disposal cost models to optimise pulp stock production · PDF fileApplication of linear programming models to optimise pulp stock production process PULP PRODUCTION ... model was used

20 TAPPSA JOURNAL | VOLUME 1 2013

Application of linear programming models to optimise pulp stock production process

PULP PRODUCTION

K.D. MOKEBEMpact Packaging South Africa (Pty) Ltd. Springs Mill

J.W. JOUBERTDepartment of Industrial and System Engineering, University of Pretoria

ABSTRACT

Floudas (1995) and Edgar, Himmelblau and Lasdon (2001) have shown that mathematical programming, especially Mixed Integer Linear Programming (MILP), can effectively be used for flow sheet selection and optimisation of existing system, as well as retrofits in production processes. However, mixed integer linear programming (MILP) and mixed integer non-linear programming (MINLP) has extensively been used because of its rigorousness and flexibility.

Presented in this article is an application of stochastic

mixed integer linear programming for optimisation of fibre preparation system. The technique determines the optimum system throughput, the network to be used, as well as the preferred landfill site, given the stream operating cost and landfill disposal cost. The technique was applied to the case study showing a potential energy saving of more than 18% (from 45 kWh per tonne to 37kWh/tonne). This amounts to about 1.12 MWh per annum which has a significant cost saving in this environment of increasing energy cost. The paper was inspired by the Advance Operational Research course presented at the University of Pretoria’s Industrial Engineering Department.

Keywords: MINLP, Stock preparation process, recycle fibre

1. INTRODUCTION

Resource conservation has gained more attention in the process industry in recent years. An efficient resource conservation strategy is beneficial since it enhances competitiveness through reduced operational costs and promotes sustainable development. Figure 3 gives a schematic representation of a production plant. In any production process, a sub system can be differentiated to use different technologies to transform the raw material into a useful product. This transformation defines the plant’s operational expenses based on the energy used, input resource utilisation and the generated waste disposal.

Recent developments (Floudas, 1995; Edgar, Himmelblau and Lasdon, 2001) have shown that

mathematical programming, especially Mixed Integer Linear Programming, can effectively be used for flow sheet selection and optimisation of the existing system, as well as retrofitting in production processes. Mixed Integer Linear Programming (MILP) and Mixed Integer Non-Linear Programming (MINLP) is mainly used because of its rigorousness and flexibility. A textbook by Edgar, Himmelblau & Lasdon (2001) (Optimisation of Chemical Process) presents different applications of mathematical optimisation to the process industry.

Recent work by Liao et al. (2007) considered the multi-period problem (unequal working hours in each period), where an MINLP model was solved to locate the minimum interplant water targets while an MILP was solved to obtain a water network that meets the water flow rate targets with the simplest network configuration. Linear

TAPPSA JOURNAL | VOLUME 1 2013 21

programming has extensively been used for synthesis, analysis and optimisation of waste water networks, with a full literature review given by Bagajewicz (2000).

This document presents the optimisation and flow sheet selection for the stock preparation process using Stochastic Mixed Integer Linear Programming (SMILP) in the pulp and paper industry. This work aims at developing an innovative method for simultaneous optimisation of resources usage and energy utilisation in process through a novel Stochastic Mixed Integer Linear Programming (SMILP) for problem formulation. The proposed method has been reported to reduce the total operating cost by 34% compared to the current operating conditions of the case study (Bagajewicz, 2001).

2. PROBLEM STATEMENT

Many authors have recognised that it is unlikely to apply deterministic models in real scenarios without decreasing considerably their performance, and have made efforts to extend deterministic approaches to situations with some type of uncertainty so as to obtain better results when their solutions are deployed in real scenarios (Floudas, 1995). The problem addressed in this paper can be stated as follows:

i) Given a different type of raw material with a varying degree of contaminant mass load and unit costii) Maximum capacity of each operating stage and the separation efficiencyiii) The yield for each of the operating stage and fixed operating cost

iv) Alternative landfills site to be used based disposal costv) The plant operating cost

Determine the optimum value of the pulp produced, tonnage, solid waste and the landfill site to be used. This must be done amidst the uncertainty regarding the operating efficiency of the new pressure screen technology that has a high average efficiency compared to the older technology. The limiting stage capacity refers to the optimal process volume of the stage. These are determined by the operation characteristics of the particular equipment in that stage.

3. CASE STUDY DESCRIPTION

Linear programming is often used in the design and operation of production systems Figure 3 (following page) shows a stock preparation system for production of stock to be supplied to the paper machine. The commonly used raw material is different grades of recycle paper. Figure 2 (below) shows the different types of recycled paper collected as well as the load of contraries that can be found inside the bales if opened and checked. Thus, this presents a practical challenge as more than 200 000 tonnes of recycled fibre is used per annum and each bale weighs about 1 tonne on average. This will require more resources and increased cost of operation. The other dilemma for paper industry is the means of disposal for contraries.

This problem is exacerbated by refusal on conditions set by landfill sites. Therefore an optimisation model is needed that will also incorporate the landfill to be used. This

Figure 2: Recyclable material grades and (bottom centre) contraries found in a bale.

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will allow for the simultaneous selection of technology and landfi ll site while taking into consideration the value added.

Figure 3 is a schematic of the pulping process stage, producing about 400 BDt per day of paper pulp. The plant’s pulping stage consists of the pulper, detrasher and drum screen for large contrary rejection. The second stage consists of three stage coarse screens for fi ner contraries. The contraries are binned and sent to different landfi ll sites depending on the disposal cost. This production uses about 400 BDt per day of different grades of raw material with the throughput of more than 350 BDt per day, depending on the availability of that particular raw material and cost.

4. MATHEMATICAL FORMULATION

The mathematical optimisation formulation is derived from the superstructure in Figure 3 and by writing mass and yield balances across the existing operations. The superstructure has 6 yield balances for operational 1, 2, and 3 respectively and three further mass balances across points M1, M2 and S1 respectively. The expected value model was used for the separation effi ciency of stage 1 and 2 but the uncertainty about stage 3 requires the use of

the stochastic linear programming model. The effi ciency tests were performed and found to vary between 95.72 and 98.18%. This was then fi tted with a normal distribution with the mean (μ) of 97.32% and standard deviation (σ) of 0.936%. The yield and node balances equation as well as the availability and process selection model are presented below:

Model Parametres = the stage reject rate as % of the feed = Number of landfi ll sites to be used = Stream operating cost, = Raw material cost and = Stage fi xed operating cost and

Decision Variables = stream fi bre fl ow rate (tonnes/day) and = Binary variable for landfi ll and process selection and

Yield, mass balance, raw material availability and process and landfi ll selection equation for the superstructure presented in Figure 3 are given as follows:

Figure 3: Stock preparation plant model in pulp and paper industry

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Mass Yield Equations:

Node Balances:

Availability of Raw Material A:

Availability of Raw Material B

Demand for C

Process selection model:

Select either Process 1 or Purchase B Select either Process 1 or Purchase B Landfi ll selection model

Where the number of landfi ll sites to be used can be specifi ed ( ).

Variable Constraints

or 1

The objective function for this case can be stated as:- The maximum of the overall system benefi t;- Minimise the waste disposal cost and - Minimise the number of landfi ll sites to be used for disposal, given the landfi ll disposal cost.

This objective must be achieved without violating any of the design and operational constraints within the pulping network (Figure 3). The maximum system benefi t to be maximised given in cost can be stated:

The cost coeffi cients for the system were obtained from open sources and electricity and waste disposal cost from municipality tariffs list, while the fi xed cost for man-hours was based on the current legal stipulated rates.

5. MILP SOLUTION UNDER UNCERTAINTY USING CHANCE CONSTRAINED INTERVENTION

The presence of uncertainty affects both feasibility and optimality. In fact, formulating an appropriate objective function raises an interesting modelling and algorithmic challenge. The solution method for this problem class has been presented by Nemirovski & Shapiro (2006) and Joubert (2012). They proposed the incorporation of uncertainty within an LP using chance constraint programming. Two approaches have been adopted in practice. The method used in this study was to convert the given SLP to chance constraints approach which can be stated as follows:

Given a Stochastic Linear Programming:

Raw Material Type

Raw Material PriceCurrency: Dollars Rands

Min Max Min MaxA $150.00 $170.00 R 1,230.00 R 1,394.00B $175.00 $180.00 R 1,435.00 R 1,476.00C $155.00 $165.00 R 1,271.00 R 1,353.00D $240.00 $260.00 R 1,968.00 R 2,132.00E $550.00 $640.00 R 4,510.00 R 5,248.00Exchange Rate(Spot)

8.2

Table 1: Raw material cost (source: www.alibaba.com)

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The problem is then reformulated using chance constraint and the appropriated intervention (Nemirovski & Shapiro, 2006, and Joubert, 2012):

It is then converted to its deterministic equivalent using a proposed method by Nemirovski & Shapiro (2006) and Joubert (2012). This allows the problem to be solved as a deterministic equivalent with an additional of non-linear constrained added due to the safety factor used. The formulation allows for the separation of the certain component of the constrained and the uncertain part. This uses the mean and the standard deviation assuming that the variable is uniformly distributed

And: the element of vector is the required value of risk (reliability)

This form of deterministic equivalent is solved for the given objective function. Microsoft Excel Premium Solver was used for this particular problem while optimising the cost benefi t function. The optimum solution was found to be a production of about 368 BDT per day. This solution did not consider the disposal costs assuming that all the waste generated on site is disposed freely. But this is not realistic, as the generated waste must be disposed at a certain site selected by management. Table 1 gives the premium solver optimum solution taking this additional constraint into consideration and process 1 and 3 as well as landfi ll 7 were selected.

Multi-objective Approach

The above process deals with multiple confl icting, objectives or design criteria. The problem was solved using the multi-objective pre-emptive approach given by Rardin (2008). The disposal cost was incorporated as the constraint and the Pareto optimal; curve was generated. The reason for incorporating the disposal cost in the constraint is due to the structure of the organisation. The

Pareto optimum solution for the given network is found in Figure 4, which shows the value added as function of waste disposal cost on the daily basis for the selected network. The Pareto approach can be used to optimise amongst many confl ict objective for any given system.


Stream No: Stream Label Solver Decision Variable

1 F1 400

2 F2 380

3 F3 20

4 F4 0

5 F5 380

6 F6 0

7 F7 380

8 F8 0

9 F9 0

10 F10 368.6

11 F11 11.4

12 F12 368.6

BINARY VARIABLES

No. Binary Selection Variable

1 y1 = on/off for Process 1 1



4 y4 = on/off for stream 4 0

5 y5 = on/off for landfi ll 1 0



Figure 4: Pareto optimum curve

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And:

the element of vector is the required value of risk (reliability)


Table 3 presents the optimal solution for multi-objectives formulation of this particular problem, which shows that the optimum production is not more than 306 tonnes per day with average disposal cost of not more than R26 per tonne (Figure 4). Any attempt to increase the throughput will just result in high disposal costs for given separation effi ciency.

No: Stream Label Solver Decision Variable

1 F1 332.72

2 F2 316.08

3 F3 16.64

4 F4 0.00

5 F5 316.08

6 F6 0.00

7 F7 316.08

8 F8 0.00

9 F9 0.00

10 F10 306.60

11 F11 9.48

12 F12 306.60

BINARY VARIABLES

No. Binary Selection Variable




4 y4 = on/off for stream 4 0




The optimum for the multi-objective pre-emptive optimisation technique was found to be 306 BDT per day with the side to be used as well as the raw material requirements. At this throughput, the waste disposal cost is about R26 per tonne per day. The model also selected both Process 1 and Process 3 for usage, as well as landfi ll 1 and 2 for disposal (Figure 5). This new proposed network has the total input power consumption of 74kWh/tonne which is a 10% reduction. The potential cost saving for technology change based on the prevailing energy price is in excess of R 0.7 million per annum.

6. CONCLUSION

Mixed Integer Linear Programming synthesis for grassroots design is very well appreciated. In contrast, network synthesis for retrofi ts has received much attention by the process integration research community as much as effl uent optimisation, despite the huge number of opportunities for implementing it in resource conservation within existing plants.

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Table 3: Multi-objective optimum solution

Figure 5: Proposed new confi guration with reduced power usage


In this work, a mathematical model based on linear programming and mixed integer programming has been developed to systematically address the network retrofi t and landfi ll selection. The focus was to select between the highly effi cient screen technology and the existing rotary screens (referred to as Turbo separator). The analysis showed that the stream value added is directly linked to the landfi ll disposal cost. The model formulation even allowed for the stochastic nature of linear programming for the screen reject effi ciency which was determines experimentally and the resulting model was solved using chance constraints. ■

7. NOMENCLATURE

Bone dry tonnes (BDT) is a volume of wood chips/pulp (or other bulk material) that would weigh one tonne (0.9072 metric tonnes) if all the moisture content was removed.

Pulp is a lignocellulosic fi brous material prepared by chemically or mechanically separating cellulose fi bres from wood, fi bre crops or waste paper.

Fibres are usually cellulosic elements that are extracted from trees and used to make materials including paper

Paper recycling is the process of turning waste paper into new paper products

Stage Reject Rate ( ri) is the ratio of the reject fl ow rate as fraction of the feed to the stage.

ACKNOWLEDGEMENT

We would like to acknowledge the support of the Mpact Springs Mill in this research.

REFERENCES

Bagajewicz, M., 2000. A review of recent design procedures for water networks in refi neries and process plants. Computers & Chemical Engineering, 24 (9), pp. 2093–2115.

Edgar, T.F, Himmelblau, D.M, Ladson, L.S, 1990. Optimisation of Chemical Process 2nd Edition, McGraw-Hill, USA. Chapter 7-8

Floudas, C.A., 1995. Nonlinear and Mixed Integer Optimisation: Fundamentals and applications. Oxford University Press, New York NY, Part 2.

Hahn, B.D., 2002. Essential Matlab for Scientists and Engineers. 3rd Edition, Pearson Maskew Miller Longman, Chapter 1 – Part II.

Joubert, J., 2012. Operations Research-Stochastic Programming Notes. Industrial and Systems Engineering, University of Pretoria.

Liao, Z. W., Wu, J. T., Jiang, B. B., Wang, J. D., Yang, Y. R., 2007. Design methodology for fl exible multiple plant water networks. Ind. Eng. Chem. (46), pp. 4954–4963.

Nemirovski, A., Shapiro, A., 2006. Convex Approximation of chance constraint programmes. Society for Industrial and Applied Mathematics (SIAM), 17 (4), pp. 969–996

Savelski, M. J. and Bagajewicz, M. J., 2000. On the optimality conditions of water utilisation systems in process plants with single contaminants. CES, 55(21), pp. 5035–5049.

Winstonne, W., 2004. Operations Research Application and algorithm. 4th Ed, Chapter 1-9.

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disposal cost models to optimise pulp stock production · PDF fileApplication of linear programming models to optimise pulp stock production process PULP PRODUCTION ... model was used