An iterative method for forecasting most probable point of stochastic demand
Abstract
EconStor is a publication server for scholarly economic literature, provided as a non-commercial public service by the ZBW.
Full text
Behnamian, J.; Ghomi, S.M.T. Fatemi; Karimi, B.; Moludi, M. Fadaei Article An iterative method for forecasting most probable point of stochastic demand Journal of Industrial Engineering International Provided in Cooperation with: Islamic Azad University (IAU), Tehran Suggested Citation: Behnamian, J.; Ghomi, S.M.T. Fatemi; Karimi, B.; Moludi, M. Fadaei (2014) : An iterative method for forecasting most probable point of stochastic demand, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 10, pp. 1-9, https://doi.org/10.1007/s40092-014-0064-8 This Version is available at: https://hdl.handle.net/10419/157403 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/2.0/
ORIGINAL RESEARCH An iterative method for forecasting most probable point of stochastic demand J. Behnamian •S. M. T. Fatemi Ghomi • B. Karimi •M. Fadaei Moludi Received: 25 January 2014 / Accepted: 5 May 2014 / Published online: 27 May 2014 ÓThe Author(s) 2014. This article is published with open access at Springerlink.com Abstract The demand forecasting is essential for all production and non-production systems. However, nowadays there are only few researches on this area. Most of researches somehow benefited from simulation in the conditions of demand uncertainty. But this paper presents an iterative method to find most probable stochastic demand point with normally distributed and independent variables of n-dimensional space and the demand space is a nonlinear function. So this point is compatible with both external conditions and historical data and it is the shortest distance from origin to the approximated demand-state surface. Another advantage of this paper is considering ndimensional and nonlinear (nth degree) demand function. Numerical results proved this procedure is convergent and running time is reasonable. Keywords Uncertainty First-order Taylor series expansion State space models Most probable point Forecasting practice Demand forecasting Introduction Forecasting can be defined as the art of predicting the occurrence of events before they actually take place (Archer 1980). Forecasting provides some information about the uncertain future for example regarding demand. Prediction of demand is one of most important problems for managers and planners. Extensive and successive changes in various aspects of the global economy in recent decades have caused prediction and its quality to be of the most critical basics of excellence of the organization. Answer to the following questions recognizes contribution of demand forecast among prediction problems: How much should be produced? How much of the resources and capacity is required? How are products produced in production planning? How much funding is required in operations? Because of the importance of such questions, most large firms have to forecast future demand for their products. Planning results would be closer to reality considering demand as a random variable. In the case where the demand is a random variable, the problem is very complex and few works have been done in this field. Most of practical results are limited to a particular cost structure (Gupta 1977). In this paper, we present an iterative method for prediction of demand with normally distributed and independent variables of n-dimensional space and the demand space is a nonlinear function, in general. In the proposed method, demand has arbitrary pricing structure. The remainder of this paper is organized as follows. In ‘‘Literature review’’, the related literature is reviewed. In ‘‘Assumptions and notations’’, assumptions and notations are introduced. ‘‘Problem description’’ and ‘‘A proposed algorithm’’ give problem description for a simple problem and results are developed for n-dimensional and nonlinear demand-state function. The performance of the proposed iterative procedure is validated with randomly generated experimental data in ‘‘Numerical example’’. Finally, ‘‘Conclusions and future works’’ is devoted to conclusions and directions for future research. A1 J. Behnamian A2 Department of Industrial Engineering, Faculty of Engineering, A3 Bu-Ali Sina University, Hamedan, Iran A4 S. M. T. Fatemi Ghomi (&)B. Karimi M. Fadaei Moludi A5 Department of Industrial Engineering, Amirkabir University of A6 Technology, Hafez Avenue No. 424, 15916-34311 Tehran, Iran A7 e-mail: [email protected]; [email protected] 123 J Ind Eng Int (2014) 10:64 DOI 10.1007/s40092-014-0064-8
Literature review Graves (1980) studies the multi-product production cycling problem which is concerned with the determination of a production–inventory policy for a single capacitated production facility dedicated to produce a family of products. He presented a heuristic method for this problem assuming stochastic demand. He assumed that demand is characterized by a known stationary distribution function. Bitran and Yanasse (1984) provided a heuristic method for multiperiod production planning with capacity constraints and service level constraints. They developed probabilistic models to deterministic approximation (non-randomized) that for examples with a high service level has relatively small error. In their model, the stochastic problem is transformed into a deterministic one by replacing the random demand with its average values. Khang and Fujiwara (1993) in their method applied previous technique of Bitran and Yanasse (1984), significantly. They formulated the problem with uncertain demand as a network flow. In their study, a deterministic approximation is obtained from the probable problem which can be solved using network flow methods. Sox and Muckstadt (1996) proposed a model and an approximate method of solving for the multi-product and multi-level production planning problem with finite horizon, capacity constraints and random demand. The model includes the linear inventory and backorder costs in objective function but the setup costs and times are not considered. The cumulative distribution of demand per period must be considered in their model. Demand of each period is not brought in their model separately. Cumulative distribution of demand in beginning of the planning horizon to each period is required. Although their proposed technique is an approximate method but in comparison with the previous methods, it could be operated in the reasonable runtime for the instances with large number of products and periods. They decomposed the problem and found near-optimal solution using the Lagrangian model. Haneveld (1988) and Peters et al. (1977) considered the production planning problem with limited production capacity and random demand for multiple products formulated as a stochastic programming. Their approach is discretization of probability distribution to formulate the problem as a linear model that can be solved with linear programming algorithm in large scale. Yokoyama (1999) proposed a method to deal with random demand. In his study, it is assumed that the demand in each period is independent and probability distribution is specified. His/her investigated problem is a single product and single level. The objective function is to minimize the expected value of total production costs, inventory costs, backorder and setup costs. In this study, a computational method has been developed using the branch and bound method. Melo and Dellaret (1996) presented production strategies for a stochastic lot-sizing problem with constant capacity. The objective is minimizing expected cost. The costs include setup costs, inventory costs and backorder costs. Brandimarte (2006) considered a stochastic version of the classical multi-item capacitated lot-sizing problem. He studies stochastic demand by scenario trees and discretization of probable values. In this model, a scenario is a sequence of tree nodes and a scenario is a deterministic parameter equivalent to stochastic parameter. The complexity of problem increases as the number of scenarios grows. So the model is solved using a heuristic method in large-scale dimensions. Ma et al. (2013) proposed three forecasting techniques which would be chosen by the retailer to minimize the sum of the bullwhip effect on product orders and inventory under different weightings in a two-level supply chain. In this study, the observations were used to develop managerial insights regarding choosing an appropriate forecasting technique after considering certain distinct characteristics of the product. Kim (2013) modeled a bilateral contract with order quantity flexibility. Under the contract, the buyer places orders in advance for the predetermined horizons and makes minimum purchase commitments. In this study, for demand forecasting in a supply chain, two techniques were considered i.e., the exponentially weighted moving average and the minimum mean square error. Zheng et al. (2010) presented a solution method using application of the Black–Scholes model incorporating stochastic processes used in financial engineering for option pricing. Indulkar and Ramalingam (2013) proposed a Monte Carlo analysis for forecasting the load of plug-in electric vehicles. This study applied the Monte Carlo method and considering the associated ranges of the various parameters and variables, the range of the load was forecasted. Christ (2011) developed an overarching linear basis function model to forecast demand. In this study, in order to ensure that all relevant demand drivers were included, the model was validated following a typical frequentist interpretation. Finally, the appropriate tests were extended for Bayesian learning. Massy (1976) proposed the stochastic evolutionary adoption model. In this study, several methods were outlined for estimating the proposed model’s parameters from panel data. Furthermore, a simulation procedure was also considered in order to project the results into a total market forecast. Jaipuria and Mahapatra (2014) proposed an integrated approach of discrete wavelet transforms (DWT) analysis and artificial neural network (ANN) denoted as DWT-ANN for demand forecasting. In this study, the proposed model was tested and validated by conducting a 64 Page 2 of 9 J Ind Eng Int (2014) 10:64 123
comparative study between Autoregressive Integrated Moving Average (ARIMA) and proposed DWT-ANN model using a data set from open literature. All other papers considered uncertain demand and applied the concept of service level. Three types of service levels are introduced in the literature review of probabilistic demand (Tempelmeier 2011). –aservice level: it is a service-level constraint to limit the total expected backlog relative to the maximum possible total expected backlog. The overall objective is to minimize the expected costs due to setups, inventory and overtime (Helber et al. 2012). –bservice level (or ‘‘fill rate’’): it is the fraction of the demand per cycle that is met immediately, i.e., without backlogging. This makes it difficult to simultaneously determine production quantities. Furthermore, the b service level does not reflect the waiting time of the customer (Helber et al. 2012). –cservice level: The attractive feature of this measure is that it reflects backlog and hence, to some extent the waiting time of the customers. It can be defined either (as above) for a specific period t, or as an average over the entire planning horizon. However, for a particular period the expected demand may be smaller than the expected backlog or even be zero. Therefore, this measure can be negative or even undefined (if the expected demand is zero) (Helber et al. 2012). Assumptions and notations The assumptions underlying the subsequent analysis include the following: – Demand-state equation consists of two parts, first part indicates the impact of various factors on demand (such as economic or social indicators, or environmental factors and etc.) and second part achieved from demand is estimated using previous data. – Demand formulation is a linear or nonlinear and ndimensional equation. – The demands in the separate discrete periods (e.g., days, weeks) are independent, normally distributed, variables but not necessarily have the same means and standard deviations. – Demand in each period is independent of other periods. The following notations, arranged alphabetically, will be used: gð:Þ:Linear or nonlinear n-dimensional function that is the difference between the anticipated demand for the various conditions and historical data Ið:Þ:A linear or nonlinear n-dimensional function that indicates the impact of various factors on demand (such as economic or social indicators, or environmental factors and etc.) hð:Þ:Anticipated demand of historical data xi:Independent and normally distributed random variable in dimension i li:Mean value of the variable xi ri:Standard deviation of the variable xi x i:Most probable demand point in the dimension i Problem description Demand is forecasted by various factors (e.g. currency, seasonal variations and etc.). On the other hand demand can also before casted using historical data. Predicted demand is matched with the environmental and historical data if the difference between these two predictions is zero. However, due to the stochastic variables, such prediction is very complicated. Therefore, a method to find the most probable point is provided in this paper. First, a linear equation with two variables will be considered for simplicity and then the results are extended to the general case. gXðÞ¼Ix 1 ðÞhx 2 ðÞ¼x1x2ð1Þ The above two random variables are assumed to be independent and normally distributed. The mean values of x1and x2are l1and l2, respectively. And r1and r2are standard deviations of these variables. In the proposed procedure, the vector Xis transformed into the independent standardized normal vector, U. Our aim was to find the point with the highest probability density or maximum likelihood. Demand space and g(X)=0 lie in the coordinate system as shown in Figs. 1and 2. Figures 3and 4show transformation of variables to Uspace system. It is proved that due to the rotational symmetry of the second-moment representation of the standardized normal distribution Z, the geometrical distance from the origin in Z-space to any point on f(Z)=0 is simply the number of standard deviations from the mean value point in X-space to the corresponding point on f(X)=0 (Choi et al. 2007). Since the point that has most probability is the point that has the minimum standard deviation (Wallace 2005), the most probable point has minimum geometrical distance from the origin in U-space to any point on g(U)=0. Suppose function g(.) in Eq. (1) with two variables x1 and x2. Transform the variables into standard normalized random variables u1and u2: J Ind Eng Int (2014) 10:64 Page 3 of 9 64 123
ui¼xilxi rxi ð2Þ Figure 5shows geometrical interpretation of the most probable point in a two-dimensional space and function g(.) expressed in Eq. 1. Figure 6shows this interpretation in a three-dimensional graph. gu 1;u2 ðÞ¼rx1:u1þlx1 rx2:u2þlx2 ¼0ð3Þ gu 1;u2 ðÞ¼rx1:u1rx2:u2lx1þlx1lx2 ¼0ð4Þ Px 1;x 2 ¼0rx1þlx1 ;0rx2þlx2 ð5Þ These calculations are complicated for n-dimensional and nonlinear problem. So we proposed an iterative method in this paper to facilitate this computation. In general, the independent variables can be assumed to be normally distributed in a multi-dimensional space. The demand surface is a nonlinear function of these variables: gXðÞ¼gx 1;x2;...;xn ðÞ T no ð6Þ The variables are transformed to a standard normalized form by Eq. (2). Demand-state surface with n-dimensional and independent, normally distributed random variables Xis gXðÞ¼gx 1;x2;...;xn ðÞ T no ¼0ð7Þ This demand-state function can be linear or nonlinear. Based on the transformation given in Eq. (2), the demandstate function given in Eq. (7) is transformed into Fig. 1 Demand space and g(X)=0 in the 3D coordinate system Fig. 2 Demand space and g(X)=0 in the 2D coordinate system 64 Page 4 of 9 J Ind Eng Int (2014) 10:64 123
gUðÞ¼grx1:u1þlx1;rx2:u2þlx2;...;rxn:unþlxn T no ¼0 ð8Þ pis the point of intersection between vector comes from origin Oand demand-state surface g(U)=0. Distance from the origin to the ppoint is the shortest distance. First-order Taylor series expansion for g(U) in the pin the U-space is: gUðÞ¼gp ðÞþ X n i¼1 ogðpÞ oui :ðuipÞð9Þ From Eq. (2) ogðUÞ oui ¼ogðXÞ oxi rxið10Þ Consider a two-variable function represented in Fig. 7. According to Fig. 7and basic concepts of mathematics: OP:rgU p ¼gU p )OP ¼~ gðUpÞ rgðUpÞð11Þ cos u1¼ ogðpÞ ou1 rgðUpÞ ð12Þ cos u2¼ ogðpÞ ou2 rgðUpÞ ð13Þ u 1¼OP cos u1ð14Þ Fig. 3 Transformation and most probable point in 3D U-space Fig. 4 Transformation and most probable point in 2D U-space J Ind Eng Int (2014) 10:64 Page 5 of 9 64 123
u 2¼OP cos u2ð15Þ Extending Eqs. (11)–(15) to general case gives the following relations: OP ¼gðUpÞ rgðUpÞ¼gp ðÞ Pn i¼1 ogðpÞ oui:p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn i¼1 ogðpÞ oui 2 rð16Þ cos ui¼ ogp ðÞ oui rgU p ¼ ogðXÞ oxirxi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn i¼1 ogðXÞ oxirxi 2 rð17Þ ui¼x ilxi rxi ¼OP cos uið18Þ x i¼lxiþOP cos uirxii¼1;2;...;nð19Þ A proposed algorithm The main steps of the proposed iterative method to forecast the demand are: 1. Define the appropriate demand-state function of Eq. (6). 2. Set the mean value point as an initial point, i.e., xi;1¼lxi;i¼1;2;...;n. Here xi;kis the ith element in the vector Xkof the kth iteration. 3. Compute the gradients of the demand-state function at this point. Fig. 5 Geometrical interpretation of the most probable point in a twodimensional space Fig. 6 Most probable point in g(U)=0 surface with two variables 64 Page 6 of 9 J Ind Eng Int (2014) 10:64 123
4. Compute the initial OP and its direction cosine. 5. Compute a new point Xkand Uk(Eqs. 19 and 18), function value, and gradients at this new point. 6. Compute the OP using Eq. (16) and the direction cosine variables from Eq. 17. 7. Repeat steps 5–7 for GE (maximum replications determined by the user) times. 8. When GE is greater than a specific number or g(X)is close to zero sufficiently, stop iterations. 9. Compute the coordinates of the point Xkor most probable point ðXÞ;that has the smallest absolute value of the demand-state function. Several nonlinear and multi-dimensional examples have been solved with this procedure. Most of the examples will converge to a special point. Numerical example The demand function is gx 1;x2;x3 ðÞ¼x3 1þx3 2x3 where x1,x2and x3are the random variables with normal distributions with means l1¼10;l2¼2;l3¼10 and standard deviations r1¼2;r2¼0:5;r3¼3, respectively (in 1,000 units). Find most probable demand point using proposed iterative method. Set the mean point as an initial point and the required convergence tolerance e¼0:00001. In the following demand-state function value and its gradients in initial point are computed. gX 1 ðÞ¼x3 1þx3 2x3¼103þ2310 ¼998 og ox1 jl1¼3l2 1¼300 og ox2 jl2¼3l2 2¼12 og ox3 jl3¼1 OP1¼gðX1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ogðl1;l2;l3Þ ox1r1 2þogðl1;l2;l3Þ ox2r2 2þogðl1;l2;l3Þ ox3r3 2 r OP1¼998 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 300 2ðÞ 2þ12 0:5ðÞ 2þð1Þ3ðÞ 2 q¼1:6632 cos/i¼ og oxijliri ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ogðl1;l2;l3Þ ox1r1 2þogðl1;l2;l3Þ ox2r2 2þogðl1;l2;l3Þ ox3r3 2 r cos/1¼0:9999 cos/2¼0:0099 cos/3¼0:0050 Applying Eqs. (18) and (19), anew point is computed. x12 ¼l1þOP1r1cos /1¼10 þ1:6632 20:9999ðÞ¼6:6737 x22 ¼l2þOP1r2cos /2¼2þ1:6632 0:50:0099ðÞ¼1:9917 x32 ¼l3þOP1r3cos /3¼10 þ1:6632 30:0050ðÞ¼10:0249 u12 ¼x12 l1 r1 ¼6:6737 10 2¼1:6631 u22 ¼x22 l2 r2 ¼1:9917 2 0:5¼0:0166 u32 ¼x32 l3 r3 ¼10:0249 10 3¼0:0083 The next iteration includes the calculation of the demand-state function and its gradient in the X2 Fig. 7 p* in a two-variable demand function J Ind Eng Int (2014) 10:64 Page 7 of 9 64 123
Again applying Eqs. (18) and (19) another new point, X3, is computed: x13 ¼l1þOP1r1cos /1¼10 þ2:7671 20:9997ðÞ¼4:4676 x23 ¼l2þOP1r2cos /2¼2þ2:7671 0:50:0223ðÞ¼1:9692 x33 ¼l3þOP1r3cos /3¼10 þ2:7671 30:0112ðÞ¼10:0932 u13 ¼x12 l1 r1 ¼4:4676 10 2¼2:7662 u23 ¼x22 l2 r2 ¼1:9692 2 0:5¼0:0616 u33 ¼x32 l3 r3 ¼10:0932 10 3¼0:0311 Table 1represents the iteration results. The iteration stops after 17 repetitions. Results of more than 75 numerical examples showed this procedure reaches to a reasonable answer in a short time. So the predicted demand is x3;17 ¼11618:4 and ~ D¼x1;173þx2;173¼1:93293þ1:63833¼11618:7 Conclusions and future works This paper proposed an iterative method to forecast the demand. Demand predicted by this procedure can be Table 1 Iteration results in the proposed method Iteration No. Notations gðXkÞOPkcos /1cos /2cos /3x1kx2kx3ku1ku2k u3k 3 86.7124 3.4888 -0.9985 -0.0484 0.025 3.0326 1.9153 10.2618 -3.4836 -0.1692 0.0872 4 24.6575 3.9268 -0.9936 -0.099 0.054 2.1964 1.8054 10.6363 -3.9017 -0.3891 0.2121 5 5.8454 4.1114 -0.9809 -0.1656 0.1016 1.9339 1.6593 11.2539 -4.033 -0.6812 0.4179 6 0.5484 4.1331 -0.975 -0.1794 0.1303 1.9396 1.6291 11.6163 -4.0301 -0.7417 0.5387 7 0.0047 4.1332 -0.9764 -0.1722 0.1297 1.928 1.6441 11.6091 -4.0359 -0.7117 0.5363 8 0.00188 4.1332 -0.9753 -0.1773 0.1311 1.9371 1.6335 11.6267 -4.0314 -0.7328 0.5422 9 0.001 4.1332 -0.9761 -0.1735 0.1301 1.9302 1.6413 11.6129 -4.0348 -0.7173 0.5376 10 0.00057 4.1332 -0.9756 -0.1763 0.1309 1.9353 1.6356 11.6234 -4.0323 -0.7289 0.5411 11 0.00031 4.1332 -0.976 -0.1743 0.1303 1.9315 1.6398 11.6157 -4.0342 -0.7203 0.5386 12 0.00017 4.1332 -0.9757 -0.1758 0.1308 1.9343 1.6367 11.6214 -4.0328 -0.7267 0.5405 13 0.00009 4.1332 -0.976 -0.1747 0.1304 1.9323 1.639 11.6172 -4.0339 -0.722 0.5391 14 0.00005 4.1332 -0.9758 -0.1755 0.1307 1.9338 1.6373 11.6203 -4.0331 -0.7255 0.5401 15 0.00003 4.1332 -0.9759 -0.1749 0.1305 1.9326 1.6386 11.618 -4.0337 -0.7229 0.5393 16 0.00002 4.1332 -0.9758 -0.1754 0.1306 1.9335 1.6376 11.6197 -4.0333 -0.7248 0.5399 17 0.00001 4.1332 -0.9759 -0.175 0.1305 1.9329 1.6383 11.6184 -4.0336 -0.7234 0.5395 gX 1 ðÞ¼x3 1þx3 2x3¼6:67373þ1:9917310:0249 ¼295:1173 og ox1 jl1¼3l2 1¼133:6148 og ox2 jl2¼3l2 2¼11:9006 og ox3 jl3¼1 OP2¼gX 2 ðÞ P3 i¼1 ogðX2Þ oxiriui2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ogðl1;l2;l3Þ ox1r1 2þogðl1;l2;l3Þ ox2r2 2þogðl1;l2;l3Þ ox3r3 2 r OP2¼295:1173 133:6148 21:6631ðÞðÞ11:9006 0:50:0166ðÞðÞð1ðÞ30:0083Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 133:6148 2ðÞ 2þ11:9006 0:5ðÞ 2þð1Þ3ðÞ 2 q OP2¼2:7671 cos /i¼ og oxijliri ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ogðl1;l2;l3Þ ox1r1 2þogðl1;l2;l3Þ ox2r2 2þogðl1;l2;l3Þ ox3r3 2 r cos /1¼0:9997 cos /2¼0:0223 cos /3¼0:0112 64 Page 8 of 9 J Ind Eng Int (2014) 10:64 123