power

, and the running time *+ .

elsp , , ∀ ∈

(3.6)

∈

elsp ≥ ∀ ∈

(3.7)

, , ∗ , ∗ *+ ∀ ∈ , ∈

(3.8)

!"# !% , ∗ , ≤ , ≤ , ∗ , ∀ ∈ , ∈

(3.9)

This is the power consumption/generation model, which consider the highly nonlinear electricity consuming characteristics. The complete integrated model was represented in previous paper (Wang et al. Ind. Eng. Chem. Res.). In this section,



Page 10 of 37

piecewise linear model is considered to approximate nonlinear characteristics. 3.2 Hinging Hyperplane Model In this section, hinging hyperplane (HH) model is introduced to approximate these nonlinear items eqs3.1 to 3.9. HH is a canonical continuous piecewise linear (CPWL) model, which is proven to have the capability to approach any continuous function with a finite dimension on a compact set in any given precision. HH model is widely used because of its simple model structure and easy-to-determine parameters. Compared to other variants, HH model is mature and sufficient for industrial applications. For the sake of readability, a brief description of HH model is given as follows. The basis function of hinging hyperplanes takes the forms as follows:

max 0, 2 3 4 where 2 [1 27 … 2# ] and 4 is the parameter vector of basis function. Notice that for notation simplicity, we do not consider the bias term. In real-world nonlinear process approximation problem, the bias term can be included easily. HH model is formulated as =

2|4 2 4; 7


(3.11)

Page 11 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


where, D" is nonlinear function output given input 2" . Clearly, it is a typical nonlinear parameter training problem. There are many effective algorithms applicable, such as the Newton’s method, conjugate gradient method, combination of these two methods24 and so on. The hyper-parameter, like the number of hinges, is tuned by K-fold cross validation strategy in this paper.25 And then, using least square methods, we can easily obtain the linear coefficients < . Embedding the well-trained HH models into the MINLP model, a new planning model is then obtained. The total electricity is shown in eqs3.1~3.3. It can be transformed into a piecewise linear representation in eq3.12.

F G,, H ∗ , ∗ F G,, H ∗ , ∗ ∀ ∈ (3.12) ∈

∈

where F is the piecewise linear model approximating the nonlinear functions

∗ , by HH model. The new relationship between electricity consumption of each equipment and production rate of CaC2 or Cl2 is given respectively in Figure 4 and 5. Figure 6 shows the relationship between CHP units power and coal consumption. For the left subpicture in Figure 4, the blue cross points represent the collected data while red line represents the prediction curve by HH model. The partitioning points and its HH curve are also depicted in the left subpicture. The right subpicture gives the relative model errors curve, where the blue dotted line represents the average relative error. Clearly, the nonlinear function has been partitioned into several subregions and a series of linear functions in these subregions. The HH and corresponding relative errors curves for the energy model of electrolytic cell and CHP are depicted in Figure 5 and 6 respectively. More details about the HH model, such as the mathematical expressions,



fitting accuracy, and relative sum of squared errors (RSSE), are provided in the Supporting Information. x 10

5

HH model collected data

3.7

1.0%

3.6 3.5

relative error

Energy consumption of unit production /(kwh/day)

3.8

3.4 3.3

0.5%

3.2 3.1 3

0.13%

80

96

141 157 (a) Production rate /(t/day)

0

180

80

100

120 140 (b) production rate /(t/day)

160

180

Figure 4. (a) HH model and (b) model errors for a calcium carbide furnace electricity consumption

1.0%

HH model collected data 8200

8000

relative error

Energy consumption of unit production /(kwh/h)

8400

7800

0.5%

7600

7400

0.08% 7200

1000

1225

2224 (a) Production rate /(kg/h)

0

2510 2673 2878

1000

1500

2000 2500 (b) production rate /(kg/h)

3000

Figure 5. (a) HH model and (b) model error for an electrolytic cell electricity consumption.

7

x 10

4

1.0%

HH model collected data

6.9 6.8 6.7

relative error

Coal consumption /Kg/h

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 37

6.6 6.5

0.5%

6.4

0.14%

6.3 6.2

105

119 131 (a) Load of CHP unit /MW

143

0

105

110

115

120 125 130 (b) CHP unit load /MW

135

140

Figure 6. (a) HH model and (b) model error for a CHP unit coal consumption.

Since all the nonlinear relationships are approached by PWL functions, the overall


Page 13 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the MINLP problem is approximated by an MILP problem. Specifically, to get the formulation, the partitioning points based method proposed by D’Ambrosio26 is applied. Suppose there are I partitioning points, such as J7 J … J# , and the corresponding value of function is J7 J … J# . Then, introduce a continuous variable KL and a binary variable ML , where 0 ≤ KL ≤ 1 N 1, … I and M# 0 . Let ML associated with the Nth interval [JO , JOP7 ] N 1, … I − 1 . For an arbitrary point J and corresponding value of piecewise linear function ∙ can be linearly expressed. #R7

ML 1

(3.13)

KL ≤ MLR7 ML N 2, … , I

(3.14)

KL 1

(3.15)

L>7 #

L>7

#

J KL ∗ JL

(3.16)

L>7 #

∙ KL ∗ JL

(3.17)

L>7

Then, up to now, the original nonlinearity · in PWL form has been transformed into linear representation form. Correspondingly, the particular formulations in planning model for electrolytic cell are as follows: #R7

M< L,, 1 ∀ ∈ , ∈

(3.18)

K< L,, ≤ M< LR7,, M< L,, ∀ ∈ , ∈ , N 2, … , I,

(3.19)

K< L,, 1

(3.20)

L>7

#

L>7



Page 14 of 37

#

,, K< L,, ∗ J< L,, L>7

(3.21)

#

F G,, H KL ∗ J< L,,

(3.22)

L>7

Where, M< L,, and K< L,, are variables corresponding to ML and KL .

F G,, H and ,, can be denoted by the partitioning points J< L,, . Similarly, the corresponding linear functions of calcium carbide furnaces and CHP units will get easily. The detailed MILP model will be given in next section.

4． Equivalent MILP model based on HH In this section, a novel MILP model is presented which integrates the HH model with the MINLP model. 4.1. Polymerization Process Model In the batch polymerization process, the production amount ",, for grade @ PVC in polymerization reactor during w week can be computed as in eq. 1. Because of the process is operated in a batch-wise manner, cy is discharge period, and each polymerization reactor’s feeding quantity F, and conversion ratio T", is constant. Then, it is linear for eq1. Clearly, the number of feeding batches U",, should be restricted in a reasonable range shown in eq2.

",, U",, ∗ F, ∗ T", ∀@ ∈ ?, ∈ ?, ∈ 0 ≤ U",, ≤ "∈=+

168 ∀ ∈ ?, ∈ 1

(19-2)

f∈

f∈

!"# !% X ≤ X, ≤ X ∀ ∈

(20)

4.3. Power Consumption and Generation We have introduced the nonlinear relationship and given a piecewise linear representation in section 3. So, the total electricity consumption can be formulated in eq21. Eqs22~25 express some constraints.

F ,, ∗ , ∗ F G,, H ∗ , ∗ ∀ ∈

(21)

elsp , , ∀ ∈

(22)

elsp ≥ ∀ ∈

(23)

, , ∗ , ∗ *+ ∀ ∈ , ∈

(24)

!"# !% , ∗ , ≤ , ≤ , ∗ , ∀ ∈ , ∈

(25)

∈

∈

∈

4.4. Raw Material Constraints Raw material constraints denoted by inventory balance and chemical principle balance, as shown in eqs26~31. The coal consumption function F G, H in terms of



Page 18 of 37

CHP units load is described with HH model as presented in section 3.

Xh, Xh, R7 ,h, − Kih, ∀j ∈ ?k, ∈ ∩ ≠ 1

(26)

Xh!"# ≤ Xh, ≤ Xh!% ∀j ∈ ?k, ∈

(27)

K?lLm, n ∗ ,, ∀ ∈

(28)

K?l, o ∗ ,, ∀ ∈

(29)

K?B, p ∗ ,, ∀ ∈

(30)

K?l, F G, H ∗ , ∗ *+ ∀ ∈

(31)

∈

∈

∈

∈

where n denotes the coefficient between the CaC2 and coke; o denotes the coefficient between CaC2 and CaO; p denotes the coefficient between Cl2 and salt. 4.5. Start−Stop Operation In PVC production process, work state of multiple parallel equipment may be changed. But, frequent start-stop operation affects the manufacturing process and causes unnecessary energy loss. The binary variable , denotes the work state of equipment u during w week. F, is introduced to indicate whether the unit u starts up or shuts down in week w. Then, the relationship can be formulated in eqs32~33.

ZF, R7 , ≥ , R7 ∀ ∈ Fd, ∈ ∩ ≠ 1

(32)

ZF, R7 , R7 ≥ , ∀ ∈ Fd, ∈ ∩ ≠ 1

(33)

Obviously, eqs32~33 cannot strictly constraine the start-stop operation. However, due F, is penalized in the objective function, the state variables will be determined reasonably then.


Page 19 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4.6. Objective Function The objective is to minimize the overall cost, including raw materials cost ?

Piecewise Linear Approximation Based MILP Method for PVC Plant

Recommend Documents