Multi Product Multi Period Aggregate Production Planning in an Uncertain Environment

Transcription

1 Global Science and Technology Journal Vol. 4. o. 1. Setember 2016 Issue. P Multi Product Multi Period Aggregate Production Planning in an Uncertain Environment Md. Mohibul Islam * and Md. Mosharraf Hossain ** In this aer a new aroach is roosed to handle the uncertain inut value in the aggregate roduction lanning model.generally aggregate roduction lanning models have been develoed by considering certain value of each arameter over a lanning horizon. But in real life these values are not remain certain over a lanning horizon rather uncertain.the main objective of this roosed araoch is to find out the minimum total aggregate roduction lanning cost considering uncertain inut value. Hence, random value is considered from secific data range of each arameter of a secific automobile facory and finally comares the results with revious different aroaches. Keywords: Random value, Data range, imrecise arameters and uncertain environment. 1. Introduction Demand forecasting can be classified as short, medium and longe range. Aggregate Production Planning (APP) is an intermediate based caacity lanning and used medium range forecast, usually 3 to 18 months. The main attention of aggregate roduction lanning is to determine roduction, inventory, and work force levels to meet fluctuating demand. The lanning horizon is often divided into eriods. Every month of a year may be considered one eriod. The form of time horizon for aggregate roduction lanning varies from comany to comany. Generally aggregate roduction lanning determines the best way to meet forecasted demand in the intermediate time by adjusting regular time roduction, overtime roduction, subcontracting levels, inventory levels, labor levels, backordering rates and other controllable variables.theoretically APP controllable variables are assumed deterministic in nature but in real-world they are not deterministic. It is found that APP inut arameters such as forcasted market demand, corresonding oerating cost and caacity are uncertain in nature owing to some information being incomlete or unobtainable. From ast studies it is found that few authors have tried to solve the aggregate roduction lanning roblem considering imrecise data. To attain their goal they have used certain value for a secific time horizon and other researchers have used triangular fuzzy number for getting APP decision. From review guideline it is found that researchers mentioned that better aggregate roduction lanning decision may *Md. Mohibul Islam (Corresonding author), Deartment of Industrial & Production Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh, mohibul05ie@yahoo.com, **Md. Mosharraf Hossain, Deartment of Industrial & Production Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh, mosharraf80@yahoo.com,

2 be obtained if APP inut data randomly consider in the model. But nobody has carried out research in such a way where APP inut data randomly considered in their model. Hence the authors of this aer are highly motivated to fill u this research ga. This is the main cause for carry out the research work on this roblem arena. The major distinguishable feature in this aer is that random value is to be considered randomly in the model. Then this random value is to be used both in the constraints and objective function iteratively for searching the minimum aggregate roduction lanning cost. According to review considering uncertainty in APP model is very much imortant for roduce more accetable result. The major significance side of this new aroach is that it incororated uncertainty in the APP model. So, APP ractioners get a clear guidline for handling uncertainty to solve their real life roblem. The rest of the aer is organized as follows: literature review is resented in section 2. Research methodology and mathematical statements are mentioned in section 3. Data collection and solving rocedure are mentioned in section 4. Resuls and findings are resented in section 5. Section 6 and 7 reresent conclusion and recommendation for further research resectively and finally the references are mentioned in the last ortion of this aer. 2. Literature Review Wang and Fang (2001) mentioned that Aggregate Production Planning (APP) is a medium range caacity lanning. It tyically encomasses a time horizon from 2 to 12 months. They mentioned that inut data of APP are fuzzy in nature. But they considered cris value and aly traezoidal membershi function to determine the APP result. Generally when APP inut data are considered as a cri value then these data do not behave fuzzy nature rather deterministic in nature. Leung et al (2003) roosed a multi-objective model which is develoed to solve the roduction lanning roblem for maximize rofit and minimize workforce level. For ractical imlications of the roosed model different managerial roduction lans are evaluated. Forecasted demand and others oerating costs are varied from time to time. This variation should be considered during making decision regarding on aggregate roduction lanning but that are ignored in their research study. Pradenas et al (2004) mentioned that aggregate roduction lanning is articularly comlex in nature for roducing different roducts. They considered demands and corresonding oerating costs are deterministic in nature but in real life these data are uncertain in nature. Wang and Liang (2004) mentioned that traditional methods are used recise data to solve APP roblem. But in real world to get aroriate APP result the inut data are need to consider imrecise owing to some information is incomlete or unobtainable. Jana and Roy (2005) discussed multi-objective linear rogramming roblem with fuzzy objective function and fuzzy constraints. For doing solution they considered triangular fuzzy numbers which are deterministic in nature. Wang and Liang (2005) mentioned that in real world to get aroriate APP result the inut data need to consider imrecise. To attain this goal they considered triangular fuzzy numbers and then fuzzy logic is alied. Jain and Palekar (2005) mentioned that most of the research on aggregate roduction lanning focused on discrete arts manufacturing. They mentioned that where intermediate inventory is not allowed and multile roducts are roduced simultaneously using comlex configurations of roduction machines traditional APP aroach may roduce erroneous results. They roosed configuration-based model from resource-based 2

3 model. But assumed deterministic inut data for APP arameters which are not always deterministic in real life. Takey and Mesquita (2006) claimed that many industrial roducts are highly seasonally varied. To meet fluctuating demands make to stock strategy is used in terms of lowest inventory cost. But they considered fix oerating costs which may variy with demand variation that is not considered in their study. Filho et al (2006) claimed that nobody integrate the objectives defined in manufacturing strategy in aggregate roduction lanning model. Hence they roosed a multi objective model that included the context of manufacturing strategy. In order to accomlish their research works they considered demand and cost arameters are deterministic in nature. Jamalnia and Soukhakian (2009) mentioned that to get aroriate result from APP roblems the inut data need to consider imrecise. But it is noticed that they assigned certain value for every secific arameter. Gulsun et al (2009) mentioned that roblems that involved a range data for a secific arameter need to take into account not only cost related objectives but also motivational or service level objectives simultaneously. They also mentioned that another imortant oint is the uncertainty and flexibility factors. They mentioned that objective function of the model can be exressed with interval numbers. But it is noticed that in their work they assumed deterministic values of the APP arameters rather than interval value. Baykasoglu and Gocken (2010) mentioned that in real world APP inut data are imrecise because some information is incomlete or unobtainable. They consider triangular fuzzy numbe for some of the arameters and fix number for others. Effati and Abbasiyan (2010) focused on linear rogramming roblems in which both the right-hand side and the technological coefficients are fuzzy numbers. They mentioned that according to Bellman and Zadeh (1970) aroach obtained criss values can be non-linear, where the non-linearity arises in constraints. They roosed two methods such as augmented lagrangian enalty method and method of feasible direction to solve this roblem. Hashem et al (2011) mentioned that some critical arameters such as customer demand, corresonding oeratin cost and manufacturing caacity are not known with certainty. So decision based on the certain value is a risk that demand might not be met with the right roducts. Mondal and Pathak (2011) mentioned that demand fluctuation is available in a dynamic market. Assigning a set of cris values for solving APP roblem is no longer aroriate. They alied fuzzy triangular member function and recommend that better result may be obtained by using random variable. Veeramani et al (2011) mentioned that fuzzy multi-ojective linear rogramming is one of the most frequently alied in fuzzy decision making techniques. They mentioned that objective functions and the constraints involve many arameters whose ossible values may be assigned by the exerts. In traditional aroaches, such arameters are fixed at some values in an exerimental or subjective manner throught the exert understands of the nature of the arameters. Unfortunately, real world situations are often not deterministic. There exist various tyes of uncertainities in social, industrial and economic systems. Yenradeea and Piyamanothorna (2011) mentioned that there is a link between marketing romotion and customer demand. Demand is always affected by marketing romotion. It is beneficial to determine both lans simultaneously. They roosed a general otimization model that can hel making decision in marketing romotion in accordance with roduction lanning in order to maximize the rofit for the comany. They assumed APP arameters are deterministic in natures which are not always deterministic. Ramezanian et al (2012) mentioned that aggregate roduction lanning is a medium-term caacity lanning to 3

4 satisfy fluctuating demand over a lanning horizon. In their aer they considered all rogramming inut data deterministic in nature and there is no randomness. Huang and Chen (2012) mentioned that enterrises around the world have increasingly emhasized aggregate roduction lanning for deciding an aroriate way to match many controllable factors.they recommend that to get roer aggregate roduction lanning result uncertainty of inut data should be considered. Chakrabortty and Hasin (2013) mentioned that market demand and curresonding oerating cost and caacities are uncertain in real life. They considered triangular fuzzy number for solving APP roblem. Madadi and Wong (2013) mentioned that aggregate roduction lanning is a rocess by which a comany determines ideal levels of caacity, roduction, subcontracting, inventory, stok out, and even ricing over a secified time horizon to meet customer s requirements. In their study quality of roducts tried to consider and all inut data are assumed deterministic during the time horizon. Mortezaei et al (2013) mentioned APP roblems with deterministic arameters are unsuitable for determining effective solution. According to their views to get aroriate APP result forecasted demand, corresonding oerating cost and caacities should be considered imrecise in nature. ing et al (2013) mentioned that for getting aroriate result of aggregate roduction lanning all characteristics such as market demand, roduction cost and subcontracting cost must be considered uncertain in nature. Cheng and Bing (2013) mentioned that roduction lanning and control has crucial imact on the roduction and business activities of enterrise.they roosed a multi-objective roduction lanning otimization model based on the oint of view of the integration of roduction lanning and control, in order to achieve otimization and control of enterrise manufacturing management. They assumed all kinds of arameters such as demand, caacity and costs are deterministic in nature. owak (2013) mentioned that uncertainties are ignored in any managerial decisions but making sound decision uncertainty and risk of uncertainty have to be considered. Damghani and Shahrokh (2014) roosed a new multi-roduct multi-eriod multi-objective aggregate roduction lanning model where they focused three objective functions, including minimizing total cost, maximizing customer services level, and maximizing the quality of end-roduct concurrently. Then, a fuzzy goal rogramming (FGP) is roosed to solve the roosed model. Forecasted demand and corresonding oerating cost generally vary from eriod to eriod but these are not considered here. Sultana et al (2014) mentioned that aggregate lanners are concerned with the quality and the timing of exected demand. They mentioned that the task of aggregate lanner is to achieve equality of demand and caacity over the entire lanning horizon. It is noticed that demand and oerating costs are varied over the lanning horizon. Chen and Huang (2014) mentioned that aggregate roduction lanning lays a critical role in suly chain management. They investigated multi roduct, multi eriod APP roblems with several distinct tyes of fuzzy uncertainties. In their work they assumed three ossible values of the arameters and alied triangular membershi function to get the solution. Throughtout the literature review it is found that most of the researchers are aggred to get aroriate aggregate roduction lanning result inut data should be considered as imrecise. 4

5 3. Research Methodology and Mathematical Statement 3.1 Research Methodoly Outline of the research methodology is mentioned as follows: Literature Review Problem Statement Model Assumtion Model Reorganized Five cases are considered Case-1 Case-2 Case-3 Case-4 Case-5 Pessimistic data Most likely data Otimistic data Weighted average data Random generated data Comarison Select better result Previous result Comarison Final result Based on ast studies limitations, data are newly assumed for generating better result. Five cases are considered as follows: Case 1: Pessimistic value, Case 2: Most likely value, Case 3: Otimistic values, Case 4: Pessimistic, Most likely and Otimistic values jointly used by alying weighted average method and finally Case 5 ( Proosed aroach): Random values are considered. Mathematical model is reorganized for adoting new data assumtion in the current model. Accordingly secondary data have been adoted in the model and it is executed in alication software matlab 2010 version for getting result. Finally comare the result with the revious results for taking decision. 5

6 3.2 Mathematical Statement Current Mathematical Model The multi-roduct multi-eriod APP roblem can be described as follows. Assume that a comany manufactures tyes of roducts to satisfy the market demand over a lanning horizon T. Based on the above characteristics of the considered APP roblem, the mathematical model herein is develoed on the following assumtions. i. The values of all arameters are certain over the next T lanning horizon. ii. Actual labor levels, machine caacity and warehouse sace in each eriod cannot exceed their resective maximum levels. iii. The forecasted demand over a articular eriod can be either satisfied or backordered, but the backorder must be fulfilled in the next eriod. The following notation is used after reviewing the literature and considering ractical situations (Wang and Liang, 2004; Masud and Hwang, 1980; Wang and Fang, 2001) : Tyes of roducts Z: Total costs ($) D nt : Forecasted demand for nth roduct in eriod t (units) a nt : Regular time roduction cost er unit for nth roduct in eriod t ($ /unit) Q nt: Regular time roduction volume for nth roduct in eriod t (units) b nt: Overtime roduction cost er unit for nth roduct in eriod t ($ /unit) O nt: Overtime roduction volume for nth roduct in eriod t (units) c nt: Subcontracting cost er unit of nth roduct in eriod t ($ /unit) S nt: Subcontracting volume for nth roduct in eriod t (units) d nt : Inventory carrying cost er unit of nth roduct in eriod t ($ /unit) I nt : Inventory level in eriod t for nth roduct (units) e nt : Backorder cost er unit of nth roduct in eriod t ($ /unit) B nt : Backorder level for nth roduct in eriod t (unit) K t : Cost to hire one worker in eriod t ($/man-hour) H t : Worker hired in eriod t (man-hour) m t : Cost to layoff one worker in eriod t ($/man-hour) F t : Workers laid off in eriod t (man-hour) i nt : Hours of labor er unit of nth roduct in eriod t (man-hour/unit) r nt : Hours of machine usage er unit of nth roduct in eriod t (machine-hour/unit) V nt: Warehouse saces er unit of nth roduct in eriod t (ft 2 /unit) W tmax : Maximum labor level available in eriod t (man-hour) M tmax: Maximum machine caacity available in eriod t (machine-hour) V tmax : Maximum warehouse sace available in eriod t (ft 2 ) T T Min Z = [a nt Q nt + b nt O nt + c nt S nt + d nt I nt + e nt B nt ] + (K t H t + m t F t ) t=1 t=1 (1) Here the first five terms are used to calculate roduction costs. The roduction costs include five comonents-regular time roduction, overtime, and subcontracts, carrying inventory and backordering cost. The later ortion secifies the costs of change in labor levels, including the costs of hiring and lay off workers. 6

7 Constraints: Constraints on carrying inventory: I n(t 1) B n(t 1) + Q nt + O nt + S nt I nt + B nt = D nt for n, t (2) Where, D nt denotes the imrecise forecast demand of the nth roduct in eriod t. In real-world APP decision roblems, the forecast demand D nt cannot be obtained recisely in a dynamic market. The sum of regular and overtime roduction, inventory levels, and subcontracting and backorder levels essentially should equal the market demand, as in first constraint Equation. Demand over a articular eriod can be either met or backordered, but a backorder must be fulfilled in the subsequent eriod. Constraints on Labor levels: i n(t 1) (Q n(t 1) + O n(t 1) ) + H t F t i nt (Q nt + O nt ) = 0 for t (3) Here in the 3 rd constraint equation reresents a set of constraints in which the labor levels in eriod t equal the labor levels in eriod t-1 lus new hires less layoffs in eriod t. Actual labor levels cannot exceed the maximum available labor levels in each eriod, as in 4 th equation. Maximum available labor levels are imrecise, owing to uncertain labor market demand and suly. i nt (Q nt + O nt ) W tmax for t (4) Constraints on Machine caacity: r nt (O nt + Q nt ) M tmax for t (5) Constraints on Warehouse sace: V nt I nt V tmax for t (6) Here r nt and M tmax denote the imrecise hours of machine usage er unit of the nth roduct and the imrecise maximum available machine caacity in eriod t, resectively. Above two equations reresent the limits of actual machine and warehouse caacity in each eriod. on-negativity Constraints on decision variables are: Q nt, O nt, S nt, I nt, B nt, H t, F t 0 for n, t (7) It is noted that current mathematical model will be used for Case 1, Case 2 and Case 3. 7

8 3.2.2 Imroved Mathematical Model for Case 4 Model assumtions i. Three values are considered for each of the arameter over the next T lanning horizon. ii. Actual labor levels, machine caacity and warehouse sace in each eriod cannot exceed their resective maximum levels. iii. The forecasted demand over a articular eriod can be either satisfied or backordered, but the backorder must be fulfilled in the next eriod. Mathematical notation is same as like current model. w 1 : w 2 : w 3 : Weight of the essimistic value Weight of the most likely value Weight of the otimistic value w 1 + w 2 + w 3 =1 and w 1 = w 3 = 2 and w 6 3= 4 because for using the most likely values 6 here is that the most ossible values usually are the most imortant ones, and thus should be assigned more weights. β = minimum accetable ossibility of converting imrecise value into recise value (0.5) Accordingly imroved objective function Min Z = t=1 [(w 1 a nt,β o w 3 b nt,β ) O nt + (w 1 c nt,β + w 2 c m nt,β (w 1 e nt,β mk m 1,β T + w 2 e m nt,β o + w 3 e nt,β + w 2 a m nt,β o + w 3 a nt,β )Q nt + (w 1 b nt,β + w 2 b m nt,β + o + w 3 c nt,β ) S nt + (w 1 d nt,β + w 2 d m nt,β T ) B nt ] + t=1[(w 1 k 1,β + w 2 k m 1,β o + w 3 d nt,β ) I nt + + w 3 k o 1,β ) H t + (w 1 m 1,β + + w 3 m o 1,β )F t )] (8) Constraints: Constraints on carrying inventory: I n(t 1) B n(t 1) + Q nt + O nt + S nt I nt + B nt = (w 1 D nt,β o w 3 D nt,β ) for n, t (9) m + w 2 D nt,β + Constraints on Labor levels: i n(t 1) (Q n(t 1) + O n(t 1) ) + H t F t i nt (Q nt + O nt ) = 0 for t (10) i nt (Q nt + O nt ) (w 1 W t max,β m + w 2 W t max,β o + w 3 W t max,β ) for t (11) 8

9 Constraints on Machine caacity: m (w 1 r nt max,β + w 2 r nt max,β m o w 2 M t max,β + w 3 M t max,β ) for t for Constraints on Warehouse sace: o + w 3 r nt max,β )(O nt + Q nt ) (w 1 M t max,β + (12) V nt I nt V tmax for t (13) Q nt, O nt, S nt, I nt, B nt, H t, F t 0 for n, t (14) Imroved Mathematical Model for Case 5 Model Assumtions: i. The values of most of the arameters are uncertain over the next T lanning horizon and follow a secific range for each eriod and simultaneously a random value will be generated from this secific range and it will vary from eriod to eriod. ii. Actual labor levels, machine caacity and warehouse sace in each eriod cannot exceed their resective maximum levels.this caacity levels also follow a secific range. iii. The forecasted demand over a articular eriod is also uncertain that will be taken from generating random number from a redefined secific range and it will vary from eriod to eriod and it can be either satisfied or backordered, but the backorder must be fulfilled in the next eriod. iv. For Every iteration random values will be generated for each of the arameter for evaluates the objective function as well as constraints function and total number of iteration will be evaluated one lac (i=1:100000) The following notation is used after reviewing the literature and considering ractical situations (Wang and Liang, 2004; Masud and Hwang, 1980; Wang and Fang, 2001) RG Random number generation : Tyes of roducts Z: Total costs ($) D nt,i : Forecasted demand for nth roduct in eriod t at i th iteration of RG (units) a nt,i : Regular time roduction cost er unit for nth roduct in eriod t at i th iteration of RG ($ /unit) Q nt, i: Regular time roduction volume for nth roduct in eriod t at i th iteration of RG (units) b nt, i: Overtime roduction cost er unit for nth roduct in eriod t at i th iteration of RG ($ /unit) O nt, i: Overtime roduction volume for nth roduct in eriod t at i th iteration of RG (units) c nt: Subcontracting cost er unit of nth roduct in eriod t at i th iteration of RG ($ /unit) S nt,i : Subcontracting volume for nth roduct in eriod t at i th iteration of RG (units) 9

10 d nt,i : Inventory carrying cost er unit of nth roduct in eriod t at i th iteration of RG ($ /unit) I nt,i : Inventory level in eriod t for nth roduct at i th iteration of RG (units) e nt: Backorder cost er unit of nth roduct in eriod t at i th iteration of RG ($ /unit) B nt,i: Backorder level for nth roduct in eriod t at i th iteration of RG (unit) K t,i : Cost to hire one worker in eriod t at i th iteration of RG ($/man-hour) H t,i : Worker hired in eriod t at i th iteration of RG (man-hour) m t,i : Cost to layoff one worker in eriod t at i th iteration of RG ($/man-hour) F t,i : Workers laid off in eriod t at i th iteration of RG (man-hour) i nt : Hours of labor er unit of nth roduct in eriod t (man-hour/unit) r nt,i: Hours of machine usage er unit of nth roduct in eriod t at i th iteration of RG V nt : Warehouse saces er unit of nth roduct in eriod t (ft 2 /unit) W tmax,i : Maximum labor level available in eriod t at i th iteration of RG (man-hour) M tmax,i: Maximum machine caacity available in eriod t at i th iteration of RG (machine-hour) V tmax : Maximum warehouse sace available in eriod t (ft 2 ) The related cost coefficients in the objective function frequently are imrecise in nature because some information is incomlete or unobtainable in a medium time horizon. These cost coefficients will be considered randomly by generating random number from redefined articular range of each arameter so accordingly; imroved objective function of the roosed model is as follows: T Min Z = t=1[a nt,i Q nt,i + b nt,i O nt,i + c nt,is nt,i + d nt,i I nt,i + e nt,ib nt,i ] + T (K t,i H t,i + m t,i F t,i ) (15) t=1 Constraints: Constraints on carrying inventory: I n(t 1) B n(t 1) + Q nt,i + O nt,i + S nt,i I nt,i + B nt,i = D nt,i for n, t (16) Constraints on Labor levels: i n(t 1) (Q n(t 1) + O n(t 1) ) + H t,i F t,i i nt (Q nt,i + O nt,i ) = 0 for t (17) i nt (Q nt,i + O nt,i ) W t max,i for t (18) Constraints on Machine caacity: r nt,i (O nt + Q nt ) M t max,i for t (19) Constraints on Warehouse sace: V nt I nt V tmax for t (20) Q nt, O nt, S nt, I nt, B nt, H t, F t 0 for n, t (21) 10

11 4. Data Collection and Solving Procedures 4.1 Data Collection Secondary data have been used to demonstrate the racticality of the roosed methodology (Wang and Liang, 2004b). The lanning horizon is 4 months long, including May, June, July, and August. The model includes two tyes of standard ball screw, namely the external recirculation tye (roduct 1) and the internal recirculation tye (roduct 2). According to the reliminary environmental information, Tables 1 6 summarize the forecast demand, related oerating cost, and caacity data used in the model. The forecast demand, relevant oerating cost, and maximum labor and machine caacity are imrecise numbers with triangular ossibility distributions from eriod to eriod. Other relevant data are as follows: 1. Initial inventory in eriod 1 is 400 units of roduct 1 and 200 units of roduct. End inventory in eriod 4 is 300 units of roduct 1and 200 units of roduct Initial labor level is 300 man-hours. The costs associated with hiring and layoff are ($8, $10, $11) and ($2.0, $2.5, $3.2) er worker er hour, resectively. 3. Hours of labor er unit for any eriods are fixed to 0.05 man-hours for roduct 1 and 0.07 man hours for roduct 2. Hours of machine usage er unit for each of the four lanning eriods are (0.09, 0.10, 0.11) machine-hours for roduct 1 and (0.07, 0.08, 0.09) machine-hours for roduct 2. Warehouse saces required er unit are 2 square feet for roduct 1 and 3 square feet for roduct 2. Table 1: Forecast Demand Data (Existing Table) Item Period D 1t (900,1000,1080) (2750, 3000, 3200) (4600, 5000, 5300) (1850, 2000, 2100) D 2t (900,1000,1080) (450, 500, 540) (2750, 3000, 3200) (2300, 2500, 2650) Table 2: Related Oerating Cost Data (Existing Table) Product a nt ($/unit) b nt ($/unit) c nt($/unit) d nt ($/unit) e nt($/unit) 1 (17, 20, 22) (26, 30, 33) (22, 25, 27) (0.27, 0.30, 0.32) (35, 40, 44) 2 (8, 10, 11) (12, 15, 17) (10, 12, 13) (0.13, 0.15, 0.16) (16, 20, 23) Table 3: Maximum Labor, Machine & Warehouse Caacity Data (Existing Table) Period W t max ( man-hours) M t max ( machine-hours) V t max (ft 2 ) 1 ( 175, 300, 320) (360, 400, 430) 10,000 2 ( 175, 300, 320) (450, 500, 540) 10,000 3 ( 175, 300, 320) (540, 600, 650) 10,000 4 ( 175, 300, 320) (450, 500, 540) 10,000 11

12 Item D 1t (900* * *0.17) = Table 4: Weighted Average Data of Forecasting Demand Period (2750*0.17+ (4600* * * *0.17) = 5300*0.17) = (1850* * *0.17) = D 2t (900* * *0.17) = (450* * *0.17) = (2750* * *0.17) = (2300* * *0.17) = Table 5: Related Oerating Costs from Weighted Average Method (Considered) Product a nt ($/unit) b nt ($/unit) c nt($/unit) d nt ($/unit) e nt($/unit) 1 (17* * *0.17) =19.83 (26* * *0.17) = (22* * *0.17) = (0.27* * *0.17) = 0.30 (35* * *0.17) = (8* * *0.17) = 9.83 (12* * *0.17) = (10* * *0.17) = (0.13* * *0.17) = 0.15 (16* * *0.17) = Table 6: Conversion of Three Values into Single Value (Considered) Period W t max ( man-hours) M t max ( machine-hours) V t max (ft 2 ) 1 ( 175* * *0.17) = ( 175* * *0.17) = ( 175* * *0.17) = ( 175* * *0.17) = (360* * *0.17) = (450* * *0.17) = (540* * *0.17) = (450* * *0.17) = ,000 10,000 10,000 10,000 For the imlementation of roosed aroach data have been modified as follows in table 7-9 are given below. Table 7: Forecasted Demand Range (Units) Item Period D 1t ( ) ( ) ( ) ( ) D 2t ( ) ( ) ( ) ( ) Table 8: Related Oerating Cost Range Product a nt ($/unit) b nt ($/unit) c nt($/unit) d nt ($/unit) e nt($/unit) 1 (17-22) (26-33) (22-27) ( ) (35-44) 2 (8-11) (12-17) (10-13) ( ) (16-23) 12

13 Table 9: Maximum Labor, Machine, and Warehouse Caacity Range Period W t max ( man-hours) M t max ( machine-hours) V t max (ft 2 ) 1 ( ) ( ) 10,000 2 ( ) ( ) 10,000 3 ( ) ( ) 10,000 4 ( ) ( ) 10, Solving Procedures Ste 1: Formulate the LP model for the APP decision roblem according to Eqs. (1) to (7) for four eriods for two roducts. Ste 2: Use the essimistic, most likely and otimistic values from table (1-3) in current model one by one and determine the minimum cost. Ste 3: Write and run the rogram according to ste 2 in matlab comuter software version 2010 and get the otimal solution. Ste 4: Convert the three values into one recise value shown in table 4-6 by using weighted average method and then use it like as ste 2 in order to get the result for case 4. Ste 5: ( Proosed aroach): Generate random number using redefined range shown in modified data in table 7 9 for the corresonding APP arameters and accordingly write and run the rogram using for loo (for 100,000 iterations) in matlab comuter software and get the otimal result. Outline of the roosed aroach is mentioned as follows: Start Generate initial random value from secific range of each arameter Minimize objective function Generate Random value again from each range Subject to constraints - Carrying nventory - Machine caacity - Warehouse sace - Labor levels o Convergence Sto Yes 13

14 5. Results and Findings 5.1 Results It is mentioned that fuzzy triangular membershi function is alied in revious research work and acoordingly three objective values are obtained. ow current obtained results considering five cases are comared with revious existing results searately as follows in table Table 10: Comarison of Results with Previous Existing Result 1 (I) Tye of Value (II) Obtained Result ($) (III) Previous Result 1($) Difference between [(II)-(III)] Change % Comared to (III) Pessimistic value % Most likely value % Otimistic value (-) 6.76% Combinely value % Random value (-) 13.96% From table 10, it is found that if revious existing result 1 is comared with the current obtained result based on considering essimistic value, most likely value and weighted average value then it is noticed that 62.34%, 4.71% and 63.62% cost is increased than existing result 1 corresondingly. Beside it is observed that if otimistic data are considered then 6.76% cost is decreased than existing result 1. On the other hand it it remarkable that if random value (Proosed aroach) is considered then 13.96% cost is decreased than existing result 1. Therefore it may be said that case no. 5 is always better in terms of reducing total aggregate roduction cost comare with other four cases. Table 11: Comarison of Results with Previous Existing Result 2 Difference Change % (II) Obtained (III) Previous (I) Tye of Value between Comared to Result ($) Result 2 ($) [(II)-(III)] (III) Pessimistic value % Most likely value (-)11.21% Otimistic value (-)20.94% Combinely value % Random value (-)27.05% From table 11, it is found that if revious existing result 2 is comared with the current obtained result based on considering essimistic value, and weighted average value (combine value) then it is noticed that 37.65% and 38.73% cost is increased than existing result 2 corresondingly. Beside it is observed that if most 14

15 likely, otimistic data and random values are considered then 11.21%, 20.94% and 27.05% cost is decreased corresondingly than existing result 1. Throughout this observation it is remarkable that if random value (Proosed aroach) is considered then 27.05% cost is decreased than existing result 2 which is the maximum cost reducing rate among these 5 consdierations. Therefore it may be said that case no. 5 (considering random number generation from a secific limit) is always better in terms of reducing total aggregate roduction cost comare with other four cases. Table 12: Comarison of Results with Previous Existing Result 3 (III) (II) Difference Change % Previous (I) Tye of Value Obtained between Comared to Result 3 Result ($) [(II)-(III)] (III) ($) Pessimistic value % Most likely value (-)18.65% Otimistic value (-)27.56% Combinely value % Random value (-)33.16% From table 12, it is found that if revious existing result 3 is comared with the current obtained result based on considering essimistic value and weighted average value ( combine value) then it is noticed that 26.12% and 27.11% cost is increased than existing result 3 corresondingly. Beside it is observed that if most likely, otimistic and random values are considered then 18.65%, 27.56% and 33.16% cost is decreased corresondingly than existing result 3. Throughout this observation it is remarkable that if random value (Proosed aroach) is considered then 33.16% cost is decreased than existing result 3 which is the maximum cost reducing rate among these 5 consdierations. Therefore it may be said that case no. 5 (considering random number generation from a secific limit) is always better in terms of reducing total aggregate roduction cost comare with other four cases. Table 13: Comarison of Results between Case 1, 2, 3 & 4 with Case 5 [III] Obtained Change % [II] Obtained Differences($) [I] Case o. results ($) from comared result ($) [II-III] Case 5 to [II] Case % Case % Case % Case % From table 13, it is found that based on roosed aroach that s mean Case no. 5 roduced always lowest cost comare to other four cases. Therefore it is said that Case no. 5 is always cost effective aroach comare to Case no.1, Case no. 2, Case no. 3 and Case no. 4 resectively. So it can be said that case 5 generate 15

16 Result( Total cost in $) different result which is relatively better from any revious aroach. It gives a unique idea and advance knowledge for ractioners as well as researchers. Sensitivity Analysis It is noted that in this research work random values are considered that have been generated randomly from a redefined secific range for evaluating the roosed aroach. There is infinite no. of random values in each of the range. So there is question how much values should be evaluated. For searching this answer consequtively from 1 to random values are evaluated. It is noticed in the early eriod that is when evaluating small amount of random values its roduce relatively worse result than others higher number. To clarify this notion evaluation of random number and their corresonding results are reresented as follows in figure 1. Figure 1: Grahical reresentation of sensitivity analysis in Bar Diagram Evaluation of Random value Random Value Results ($) 5.2 Findings Throughout the analysis and observation it is seen that roosed aroach Case no. 5 (Random no. generation from a secific limit) is always roducing better result comare to other four aroaches of Case 1, Case 2, Case 3 and Case 4. It is found 16

17 that alying Case 5 total Aggregate Production Planning cost around 47.00%, %, 7.72% and 47.41% is decreased comare to Case 1, Case 2, Case 3 and Case 4 resectivly. Besides if result based on Case 5 is comared to reviously obtained results it is found that around 13.96%, 27.05% and 33.16% of total aggregate roduction lanning cost is decreased than revious results 1, 2 and 3 resectively. To sums u it is said that roosed aroach Case no. 5 is always resonsible for roducing better result comare to others aroaches. Hence, the authors want to conclude that roosed aroach not only cost effective aroach but also a novel idea to solve the aggregate roduction lanning roblem under considering uncertain environment. 6. Conclusion In this research work the roosed aroach draws the attention of the APP ractioner how to uncertain value is considered for getting APP decision. It focused advance knowledge for the researchers as well as ractioner. Presently most of the researchers as well as ractioner used certain value for getting APP decision regarding on uncertain environment. But by this roosed aroach APP ractioners can be used their uncertain value in their uncertain environment directly. Iteratively random value is used randomly for evaluating the APP model until the satisfactory result is found out. As it is done by matlab comuter software so a lot of random value can be evaluated within very short time. Hence the result by this aroach may be more accetable to ractioners and to researchers. Finally it may be concluded that this new aroach may be novel idea and oen a new dimension for the researchers as well as ractioners in future for handling uncertain environment for other field also. 7. Recommendations for Further Research Research is continuous and never ending rocess. Researchers always try to find out a unique result overcoming ast limitations. But still there remains scoe to carry out the research work on the same area for imroving the result. In this research work author has been tried to overcome some limitations of ast research work to get better APP result. The major limitation of this research work is that for justifying roosed aroach secondary data have been used. Besides, only Linear Programming (LP) is alied to evaluate the roosed aroach but there are various tyes of algorithm such as genetic algorithm, article swarm algorithm, ant colony otimization that may be used to evaluate the roosed aroach for getting better result. References Baykasoglu, A, Gocken, T, 2010, Multi-objective aggregate roduction lanning with fuzzy arameters, Advance in Engineering Software, vol. 41, Bowman, EH, 1956, Production scheduling by the transortation method of linear rogramming, Oerations Research, vol.4, Bowman, EH, 1963, Consistency and otimality in managerial decision making, Management Science, vol. 9, Charnes, A, Cooer, WW, 1961, Management models and industrial alications of linear rogramming ew York: Wiley 17

18 Chakrabortty, RK, Hasin, MAA, 2013, Solving an aggregate roduction lanning roblem by using multi-objective genetic algorithm ( MOGA) aroach, International Journal of Industrial Engineering Comutations, vol. 4, Cheng, W, Bing, LX, 2013, Integrated roduction lanning and control: A multiobjective otimization model, Journal of Industrial Engineering and Management, vol. 6, no. 4, Chen, SP, Huang, WL, 2014, Solving Fuzzy Multiroduct Aggregate Production Planning Problems Based on Extension Princile, International Journal of Mathematics and Mathematical Sciences, vol (2014), Article ID , 18 ages Damghani, KK, Shahrokh, A, 2014, Solving a ew Multi-Period Multi-Objective Multi-Product Aggregate Production Planning Problem Using Fuzzy Goal Programming, Industrial Engineering & Management Systems, vol. 13, no. 4, Effati, S, Abbasiyan, H, 2010, Solving Fuzzy Linear Programming Problems with Piecewise Linear Membershi Function, Alications and Alied Mathematics: An international Journal, vol. 05, no. 10, Filho, ATDA, Souza, FMCD, Almedia, ATD, 2006, A Multi criteria Decision Model for Aggregate Planning Based on the Manufacturing Strategy, Third International Conference on Production Research Americas Region 2006 (ICPR-AM06) Gulsun, B, Tuzkaya, G, Tuzkaya, UR, Onut, S, 2009, An Aggregate Production Planning Strategy Selection Methodology based on Linear Physical Programming, International Journal of Industrial Engineering, vol.16, no. 2, Hasem, SMJMAE, Malekly, H, Aryanezhad, MB, 2011, A multi-objective robust otimization model for multi-roduct multi-site aggregate roduction lanning in a suly chian under uncertainty, Int. J. Production Economics, vol.134, Holt, CC, Modigliani, F, Simon, HA, 1955, Linear decision rule for roduction and emloyment scheduling, Management Science, vol. 2, Huang, W L, Chen, SP, 2012, Otimal Aggregate Production Planning with Fuzzy Data, World Academy of Science, Engineering and Technology, vol. 6, Jamalnia, A, Soukhakian, MA, 2009, A Hybrid Fuzzy Goal Programming Aroach with Different Goal Priorities to Aggregate Production Planning, Comuters & Industrial Engineering, vol. 56, Jana, B, Roy, TK, 2005, Multi-objective Fuzzy Linear Programming and its Alication in Transortation Model, Tamsui Oxford Journal of Mathematical Sciences, vol. 21, no. 2, Jain, A, Palekar, US, 2005, Aggregate roduction lanning for a continuous reconfigurable manufacturing rocess, Comuters & Oerations Research, vol. 32, Jones, CH, 1967, Parametric roduction lanning, Management Science, vol.13, Leung, SCH, Wu, Y, Lai, KK, 2003, Multi-site aggregate roduction lanning with multile objectives: A goal rogramming aroach, Production Planning & Control: the Management of Oerations, vol. 14, no. 5, Maciej,, 2013, An Interactive Procedure for Aggregate Production Planning, Croatian Oeration Research Review (CRORR), vol. 4 18

19 Madadi,, Wong, KY, 2013, A deterministic aggregate roduction lanning model considering quality of rouducts, IOP Conf. Series: Material Science and Engineering 46 (2013) Masud, SM, Hwang, CL, 1980, An aggregate roduction lanning model and alication of three multile objective decision methods, International Journal of Production Research, vol.18, Mondal, SS, Pathok, S, 2011, Possibilistic Linear Programming Aroach to the Multi-item Aggregate Production Planning, International Journal of Pure and Alied Sciences and Technology, vol.7, no. 2, avid, M, orzima, Z, Tang, SH, 2013, Multi-objective aggregate roduction lanning model with fuzzy arameters and its solving methods, Life Science Journal, vol.10, no.4, ing, Y, Liu, J, Yan, L, 2013, Uncertain aggregate roduction lanning, Soft Comut, vol. 2013, no.17, Pradenas, L, Penailillo, F, Ferland, J, 2004, Aggregate roduction lanning roblem a new algorithm, Electronic otes in Discrete Mathematics, vol.18, Ramezanian, R, Rahmani, D, Barzinour, F, 2012, An aggregate roduction lanning model for two hase roduction systems: Solving with genetic algorithm and tabu search, Exert Systems with Alications, vol. 39, no. 1, Saad, G, 1982, An overview of roduction lanning model: structure classification and emirical assessment, International Journal of Production Research, vol. 20, Singhal, K Adlakha, V, 1989, Cost and shortage trade-offs in aggregate roduction lanning, Decision Sciences, vol. 20, Sultana, MS, Shohan, S, Sufian, F, 2014, Aggregate Planning Using Transortation Method: A Case Study in Cable Industry, International Journal of Managing Value and Suly Chains (IJMVSC), vol.5, no. 3, Setember 2014 Takey, FM, Mesquita, MA, 2006, Aggregate Planning for a Large Food Manufacturer with High Seasonal Demand, Brazilian Journal of Oerations & Production Management, vol. 3, no. 1, Taubert, WH, 1968, A search decision rule for the aggregate scheduling roblem, Management Science, vol. l4, Veeramani, C, Duraisamy, C, agoorgani, A, 2011, Solving Fuzzy Multi-Objective Linear Programming Problems with Linear Membershi Functions, Australian Journal of Basic and Alied Sciences, vol. 5, Issues 8, Wang, RC, Fang, HH, 2001, Theory and Methodoloy Aggregate roduction lanning with multile objectives in a fuzzy environment, Euroean Joural of Oerational Research, vol. 133, Wang, RC, Fang, HH, 2001, Aggregate roduction lanning with multile objectives in a fuzzy environment, Euroean Journal of Oerational Research, vol. 133, Wang, RC, Liang, TF, 2004, Alication of fuzzy multi-objective linear rogramming to aggregate roduction lanning, Comuters & Industrial Engineering, vol. 46, Wang, RC, Liang, TF, 2005, Alying ossibilistic linear rogramming to aggregate roduction lanning Int. J. Production Economics, vol. 98, Yenradeea, P, Piyamanothorna, K, 2011, Integrated aggregate roduction lanning and marketing romotion: model and case study, International Journal of Management Science and Engineering Management, vol.6, no. 2,