An Alternative Model of Aggregate Production Planning for Cement Company: Solving with Particle Swarm Optimization

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1 An Alternative Model of Aggregate Production Planning for Cement Company: Solving with Particle Swarm Optimization Sayed Rezwanul Islam 1*, Md. Arifuzzaman 2 and Nokib Parvez 3 The Aggregate Production Planning (APP) is a schedule of the organization s overall operations over a planning horizon to satisfy demand while minimizing costs. It is considered as the baseline for any further planning and formulating the master production scheduling, resources, capacity and raw material planning. For APP of Cement Company, a large number of optimization criterion- regular time production, overtime production and inventory level of cement, clinker, limestone, gypsum and subsidiary ingredients play a very important role. Complexities have arisen when practitioners try to balance these large number of constraints during the planning horizon. In this research work, authors try to construct a mathematical model which reduce complexities and make an effective balance among different constraints. The model then solved by well-known heuristics algorithm named particle swarm optimization (PSO) and compare the result among various types of PSO algorithm. Authors also find out the limitations of PSO to solve APP problem. Field of Research: Supply Chain Management 1. Introduction Aggregate Production Planning (APP) is always concern about the allocation of resources of the company to meet the demand forecast. It aims to determine the production, the inventory and the workforce levels of a company on a finite time horizon (normally 3 to 18 months) simultaneously. In APP, the researchers take into consideration- the employment levels and charges, inventory levels and charges, output rates of machines and subcontracting to optimize the plan. A good aggregate production plan encompasses all levels of production costs, inventory costs, packaging costs, layoff costs & hiring costs. In aggregate production planning 80% decisions depends on cost only. Hence, the authors have considered only one objective function of minimizing cost. There are many solving procedure for APP problem but in this research work authors have used different types of particle swarm optimization (PSO) algorithms and then compare results among them. Recently, PSO algorithm has been becomes available and promising techniques for real world optimization problems (Ramazanian et al 2011). Authors have considered large number of constraints constraints of regular time production, overtime production, packaging and inventory level. Here, inventory costs include- inventory carrying cost of cement, inventory carrying cost of clinker, inventory carrying cost of limestone, and inventory carrying cost of subsidiary ingredients. In this research work, first the authors have *Corresponding Author: Sayed Rezwanul Islam, Department of Industrial & Production Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh. syedrezwanulislam@gmail.com Md. Arifuzzaman, LEAN Department, Ventura Leatherware Mfy (BD) Ltd. Bangladesh. Arifipe10@yahoo.com Nokib Parvez, Production Planning & Operations, Shanta Denims Ltd. Dewhirst Group, Bangladesh. parvezipe10@yahoo.com 1

2 formulated the model of aggregate production planning and then the PSO algorithm is used to find the optimal solution of the problem. The main objective of this research work is to fill out the gaps that are left out by other researchers. This study arranged as follows- Part 2 presents the literature review to identify the research scope on aggregate production planning. Part 3 explains the research methodology and the mathematical statement. Part 4 presents the Primary Data. Part 5 presents the solution procedures. In this part, different types of particle swarm optimization algorithm are explained in detail which are utilized to solve the APP problem. Part 6 presents summarized results. Part 7 presents the research findings which is breakdown into two categories. First category presents the findings of APP and second category presents the findings of different PSO algorithm. Part 8 presents the conclusions of the study. 2. Literature Review In the field of aggregate production planning, different researchers construct models for different perspectives. Researchers always try to construct precise model of aggregate production planning than the previous one. Wang and Fang (2001) stated that Aggregate Production Planning (APP) is a medium range capacity planning method that typically encompasses a time horizon from 2 to 12 months. The authors mentioned that APP parameters are fuzzy in nature. But in this model they have considered crisp value and apply trapezoidal membership function to determine the APP result. But when APP input parameters are considered as a crisp value then these parameters do not behave fuzzy nature rather than deterministic in nature. Leung et al (2003) developed a multi-objective model which is developed to solve the production planning problems. In this model the profit is maximized but production penalties resulting from going over / under quotas and the change in workforce level are minimized. Wang and Liang (2005) developed a fuzzy multi-objective linear programming (FMOLP) model for solving the multi-product APP decision problem in a fuzzy environment. This model considers a large number of optimization criterion attempts to minimize total production costs, carrying and backordering costs and rates of changes in labor levels considering inventory level, labor levels, capacity, warehouse space and the time value of money. Jana and Roy (2005) developed a multi-objective linear programming problem with fuzzy objective function and fuzzy constraints. In this model they have considered crisp value and apply triangular membership function to determine the APP result which are deterministic in nature. Aliev et al (2007) developed a fuzzy integrated multi-period and multi-product aggregate production and distribution model in supply chain, in which the model was formulated in terms of fuzzy programming and then the solution was solved by genetic algorithm. Liang (2007) proposed an interactive possibilistic linear programming approach to solve multiproduct and multi-time period APP problems with multiple imprecise objectives and cost coefficients by triangular possibility distributions in uncertain environments. This imprecise multi-objective APP model designed here attempts to minimize total production costs and changes in work-force level with reference to imprecise demand, cost coefficients, available resources and capacity. Feng and Rakesh (2010) developed an integrated optimization of logistics and production costs associated with the supply chain members based on the scenario approach to handle the uncertainty of demand. This model considered the expected total costs, cost variability due to demand uncertainty, and expected penalty. Ramazanian and Modares (2011) developed a multi-objective goal 2

3 programming model for a multi-product multi-step multi-period APP problem in the cement industry. The model was reformulated as a single objective by goal programming approach. The strength of this model is that it balance the constraints of the whole production system. In this paper the authors also propose a PSO variant whose inertia weighted was set as a function. But the authors do not consider the inventory level of clinker, limestone, subsidiary ingredients and gypsum. And the authors also do not balance the inventory level of cement with the inventory level of clinker, limestone, subsidiary ingredients and gypsum. Gani and Assarudeen (2012) stated that fuzzy numbers and fuzzy values are widely used in engineering applications because of their suitability for representing uncertain information. In their research work the authors mentioned that in standard fuzzy arithmetic operations faces some problem in subtraction and division operations hence in their study a new operation on Triangular Fuzzy Numbers is defined, where the method of subtraction and division has been modified. Ramezanian et al (2012) developed a multi-product, multi-period APP model which aims to minimize costs and instabilities in the work force and inventory levels. In their research work, they concentrated on multi-period, multi-product and multi-machine systems with setup decisions. The model then solved by Mixed Integer Linear Programming (MILP). Baltas et al (2013) introduced a PSO variant to a service design and diversification problem. They have designed and implemented genetic algorithm and PSO to stated preference data derived from conjoint consumer preferences for service attributes in a retail setting. This method has valuable implications for managers aiming to improve how they design their services. Chakrabortty and Hasin (2013) presented an interactive MOGA approach to determine the optimum aggregate production plan for meeting forecasted demand by adjusting regular and overtime production rates, inventory levels, labor levels, subcontracting and backordering rates, escalation factor in the each of the cost categories over a period of time and other controllable variables. In this model they have considered crisp value and apply triangular membership function to determine the APP result. Ning et al (2013) developed multi product APP in fuzzy random environment, in which the market demand, production cost, subcontracting cost, inventory carrying cost, backorder cost, product capacity, sales revenue, maximum labor level, maximum capital level, etc. were all characterized as fuzzy random variables. Damghani and Shahrokh (2014) developed a new multi-product multi-period multi-objective APP model where they focused three objective functions, including minimizing total cost, maximizing customer services level, and maximizing the quality of end-product concurrently. In this model a large number of optimization criterion was considered which includes the quantity of production, available time, work force levels, inventory levels, backordering levels, machine capacity, warehouse space and available budget. Then the proposed model was solved by fuzzy Goal programming approach. Chen and Huang (2014) developed a multi-product, multi period APP model with several distinct types of fuzzy uncertainties. In this research work they have assumed three possible values of the parameters and applied triangular membership function to get the solution. Tsafarakis et al (2013) presented a new hybrid PSO approach to design an optimal industrial product line. The hybrid PSO has searched for an optimal product line in a large design space which consists of discrete and continuous design variables. The approach was illustrated through an application to a simulated dataset of industrial cranes. It also yielded important implications for strategic customer relationship and production management.with many successful applications in various domain problems, PSO has shown that it is a considerably promising, efficient and robust technique for practical applications. For examples, PSO had been successfully applied to scheduling problems (Chen 2011; Liao et al 2007), game theory problems (Lung 3

4 &Dumitrescu 2009; Pavlidis et al 2005), optimization on continuously changing environments (Parsopoulos & Vrahatis 2001), and detection of periodic orbits (Skokos et al 2005). Though PSO has shown a considerably promising, efficient and robust technique for practical applications, it has still some limitations. To overcome the limitations different researchers propose different adaptive PSO. Kennedy and Eberhart (1995) first have used the concept of inertia weight to increase the convergence rate of the standard PSO and introduced it as global best PSO. Compared to the classical PSO, GBPSO uses an inertia weight w and does not require the personal best weight c1 and global best weight c2 be set to 2. Decreasing weight particle swarm optimization (DWPSO) is similar to GBPSO, but the inertia weight is decreased linearly over time. The idea behind DWPSO is to focus on diversity in early iterations and convergence in late iterations (Hu & Eberhart, 2002). Ratnaweera et al (2004) stated that Time-varying Acceleration Coefficients PSO (TVACPSO) does not only change the inertia weight w, but also the acceleration coefficients, i.e., the personal best weight c1 and global best weight c2, over time. The idea is to have a high diversity for early iterations and a high convergence for late iterations. Ramazanian and Modares (2011) stated that the search process of PSO is nonlinear and highly complicated, linearly and nonlinearly decreasing inertia weight with no feedback taken from the global optimum fitness cannot truly reflect the actual search process. So, when the global fitness is very large, the particles are far away from the optimum point. Hence, a big velocity is needed to search the solution space and so as to the inertia weight must be larger values. Conversely, when the particles are very close to the global best solution, then the inertia weight must be very small. Here, the inertia weight is set as a function of global optimum fitness during search process of PSO algorithm. The above literature review give a comprehensive view of different contributions of different researchers in the field of aggregate production planning and particle swarm optimization in recent time. From the literature review, it is found that most of the researchers now try to construct fuzzified model to consider the impreciseness of human judgment. But this thesis work focus on the aggregate production planning of a cement company and only few works have done on cement company in the field of APP. From the through literature review and from the best of the authors knowledge none of the researchers do not consider the inventory level of clinker, limestone, subsidiary ingredients and gypsum. If the inventory level of cement, limestone, subsidiary ingredients, gypsum and clinker is not properly balanced then the APP model would not represent the actual situation of a cement plant. But this research work is first of its kind which try to construct a model of APP which will make a balance among the inventory level of cement, clinker, limestone, subsidiary ingredients and gypsum. Authors also try to reduce the inequality constraints by using the concept of raw mix concept. So this thesis work focus on constructing a mathematical model of APP which will focus on regular time production, overtime production, packaging cost, inventory level of cement, clinker, limestone, subsidiary ingredients and gypsum. 3. Research Methodology and Mathematical Statement 3.1. Research Methodology Following steps are followed during this research work: Step 3.1.1: A mathematical model of single product and multi-period aggregate production planning was constructed. To construct this mathematical model following things are done: 4

5 i. Problem was realized ii. Assumptions and notation were made iii. Problem was formulated and iv. Constraints were formulated Step 3.1.2: Finally the model was solved by different types of particle swarm optimization in C ++ programing Mathematical Statement The single product APP decision problem examined here can be described as follows: Assume that any particular company manufactures a single product (cement) to satisfy the market demand over a planning horizon T. APP model focuses on developing an optimum APP plan that meet the forecast demand by adjusting regular time and overtime production, inventory levels and other controllable variables. Based on the above characteristics of the considered APP problem, the mathematical model herein is developed on the following assumptions: I. The values of all parameters are certain over the next T planning horizon. II. Actual cement mill capacity, kiln capacity and inventory level of cement, clinker, limestone, subsidiary ingredients and gypsum in each period cannot exceed their respective maximum levels. III. The forecasted demand of cement over a particular period must be fulfilled. The following notation is used after reviewing the literature and considering practical situations i = Product type, i = 1,2,..,n t = planning period, t = 1,2,.,m C : Regular time production cost (per metric ton) C : Overtime production cost (per metric ton) C : Inventory cost of Limestone (per metric ton) C : Inventory cost of subsidiary ingredients (per metric ton) C : Inventory cost of clinker (per metric ton) C : Inventory cost of cement (per metric ton) C : Packaging cost of cement (per metric ton) KC : Maximum capacity of kiln in period t (MT) MC : Maximum capacity of cement mill in period t (MT) MCR : Maximum regular time capacity of cement mill in period t (MT) MCO : Maximum overtime capacity of cement mill in period t (M) LC : Maximum capacity of Limestone storage in period t (MT) SC : Maximum capacity of Subsidiary ingredients storage in period t (MT) GC : Maximum capacity of gypsum storage in period t (MT) D : Forecasted demand of cement in period t (MT) D : Forecasted demand of clinker in period t (MT) D : Forecasted demand of limestone in period t (MT) D : Forecasted demand of subsidiary ingredients in period t (MT) D : Forecasted demand of gypsum in period t (MT) : Previous inventory of cement in period t (MT) : Previous inventory of clinker in period t (MT) 5

6 : Previous inventory of subsidiary ingredients in period t (MT) : Previous inventory of limestone in period t (MT) : Previous inventory of gypsum in period t (MT) : Current inventory of limestone (metric ton) in period t (MT) : Current inventory of subsidiary ingredients (metric ton) in period t (MT) : Current inventory of clinker (metric ton) in period t (MT) : Current inventory of cement (metric ton) in period t (MT) : Current inventory of gypsum (metric ton) in period t (MT) : Amount of cement produced (metric ton) in period t (MT) : Amount of clinker produced (metric ton) in period t (MT) : Amount of regular time cement production in period t (MT) : Amount of over time cement production in period t (MT) : Amount of limestone purchased in period t (MT) : Amount of gypsum purchased in period t (MT) : Amount of subsidiary ingredients purchased in period t (MT) : Amount of raw mix available for cement production in period t (MT) : Amount of clinker used for cement production in period t (MT) : Amount of clinker used for regular time cement production in period t (MT) : Amount of clinker used for overtime cement production in period t (MT) : Amount of Limestone used for cement production in period t (MT) : Amount of subsidiary ingredients used for cement production in period t : Amount of raw mix used for cement production in period t (MT) : Amount gypsum used for cement production in period t (MT) S%: Percent subsidiary ingredients present in raw mix (per MT) %a: Percent of clinker produced from raw mix (per MT) %L: Percent of limestone present in raw mix (per MT) G%: Percent of gypsum present in cement (per MT) MinZ= ( )+ ( )+ ( )+ [( ) +( )+ ( )+ ( )+ ( )] (1) Here, first two terms represents the production cost. It includes regular time production cost and overtime production cost. Third term represents the packaging cost. And the later portion represent the inventory cost of different ingredients of the time horizon. Constraints: Constraints on carrying inventory: Constraints on carrying inventory of cement: PC + I = D (2) Constraints on carrying inventory of clinker: PCL + I = D (3) Constraints on carrying inventory of Limestone: Q + I = D (4) Constraints on carrying inventory of Subsidiary Ingredients: Q + I = D (5) Constraints on carrying inventory of Gypsum: Q + I = D (6) 6

7 Capacity constraints: Q + cap (7) Q + cap (8) PCL KC (9) PC MCR (10) PC MCO (11) PC MC (12) Q + cap (13) Equation (7) represents the capacity of the limestone storage. Equation (8) represents the capacity of the subsidiary ingredients storage. Equation (9) represents the maximum capacity of the kiln by which clinker is produced. Equation (10) represents the maximum capacity of regular time cement production by the cement mill. Equation (11) represents the maximum capacity of overtime cement production by cement mill. Equation (12) represents the capacity of total amount of cement production by cement mill in period t. Equation (13) represents the capacity of the gypsum storage. Constraints on balance: A = ( Q + ) + ( Q + )(14) U A (15) U = L% U (16) U Q + (17) U = S% U (18) U Q + (19) PCL = C% U (20) U PCL + (21) U = U + U (22) PC = b% U (23) PC = b% U (24) PC PC (25) PC = PC + PC (26) MC =MCR +MCO (27) U = G% PC (28) U Q + (29) Equation (14) represents the total amount of raw mix available in period t. Equation (15) represents the amount of raw mix utilization in period t. Equation (16) and (17) represent the amount of limestone utilization in period t. Equation (18) and (19) represent the amount of subsidiary ingredients utilization in period t. Equation (20) represents the total amount of clinker production in period t. Equation (21) and (22) represent the utilization of clinker in period t. Equation (23) and (24) represent the regular time and overtime cement production.. Equation (26) represents the total amount of cement production in period t. Equation (27) represents the cement mill capacity. Equation (28) and (29) represent the utilization of gypsum for cement production in period t. 7

8 Non-negativity constraints: PC,PC,PC,I,I,I,I,I,,,,,,A,U,U,U,U,PCL 0 (30) 4. Data Collection In this research work, authors have used primary data. The required data was taken from a leading Cement Company in Bangladesh. Tables 4.1 and 4.2 represent cost data and other relevant data. Table 4.1: Cost Data Name Tk. Production cost (regular time) per metric ton, 4000 Production cost (overtime) per metric ton, 0.00 Inventory cost of cement per metric ton, 15 Inventory cost of clinker per metric ton, 17 Inventory cost of limestone per metric ton, 8 Inventory cost of subsidiary ingredients per metric ton, 16 Inventory cost of gypsum per metric ton, 19 Packaging cost per metric ton, 304 Table 4.2: Other Relevant Data Name Percent Unit(MT) Forecasted demand of cement Forecasted demand of clinker Forecasted demand of limestone Forecasted demand of subsidiary ingredients Forecasted demand of gypsum Amount of limestone present in raw mix, L% 80 Amount of subsidiary ingredients present in raw mix, S% 20 Amount of clinker produced from raw mix, C% 65 Percentage of clinker in cement 95 Percentage of gypsum in cement, G% 5 Capacity of limestone storage, 200,000 Capacity of subsidiary ingredients storage, 50,000 Capacity of gypsum storage, Maximum capacity of regular time production of cement, 150,000 Maximum capacity of overtime production of cement, 0.00 Maximum capacity of kiln, 120, Particle Swarm Optimization The behaviour of PSO can be envisioned by comparing it to bird swarms searching for optimal food sources, where the direction in which a bird moves is influenced by its current movement, the best food source it ever experienced, and the best food source any bird in the swarm ever experienced. Birds are driven by their inertia, their personal knowledge, 8

9 and the knowledge of the swarm. In terms of PSO, the movement of a particle is influenced by its inertia, its personal best position, and the global best position. PSO has multiple particles, and every particle consists of its current objective value, its position, its velocity, its personal best value, that is the best objective value the particle ever experienced, and its personal best position, that is the position at which the personal best value has been found. In addition, PSO maintains the global best value that is the best objective value any particle has ever experienced, and the global best position, that is the position at which the global best value has been found. In case of standard PSO, the following iteration to move the particles: (n +1)= (n) + (n+1), (5.1) n= 0, 1, 2,.N-1 Where, is the position of particle i, n is the iteration, n = 0 refers to the initialization, N is the total number of iterations, and is the velocity of particle i. In classical PSO, the velocity of the particle is determined using the following iteration: (n +1)= (n) +2 ( ) ( )[ ( ) ( ) ( ) ( )] + 2 ( ) ( )[ ( ) ( ) ( )] (5.2) n= 0, 1, 2.N-1 Where the personal best is position, and is the global best position. ( ) ( ) ( ) ( ) Calculates a vector directed towards the personal best position, and calculates ( ) ( ) ( ) a vector directed towards the global best position. Both ( ) and ( ) are random vectors that contain values uniformly distributed between 0 and 1. The notation ( ) ( ) is meant to denote that a new random vector is generated for every particle i and iteration n. PSO can focus on either convergence or diversity at any iteration. To focus on diversity means particles are scattered, searching a large area coarsely. To focus on convergence means particles are close to each other, searching a small area intensively. A promising strategy is to focus on diversity in early iterations and convergence in later iterations. The global best PSO use the following iteration to determine the velocities: (n +1)= (n) + ( ) ( )[ ( ) ( ) ( ) ( )] + ( ) ( )[ ( ) ( ) ( )] (5.3) n= 0, 1, 2.N-1 Decreasing weight PSO use the following iteration to determine the velocities: (n +1)= (n) + ( ) ( )[ ( ) ( ) ( ) ( )] + ( ) ( )[ ( ) ( ) ( )](5.4) n= 0, 1, 2.N-1 Where the inertia weight w at every iteration n is calculated using the following equation: w (n) = ( ) (5.5) Where w (n) is the inertia weight at iteration n, is the inertia weight designated for the first iteration, and is the inertia weight designated for the last iteration N. The inertia 9

10 weight w at every iteration n depends on the total number of iterations N. Therefore, changing the total number of iterations N will change the behaviour of the algorithm at every iteration. Time Varying Acceleration PSO (TVACPSO) uses the following iteration to determine the velocities: (n +1)= (n) + ( ) ( ) ( )[ ( ) ( ) ( ) ( )] + ( ) ( ) ( )[ ( ) ( ) ( )](5.6) n= 0, 1, 2,.N-1 Where the inertia weight w at every iteration n is calculated using the following equation: w (n) = ( ) (5.7) Where the personal best weight and the global best weight at every iteration n are calculated using the following equations: ( ) = ( ) (5.8) ( ) = ( ) (5.9) where ( ) is the personal best weight at iteration n, ( ) is the global best weight at iteration n, is the personal best weight designated for the first iteration, is the personal best weight designated for the last iteration N, is the global best weight designated for the first iteration, and is the global best weight designated for the last iteration N. Adaptive PSO uses the following iteration to determine the velocities: (n +1)= (n) + ( ) ( ) ( )[ ( ) ( ) ( ) ( )] + ( ) ( ) ( )[ ( ) ( ) ( )](5.10) n= 0, 1, 2,.N-1 Here, the inertia weight w is adapted as follows: = (5.11) ( ( ) 10

11 6. Summarized Results Proceedings of 14th Asian Business Research Conference The summarized results obtained from different PSO algorithm are shown in table 6.1. Table 6.1: Summarized Result Initial Inventory (MT.) Current Inventory (MT.) Total cost (TK.) 7. Findings Name Standard Global Best Decreasing Adaptive PSO PSO PSO Weight PSO TVAC PSO Cement Clinker Limestone Subsidiary Ingredients Gypsum Cement Clinker Limestone Subsidiary Ingredients Gypsum The APP decision problem presented here was solved by different types of PSO. Program was constructed in and run in CODEBLOCK 2014 by a PC with configuration Intel (R) Core(TM) i3-2310m 2.10 GHz, 2 GB RAM. We have used =0.9, =0.4, c1s =2.5, c2s =0.5, c1e =0.5 and c2e = 2.5 in this research work. The findings of this research work are discussed in two steps. First step has shown the findings of the APP model and second step has shown the findings of different particle swarm optimization algorithms. These are summarized below: 7.1. Findings of APP Model i. In this research work, authors used the concept of raw mix to balance the constraints. This concept reduce the inequality constraints. So the practitioners will expect a good result from this model. ii. This model not only highlight the inventory level of cement but also highlight the inventory level of clinker, subsidiary ingredients, limestone and gypsum which will present a clear view of a cement company during the planning horizon. iii. The optimal value of this APP problem is Tk which is achieved by decreasing weight PSO. TVAC PSO also finds the solution which is very close to the optimal solution Findings of Different Particle Swarm Optimization Algorithm For this research work three particles are assumed. The positions of these particles are defined as follows: P1 { x(7)= 41000, x(10)= , x(13)=6600, x(17)=189200, x(20)=122900} P2 { x(7)= 40900, x(10)= , x(13)=6500, x(17)=189000, x(20)=122950} P3 { x(7)= 40100, x(10)= , x(13)=6550, x(17)=189100, x(20)=122800} 11

12 7.2.1.Behaviour of Different Particles of Different PSO According to Objective Function Value at Different Iterations Number Standard Particle Swarm Optimization Standard PSO $559,000, $558,800, $558,600, $558,400, $558,200, $558,000, particle 1 particle 2 particle 3 Figure 7.1. Objective function value of Standard PSO at different iterations number Particle 1 give better result than the other particles. Particle 2 and Particle 3 cannot improve their objective function value satisfactorily as Particle 1. But after 500 iterations, standard PSO yet not find the optimal result because of the slowness of the convergence rate Global Best Particle Swarm Optimization GLOBAL BEST PSO particle 1 particle 2 particle 3 $556,000, $554,000, $552,000, $550,000, $548,000, $546,000, $544,000, Figure 7.2. Objective function value of Global best PSO at different iterations number Particle 3 give better result than the other particles. When the number of iterations increased, a decreasing slope is observed. After 200 iterations, all the particles go to the steady state and cannot improve the objective function value. 12

13 Adaptive Particle Swarm Optimization $557,600, $557,550, $557,500, $557,450, ADAPTIVE PSO particle 1 particle 2 particle 3 $557,400, Figure 7.3. Objective function value of Adaptive PSO at different iterations number After 30 iterations all the particles have shown the same results and go to the steady state Time Varying Acceleration Coefficients Particle Swarm Optimization TVAC PSO particle 1 particle 2 particle 3 $560,000, $550,000, $540,000, $530,000, Figure 7.4. Objective function value of TVAC PSO at different iterations number After 30 iterations, all the particles have shown the same result. But after every iterations the particles also improve their objective function value. After 400 iterations the particles go to the steady state Decreasing Weight Particle Swarm Optimization $560,000, $540,000, DECREASING WEIGHT PSO particle 1 particle 2 particle 3 $520,000, Figure7.5. Objective function value of Decreasing weight PSO at different iterations number 13

14 After 60 iterations all the particles have shown the same results. After 60 iterations, a number of increasing and decreasing slope are observed. At iterations 60, optimal result is found. After 6o iterations, the particles diverge from the optimal result Convergence Rate of Different Particle Swarm Optimization Standard Particle Swarm Optimization $559,000, $558,900, $558,800, $558,700, $558,600, $558,500, $558,400, Figure7.6. Convergence rate of different particles of Standard PSO The convergence rate of different particles to the optimum solution is very poor in Standard PSO. Particle 2 and particle 3 converge to the optimal solution but particle 1 diverge from the optimal solution Global Best Particle Swarm Optimization $200, Standard PSO average particle 1 particle 2 particle 3 Global best PSO $- $(200,000.00) $(400,000.00) particle 1 particle 2 particle 3 Figure7.7. Convergence rate of different particles of Global best PSO Though the convergence rate of different particles to the optimum solution is higher than standard PSO but still it is poor. Particle 1 and particle 2 converge to the optimal solution but particle 3 diverge from the optimal solution. 14

15 Adaptive Particle Swarm Optimization $15, $10, $5, Adaptive PSO $- $(5,000.00) Figure7.8. Convergence rate of different particles of Adaptive PSO The convergence rate of adaptive PSO is very high. At 30 iterations all the particles have shown the same objective function value Time Varying Acceleration Coefficients Particle Swarm Optimization $60, $40, $20, $- $(20,000.00) $(40,000.00) particle 1 particle 2 particle 3 Figure 7.9. Convergence rate of different particles of TVAC PSO The convergence rate of TVAC PSO is also very high. After 40 iterations all the particles have shown the same objective function value Decreasing Weight Particle Swarm Optimization $100, $- $(100,000.00) $(200,000.00) $(300,000.00) $(400,000.00) TVAC PSO particle 1 particle 2 particle 3 Decreasing weight PSO particle 1 particle 2 particle 3 Figure7.10. Convergence rate of different particles of Decreasing Weight PSO 15

16 The convergence rate of decreasing weight PSO is also very satisfactory. After 70 iterations all the particles have shown the same objective function value. 8. Conclusions The Aggregate production planning (APP) problem for cement industry was considered in this research work. A mathematical model is designed appropriated to the real factory needs. In heavily competitive situation it is becoming much more difficult to keep pace with increasing competition. So in future multiple plants, multi-product, multi-period scenario would be better. Here, only one objective function is considered. But in practice, firms may be operating under competing criteria, which is management by objectives in which the primary objective may not necessarily be only minimum costs. There are so many parameters that influence costing and other decisions which should be included to get a more precise model. In recent time, Cement Company use alternative fuel for cement production which reduce the emission of and at the same time reduce the production costs. So AF s is future scope of research in APP. This research work has shown that not only the standard PSO approach but also the different types of modified PSO approach have still some gaps in APP application. From this research work, it is also evident that the optimal solution depends a lot on inertia weight. By changing the inertia weight the optimal result is changed which is the major limitation of PSO. In APP problem, a large number of equality and inequality constraints have to be satisfied. But in PSO, if the particles achieve large velocity then it would be very difficult for the particles to update their positions. Though, PSO approach can generate partial optimal result within very short time but it is not useful for solving APP decision problems. Increasingly sophisticated parameters and assumptions into an APP problem only make it hindering and impracticable for practical applications. Here, the authors also have shown how the constraints can be effectively balance by using the raw mix concept. In APP, proper adjustment of constraints lead to the optimal result which is achieved by raw mix. Therefore, this study proposed a comprehensive APP model that is easy to adjust and to be optimized by different algorithms. References Aliev, RA, Fazlollahi, B, Guirimov, BG & Aliev, RR 2007, Fuzzy-genetic approach to aggregate production-distribution planning in supply chain management, Information Sciences, 177, Baltas, G, Tsafarakis, S, Saridakis, C & Matsatsinis, N 2013, Biologically inspired Approaches to strategic service design: Optimal service diversification through evolutionary and swarm intelligence models, Journal of Service Research, 16, Chakrabortty, RK & Hasin, M. AA 2013, Solving an aggregate production planning Problem by using multi-objective genetic algorithm (MOGA) approach, International Journal of Industrial Engineering Computations 4 (2013) Chen, SP & Huang, WL 2014, Solving Fuzzy Multiproduct Aggregate Production Planning Problems Based on Extension Principle, International Journal of Mathematics and MathematicalSciences, vol. 2014, Article ID , 18 pages 16

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