# INTRODUCTION
omputational intelligence, the technical umbrella of Fuzzy Logic and Artificial Neural Networks (ANNs) have been recognized as powerful tools which is tolerant of imprecision and uncertainty and can facilitate the effective development of models by integrating several processing models. They explore alternative representation schemes, using, for instance, natural language, rules, semantic networks, or qualitative models. Fuzzy logic controllers have their origin with the E. H. Mamdani (Mamdani, 1977) researches, based on theories proposed by L. Zadeh (Zadeh, 1965). These controllers have founded space in many learning, research and development institutions around the world, being today an important application of fuzzy set theory.A great appeal of fuzzy technology in control is the possibility to operate with uncertainties and imprecision. It can consider an uncertainty on definition of input and output variables (Lee, 1995).The fuzzy controllers are robust and highly adaptable, incorporating knowledge that sometimes are not achieved by other systems. They are also versatile, mainly when the physical model is complex and has a hard mathematical representation. In fact, fuzzy controllers are especially useful in non-linear systems and plants with a high level noise. The conventional controllers deal with nonlinearities of physical systems by approaching, considering the systems simply linearly, linear in parts, or describing them by extensive lookup tables that try to map the process inputs and outputs (Chin-Fan-Lin, 1994), (Cirstea, 2002). Fuzzy logic is a powerful technique for solving a wide range of industrial control and information processing applications (Akinyokun, 2002). Urbanization is one of the most evident global changes worldwide. The rapid and constant growth of urban population has led to a dramatic increase in urban solid waste production, with a crucial socio-economic and environmental impact. However, the growing concern for environmental issues and the need for sustainable development have moved the management of solid waste to the forefront of the public agenda.Recycling technology has evolved as one of the most useful facilities that help us to maintain an environmentally friendly society. All over the world, the need of recycling is heightened by the increasing awareness of product consumer, on the need to maximize the bundle of benefits from the products brought by them. Consequently, the government is encouraging recycling and recycling practices (Goulias, 2001). For example, a strong indication of the -en d-of-life directive? endorsed by the European Union states that if resources are not recycled, a period may come that very limited resources would be available for mankind (Treloar et al. 2003). If urgent steps are not taken, we may need to suffer for the shortages of these important resources. Waste paper is an example of a valuable material that can be recycled. Paper recycling has been around as long as paper itself. Paper companies have always recognized the environmental and economic benefits of recycling. Recycled paper reduces water pollution by 35%, reduces air pollution by 74%, and eliminates many toxic pollutants (TAPPI, 2001). In recent years, paper recycling has become popular with everyone as a way to help protect our environment by reusing our resources and conserving landfill space (TAPPI, 2001).Recycling is a series of activities, including collection, separation, and processing, by which products or other materials are recovered from or otherwise diverted from the solid waste stream for use in the form of raw materials in the manufacture of new products. Recycling is one of the best C ways for citizens to make a direct impact on the environment. Recycling reduces greenhouse gas emissions that may lead to global warming. Recycling also conserves the natural resources on Earth like plants, animals, minerals, fresh air and fresh water. Recycling saves space in the landfills for future generations of people. A sustainable future requires a high degree of recycling (Misman et al. 2008). Recycling industries face serious economic problems that increase the cost of recycling (Kumaran, 2001). By way of proffering solution, a number of management strategies are adopted over time. Such strategies may include business process re-engineering, downsizing, system restructuring, lean manufacturing, etc. All these strategies are aimed towards optimizing the system variables through minimization of costs and maximization of profits.The profitability of recycling industries has been known to be highly dependent on the effective management of resources and management practices, (Craighill and Powell, 1996) (Cunningham, 1969). Taking the paper recycling industry as a case study, the recycling of paper has contributed in no small measure to the conservation of material and consequent low cost of production.The ultimate resultant effect of the recycling process is high profit generated due to the low cost of production, hence, the link between profitability and recycling industry. Since the ultimate aim of any capitalist industry is to make profit, the concept of profitability is of great significance in science and engineering (http://www.recyclingtoday.com, http://www.recyclinginternational.com,).The objective of this research is to develop a fuzzy logic model using fuzzy logic technology and apply the model for effective control of profitability in paper recycling to determine the degree of influence of various processing components on the profit generated by an industry. We derive the cost of production from the following costs; (i) cost incurred to recycle the desired quantity of waste paper materials (secondary fiber), (ii) salaries, (iii) wages, (iv) rent, (v) depreciation on machines and equipment, (vi) sundry expenses, etc. We determine the selling price based on the production cost and then generate profit based on the relationship between the above mentioned functions. A relationship is developed between thestated input(s) (cost of production and selling price) and output(s) (profits) with the help of the fuzzy logic model. This will facilitate optimization of paper recycling production.The proposed model will help to arrive at specified desired output in the optimization approach for solid waste paper recycling. To reinforce the proposed approach, we have applied it to a case study performed on Paper recycling industry in Nigeria. A computer simulation using the Matlab®/Simulink and its Fuzzy Logic Tool Box is designed to assist the experimental decision for the best control action. The obtained simulation and implementation results are investigated and discussed.Matlab is an integrated technical computing environment that combines numerical computation, advanced graphics visualization and a high-level programming language, Simulink is built on top of Matlab, and is an interactive environment for modeling, analyzing and simulating a wide variety of dynamic systems. The Fuzzy Logic Toolbox provides tools for user to create and edit fuzzy inference systems, or integrate the fuzzy systems into simulations with Simulink. (Wongthatsanekorn, 2009) developed a goal programming model for plastic recycling system in Thailand. (Kufman, 2004) carried out the analysis of technology and infrastructure of paper recycling. (Kumar et al, 2008) designed a goal programming model for paper recycling to assist proper management of the paper recycling logistic system, while (Udoakpan, 2002) carried out the financial implication of establishing a paper recycling plant in Nigeria. (Oke et al, 2006) designed a fuzzy logic model to handle the profitability concept in a plastic industry.In section 2 of the paper the mathematical model of the system is presented while in Section 3 the research methodology is presented. Section 4 presents the model experiment while in Section 5 results of findings are discussed. Finally in Section 6, some recommendations are made and conclusion is drawn.
# II. MATHEMATICAL MODEL
The mathematical model of the proposed work is based on the major components in the concept of profitability using a case in the paper recycling industry. These components include; Selling Price (SP), Cost Price (CP), Quantity Recycled (QR), and Profit (Y) The relationship among these components as used in the concept of profitability is illustrated in figure 1.The relationships among components of profitability in figure 1 show Selling Price (SP) which is the selling price per item of the quantity of recycled product. Cost Price (CP) which is the cost price per item of the quantity of recycled product. The cost price is made up of all expenses incurred directly during the recycling production processes ) STRNG = Strange (%) which is the difference between the machine speed and the reelers speed Thus, QR = PM4 (m/m) x T (m) x SUBW (m) x SUBP (g/m 2 ) x STRNG (%) (1) Profit (Y) is the profit made and is the difference in Selling Price (SP) and Cost Price (CP) multiplied by the quantity of recycled product (QR). This is given as:
Y = SP (QR) -CP (QR) (2) Thus, Y = [(SP -CP) (PM4 (m/m) x T (m) x SUBW (m) x SUBP (g/m 2 ) x STRNG (%))](3)
III.
# RESEARCH METHODOLOGY
The fuzzy inference system for effective control of profitability in paper recycling is shown in Figure 2. This system involves three main processes; fuzzification, inference and defuzzification. The knowledge base contains the following: (i) rule-base -that contains knowledge used to characterize Fuzzy Control Rules and Fuzzy Data Manipulation in an FLC, which are defined based on experience and engineering judgment of an expert. In this case, an appropriate choice of the membership functions of a fuzzy set plays a crucial role in the success of an application. The rules are in the form of IF -THEN (production rules).
(ii) data-base: Fuzzy variables are defined by fuzzy sets, which in turn are defined by membership functions. The knowledge base design of profitability control in paper recycling production is made up of both static and dynamic information about the decision variables and about the different factors that influence recycling decision for controlling paper recycling production for profit optimization.
There are qualitative and quantitative variables which must be fuzzified , inferred and defuzzified. Fuzzification of data is carried out on the transformed data by selecting input parameters into the horizontal axis and projecting vertically to the upper boundary of membership function to determine the degree of membership. This is then used to map the output value specified in the individual rules to an intermediate output measuring fuzzy sets. Parameters used in fuzzy logic model are, Cost Price (CP), Selling Price (SP), and Quantity of Recycled product (QR). These parameters constitute the fuzzy logic input variables used to generate the fuzzy logic model, while the linguistic variable for the model is (SP)(QR) -(CP)(QR) which is the difference between the selling price and cost price of the quantity recycled.
(SP)(QR) -(CP)(QR) = ZE, -Zer o-error? term(ZE) (No profit no loss) (S P )(Q R ) -(C P )(Q R ) = NE, -Neg ative-error? term (NE) (Loss) (S P )(Q R ) -(C P )(Q R ) = PO, -Positive-error? term (PO) (Profit)
If we consider the model over a period of time, we have: if 25 <= x < 50 (16) (50-x)/25 0 positive_small, and positive_big, respectively. The output is defined by fuzzy sets, high_loss, low_loss, loss, no_profit_no_loss, profit, low_profit and high_profit.The linguistic expression E and CE variables and their membership functions are evaluated using triangular membership function as presented in equations ( 5) to (11). Triangular curves depend on three parameters a 1 , a 2 , and a 3 and are given by equation ( 4), a 2 defines the triangular peak location, while a 1 and a 3 define the triangular end points. During the process linguistic labels (values) are assigned to the error and change in error indicating the associated degree of influence of membership for each linguistic term that applies to that input variable. Degrees of membership (U x ) are assigned to each linguistic value as expressed in equations ( 5) to (11) negative small, negative, zero, positive, positive small and positive big. Linguistic values are assigned to the linguistic variable, error ((SP)(QR) -(CP)(QR)) of profitability as shown in equation (5). In equations ( 6) to (11), each linguistic value is assigned a label emphasizing the degree of the value assigned in (1). For example, equation ( 6) evaluates the degree of positive small of the error and change in error, if the value of error is for instance, 55, the degree of influence will evaluate to 0.65 (65%) severity, whereas, 75 evaluates to 0.75 (75%). Fuzzy logic toolbox in Matlab 2007 is employed in this project to model the design. Graphical users interface (GUI) tools are provided by fuzzy logic box (Fuzzy Logic Toolbox Users' guide, 2007). The graphical formats which show the fuzzy membership curves for error, change in error and the output are depicted in figures 3, 4 and 5 respectively, where triangular membership functions are used to describe the variables. linguistic value of fuzzy output membership function in figure 5 is assigned a label emphasizing the degree of the value assigned as in equations 12-.18 Using derivation based on expert experience and control engineering knowledge; the experience of an expert who has been working at the Star Paper Mill located in Aba, Nigeria for over 18 years was used to obtain the rule base. The expert also assisted in defining the fuzzy rules and the fuzzy set. There are 2 inputs in the knowledge base namely; error and change in error, with 7 fuzzy sets each as antecedent parameters and 7 fuzzy sets each as consequent parameters. From the expert knowledge, these are used to generate 49 rules for the rule base defined for the decision-making unit. Some of the rules are presented in Table 1. The Rule matrix for the fuzzy control rules is shown in The process of drawing conclusions from existing data is called inference. For each rule, the inference mechanism looks up the membership values in the condition of the rule. Fuzzy input are taken to determine the degree to which they belong to each of the appropriate fuzzy sets via membership functions. The aggregationoperation is used to calculate the degree of fulfillment or firing strength, ? n of the condition of a rule n. A rule, say rule 1, will generate a fuzzy membership value ? E1 coming from the errorand a membership value ? EC1 coming from the change in error measurement. ? E1 and ? EC1 are combined by applying fuzzy logical AND to evaluate the composite firing strength of the rule. The rules use the input membership values as weighting factors to determine their influence on the fuzzy output sets of the final output conclusion. The degrees of truths (R) of the rules are determined for each rule by evaluating the nonzero minimum values using the AND operator. Only the rules that get strength higher than 0, would -fi re? the output. The Root Sum Square (RSS) inference engine is employed in this research which has the formula,
? ZE (x) = (x + 33)/33 0 if x > 33 if 0 <= x < 33 (33 -x)/33 0 if -33 <= x < 0 (8) if x < -33 if a 1 <= x a 3 if a 2 <= x = 100 -PB? if -67 <= x < -33 -NS? if -33 <= x < 0 -NE? if 0 <= x < 33 -ZE?(6)(Q R )-(C P )(Q R ) = NB AND d[(S P )(Q R )-(C P )(Q R )]/dt = NB THEN Output = HighLoss 4. IF (S P )(Q R )-(C P )(Q R ) = NB AND d[(S P )(Q R )-(C P )(Q R )]/dt = ZETHEN Output = NProfitNLoss 8. IF (S P )(Q R )-(C P )(Q R ) = NS AND d[(S P )(Q R )-(C P )(Q R )]/dt = NB THEN Output = HighLoss 14. IF (S P )(Q R )-(C P )(Q R ) = NS AND d[(S P )(Q R )-(C P )(Q R )]/dt = PB THEN Output = Profit 18. IF (S P )(Q R )-(C P )(Q R ) = NE AND d[(S P )(Q R )-(C P )(Q R )]/dt = ZETHEN Output = NProfitNLoss 19. IF (S P )(Q R )-(C P )(Q R ) = NE AND d[(S P )(Q R )-(C P )(Q R )]/dt = PO THEN Output = LowProfit 23. IF (S P )(Q R )-(C P )(Q R ) = ZE AND d[(S P )(Q R )-(C P )(Q R )]/dt = NS THEN Output = LowLoss 24. IF (S P )(Q R )-(C P )(Q R ) = ZE AND d[(S P )(Q R )-(C P )(Q R )]/dt = NE THEN Output = Loss 28. IF (S P )(Q R )-(C P )(Q R ) = ZE AND d[(S P )(Q R )-(C P )(Q R )]/dt = PB THEN Output = HighProfit 32. IF (S P )(Q R )-(C P )(Q R ) = PO AND d[(S P )(Q R )-(C P )(Q R )]/dt = ZETHEN Output = ProfitRSS = ?? R 2 = ? (R 1 2 + R 2 2 + R 3 2 + ,?., + Rn 2 ) (19)
Where R 1 , R 2 , R 3 ?. R n are strength values of different rules which share the same conclusion. RSS method combines the effects of all applicable rules, scales the functions at their respective magnitudes, and computes the "fuzzy" centroid of the composite area. This method is more complicated mathematically than other methods, but is selected for this work since it gives the best weighted influence to all firing rules (Saritas and Sert 2003). From table 1 for instance, the membership function strength values are evaluated as,
HighProfit = ? (R 28 2 + R 42 2 + R 49 2 ) LowProfit = ? (R 19 2 + R 37 2 + R 41 2 R 45 2 ) Loss = ? (R 24 2 + R 36 2 ) 6) Defuzzification
Defuzzification of data into a crisp output is a process of selecting a representative element from the fuzzy output inferred from the fuzzy control algorithm. A fuzzy inference system maps an input vector to a crisp output value. In order to obtain a crisp output, we need a defuzzification process.
The input to the defuzzification process is a fuzzy set (the aggregated output fuzzy set), and the output of the defuzzification process is a single number. Many defuzzification techniques are proposed and four common defuzzification methods are center-of-area (gravity), centerof-sums, max-criterion and mean of maxima. According to (Obot, 2008), max-criterion produces the point at which the possibility distribution of the action reaches a maximum value and it is the simplest to implement. The center of area (gravity)is the most widely used technique because, when it is used, the defuzzified values tend to move smoothly around the output fuzzy region, thus giving a more accurate representation of fuzzy set of any shape (Cochran and Chen, 2005). The technique is unique, however, and not easy to implement computationally. Center of gravity (CoG) often uses discretized variables so that CoG, y? can be approximated to overcome its disadvantage as shown in equation ( 23) which uses weighted average of the centers of the fuzzy set instead of integration. This approach is adopted in this research because it is computationally simple and intuitively plausible.
??(x i ) x i ??(x i ) y? =(21) (20) (22)
Where x i is a running point in a discrete universe, and ?(x i ) is its membership value in the membership function. The expression can be interpreted as the weighted average of the elements in the support set.
# IV. MODEL EXPERIMENT
The study adopts Matlab/Simulink and its Fuzzy Logic tool box functions to develop a computer simulation showing the user interface and fuzzy inference to assist the experimental decision for the best control action. Results of evaluation of fuzzy rule base inference for two ranges of inputs, Error These particular input conditions indicate positive value of 25.7% (25.7% Profit) therefore profit is expected with 25.7% possibility and required system response. Table 4 Rule base evaluation for error and change in error at +95 and +95 Table 4 shows that, if rules 41, 42, 48 and 49 fire from the rule base in figure 1
# RESULT AND DISCUSSION
In fuzzy logic implementation, the selection of membership functions and rule base determine the output. Hence, by selecting a triangular membership function, the variables in the system are manipulated and represented judiciously. Also, the rule base is selected from the experience of system expert. Fuzzy logic represents partial -t ruth? or partial -false? in its modeling. From the study, apart from assigning linguistic variables such as low-loss, no-profit-no-loss, lowprofit,to the profitability, the degree of influence or severity of each linguistic variable is evaluated. Tables 3 and 4 This implies that the selling price of the recycled quantity is more than the cost price of the recycled quantity; therefore profit is expected with 25.7% possibility and a low-profit with 63.7% possibility. Considering the degree of relationship between linguistic label and value of fuzzy output membership function, say -Loss?, when its value equals 1.0, it indicates that the cost price of the recycled quantity is more than the selling price of the recycled quantity and that industry will run at a loss with 100% possibility. When the fuzzy output value is 0.6, it indicates 60% possibility of loss.Considering the relationships strength among fuzzy outputs in figure 5, it indicates that only when -no-profit-no-loss? output value equals 1.0, (100%) that we can conclude that the selling price of the quantity recycled is indeed equal to the cost price and the industry is likely to run at no profit no loss. Relating -n oprofit-no-loss' with -profit? for instance, when the value of -no -profit-loss? output is 0.4 showing possibility 40%, its indicates that there is 0.6 (60%) possibility of profit. This implies that it is not likely that the industry will run at no profit no loss altogether when the selling price is only less than or equal to the cost price by 40%. Relating -n o-profitno-loss? with ?loss? with the relationship strength of 0.5 (50%), it shows no profit no loss with 0.5 (50%) possibility and 0.5 (50%) of loss in this case. Several responses can be observed during the simulation of the system. The system is tuned by modifying the rules and membership functions until the desired system response (output) is achieved. The system can be interfaced to the real world via Java programming language.
# VI. CONCLUSION
It is important to make evident the great potential that fuzzy logic has to offer, such as the need for the mathematical model. Fuzzy Logic Controllers can provide more effective control of non-linear systems than linear controllers, as there is more flexibility in designing the mapping from the input to the output space. Fuzzy logic is capable of resolving conflicts by collaboration, propagation and aggregation and can mimic humanlike reasoning. Another advantage that the fuzzy logic offers is that an autotuning algorithm can be applied to the system, by the means of this reasoning. In this way, the system can learn the control parameters to take. In our study, we represent the mathematical expression profitability components using linguistic variables. We consider -e rror? and -c hange in error? approaches in a vague, ambiguous and uncertain situation. It is shown that fuzzy logic is able to represent common sense knowledge and address the issue of vagueness, ambiguity and uncertainty (Obot, 2008) as it is used to find the exact degree of profit, loss, low-profit, etc in the profitability of an industry. To this end, fuzzy logic can be used to control and ensure the desired output in a model since it can tolerate wide variation in input variables. Fuzzy logic control model shows that profit can be achieved at various levels, but maximum profit is achieved when the selling price is more than the cost price by 100% (1.0). Also loss can be incurred at different levels when the production cost is more than the selling price. The exact level and exact loss or profit has been clearly defined by fuzzy logic control system thereby resolving the conflict of uncertainty and vagueness. Our case study reveals that production cost depends on waste paper cost, the parameters of paper recycling process and other costs associated with paper recycling. Therefore it is recommended that the parameters affecting production cost, and determine the output of paper recycling be modified so as to reduce production cost and achieve maximum profit. Since the ultimate aim of any capitalist industry is to make profit, the concept of profitability is of great significance and it is evident that the fuzzy logic model developed, if implemented is an effective tool to effectively control profitability in paper recycling to achieve maximum profit. For further optimization of results of our work, hybridization of fuzzy logic and neural network or fuzzy logic and genetic algorithm is recommended for future research.
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34![More error terms (SP)(QR) -(CP)(QR) = << N, -Ne gative Small error? (NS) (Low Loss) (SP)(QR) -(CP)(QR) = >>P, -Posi tive Small error? (PS) (Low Profit) (SP)(QR) -(CP)(QR) = <<<< N, -N egative Big error? (NB) (High Loss) (SP)(QR) -(CP)(QR) = >>>> P, -Positive Big error? (PB) (High Profit) More change in error terms d{(SP)(QR)-(CP)(QP)}/dt = << N, -Ne gative Small error-change? (NS) (Low loss over time) d{(SP)(QR)-(CP)(QP)}/dt = >> P, -P ositive Small error-change? (PS) (Low profit over time) d{(SP)(QR)-(CP)(QR)}/dt = <<<< N, -Neg ative Big error-change? (NB) (High Loss over time) d{(SP)(QR)-(CP)(QR)}/dt = >>>> P, -Positive Big error-change? (PB) (High Profit over time)In this paper, the universes of discourse for error (E),change in error (CE) and Output are chosen to be [-100, 100], [-100, 100], and [-100, 100] respectively. Both sets of the linguistic values for the linguistic variables E and CE are {NB, NS, N, Z, P, PS, PB}, and the set of linguistic values for Output is {HL, LL, L, NPNL, P, LP, HP}, where NB, NS, N, Z, P, PS, and PB represent negative_big, negative_small, negative, zero, positive,](image-5.png "3 ) 4 )")
2
20if x < 25? LProfit (x) =(x-25)/2525 <= x < 50(25-x)/25if 50 <= x < 75(17)0if x > 750if x < 50? HProfit (x) =(x-50)/25 (100-x)/25if 50 <= x < 75 if 75<= x < 100(18)0if x > 100Table 1 Fuzzy rule base Rule No. Rules 1. IF (S P )
2ENBNSNZPPSPBNBHLHLHLNPNLLPLLPNSHLLLLNPNLLLNPNLPNHLLLLNPNLLLPLPZHLLLLNPNLLPLPHPPLLLNPNLPPLPHPPSLLPPLPLPLPHPPBNPNLPLPLPHPPHP5) Fuzzy Inference
vagueness.PremiseConclusion PartMinimumRuleVariablesof rulevalueNo.(non zero)Error Changein error410.20.2LowProfit0.2420.20.8HighProfit0.2480.80.2Profit0.2490.80.8HighProfit0.8(24)5= 25.7% Profit(25)
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