2009) 4 (22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد من شهر الى شهر حتىالتنبؤ N من السنین لتخطیط مصنع انتاجي سمیر عبد الوهاب فهد جامعة بغداد، ابن الهیثم -كلیة التربیة ،قسم الحاسبات الخالصة من N شهري لتخطیط االنتاج، الخزین، القوى العاملة، المبیعات واالسعار حتى نبؤ هذا البحث یوفر طریقة ت . ات الطالب المستقبلير السابق مع األخذ باألعتبار تنبؤكل القرارات الشهریة سوف تعتمد على القرارات للشه. السنین تقنیة برمجة الحاسوب لتعظیم مدیر المصنع یستطیع تشغیل البرنامج في أي شهر من السنة، هذه الطریقة انجزت ب االرباح IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Month – to – Month Until N Years Prediction for Planning a Productive Firm S.A.Fahad Departme nt of Computer , Ibn-Al-Haitham College of Education , Unive rsity of Baghdad Abstract This p aper offers a monthly p rediction method for p lanning p roduction, inventory , work- force, sales and p rices until N y ears. Each monthly decision will depend on last month, decisions and take in consideration the future forecasted demand. The manager can run the p rogram in any month within a y ear. This method is executed by comp uter p rogramming technique to maximize p rofits. 1. Introduction In the field of p roduction, inventory and man p ower control, H.M .M .S. in their text [1] developed a dy namic model to p lan aggregate p roduction rate of a fir m and setting the size of its work-force which frequently both comp lex and difficult. The quality of these decisions can be of great importance to t he profitability of an individu al comp any, and when viewed on a national scale these decisions have a significant influence on the efficiency of the economy as a whole. They formalized a quadratic fun ction cost as summation of the following costs: a. a. Regular p ayroll cost = c1Wt + c13 (1.1) b. Hiring and Layoff costs = c2(Wt – Wt – 1 – c11) 2 (1.2) c. Over time and Idle time costs = c3 (Pt – c4 Wt) 2 + c5Pt – c6Wt + c12 Pt Wt (1.3) d. Inventory related costs = c7 [It – (c8 + c9 St)] 2 (1.4) the function subject t o the following restriction It  It – 1 + Pt – St (1.5) where Pt = p roduction rate required in p eriod t. It = level of inventory at t he end of p eriod t. Wt = level of work-force required during p eriod t. St = shipment in month t. c1 - c13 numer ical const ants which must be evaluated from hist orical costs. By using p artial drivitive with this function they have got a linear decision rule for Pt and It. 2.Prediction Model Recently , the model of H.M.M .S. was developed by introducing p rice variable to influence on the ordering p att ern (see [2]), hop efully to move heavy demand away from peak p eriods and smoothing Pt, It and Wt and reducing costs. He used the following inverse p rice- demand. Ot = a – bt p t (2.1) IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 where Ot = the forecast ed order. a = maximum productive capacity . bt = the measure of change in demand per unit change in p rice. a = op timal value of labour p roductivity x initial level of work-force x p ossible maximum shift ratio x v i.e. a = c4 W0 x N x v where number of shifts possible Per day N number of shifts worked Per day  v = a factor to comp ensete for unknown comp onents in the p roductive cap acity and for any large forecast ed demands in the interval t = 1 to t = 12. So equation (2.1) becomes: Ot = c4 W0 x N x v – bt p t (2.2) By subst ituting equation (2.2) in equation (1.4) abov e yield to Inventory connected costs = c7 [ It – c8 – c9 (c4W0 x N x v – bt p t)] 2 (2.3) As a result of using p rice variable (p t) the manufacturer bears t he following cost Op p ortunity cost = Q·Pc – T t t 1 p   (c4 W0 x N x v – bt p t) (2.4) where Pc = the (const ant) selling p rice. Q = the tot al quantity that would have been sold during the period t = 1 to t = T. The total cost function is a summation of the equations (1.1), (1.2), (1.3), (2.3) and (2.4) T T t 1 c    {(c1– c6)Wt + c13 + c2(Wt – Wt – 1 – c11)2 + c3 (Pt – c4 Wt)2 + c5 Pt + c12 Pt Wt +c7 [It –c8 – c9 (c4 W0 x N x v – bt p t)] 2 – p t (c4 W0 x N x v – bt p t)} + Q·Pc (2.5) subject to the following restriction It = It – 1 + Pt – c4W0 x N x v + bt p t (2.6) By differentiating CT with resp ect to Wr, Ir and p r result a linear decision rules as follows: Pt = g1 – g2 Wt – 1 + g3 Wt – g2 Wt + 1 (2.7) It = C26(t) + C27(t) Wt – 1 – C28(t) Wt + C29(t) Wt + 1 – C30(t) Wt + 2 (2.8) p t = C36(t) – C37(t) Wt – 1 + C38(t) Wt – C39(t) Wt + 1 + C40(t) Wt + 2 (2.9) By subst ituting the decision variables Pt, It and pt above in equation (2.6) obtain for t > 1 C27(t) Wt – 2 – C41(t) Wt – 1 + C42(t) Wt – C43(t) Wt + 1 + C44(t) Wt + 2 = c4W0 x N x v – C45(t) (2.10) and for t = 1 C47(1) W1 – C48(1) W2 + C49(1) W3 = c4W0 x N x v – I0 + C46(1) W0 – C50(1) (2.11) From equations 2.10 and 2.11, we have got 12-p eriod of simultaneous linear equations to be solved for op timizing values of Wt and by addin g two more unknowns in the end of 12- p eriods W10 = W11 = W12 and ap p lying the Gauss-Jordan method y ields to obtain Wt, t = 1 to 14. The researcher design ed his p rogram (p red.) to comp ute decision variables for one y ear from month 1 t o 12 as well as cost and p rofit. This p rogram is v ery useful to m anager or the planner for a short-time when use it in the end of a year for p reparation of budgets for next y ear as well as offer an ind ications about the size of a decision v ariables rules. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 3.Month-to-Month until N Years Prediction: 3.1 Long-Te rm Prediction As there is an short -term p rediction, there is a long term p rediction. Harvey, M .Wagner (3,p .383) goes farther than that and say s: Unquestionably most, if not a ll, d ecision-making is part of an unending history o f actions. Earlier choices have affected the present, current decisions will influence the future, and so on. In this light, all models must be viewed as imbedded in an unbounded horizon. Accordin g to that the cost function (2.5) above becomes N 12 t T 1 t 1 C    (3. 1.1) where N = number of years Also, for the time-series d ecision quantitative variables Pt, It d efined in equations (2.7) and [2.8] resp ectively which were applicable for any t, beco me N 12 t T 1 t 1 P    (3. 1.2) N 12 t T 1 t 1 I    (3. 1.3) 3.2 Month-to-Month until N Years Prediction: The best p rediction is when the present p rediction is very close to immediately p recedin g p eriod and predecessor p eriods. The shorter the interval between successive reviews and the greater the detail, the more likely are forecasts made by judgment and intuition to be unduly influenced by recent events, [4]. a.The p receding p eriod became real decisions which include inventory (It – 1) and work- force size (Wt – 1) and these variables would sharluded in p resent time (t) to p redict the decision var iables according to the equations (2.6), (2.7), (2.8), (2.9), (2.10), (2.11). b.For the p redecessor p eriods would share in p resent p eriod (t) to p redict the decision variables when the sy st em of equations (2.10) and (2.11) requir es valu es of for ecasted demand for p eriods (months) t to t + 11, (v alues of bt in equation (2.2)). So, to obtain values for the decision variables for one month we would need 12 monthly values of forecasted demand for: 12 months p redicted values we would need 23 values of for ecasted monthly demand: for 24 months p redicted values we would need 35 valu es of forecasted demand, and so on. This method will let the decision variables keep up with forecasted demand throughout p lanning horizon. 3.3Runni ng the Program in any Month: The planner knows the prediction is p rediction and not alway s comp atible with changes in the market such as actual sales greater than or less than p redicted sales y ield to actual inventory less than or greater than p redicted inventory and influence the work-force size. The It and Wt beco mes initial inventory and initial work-force resp ectively for p eriod t + 1. So rerun the p rogram fro m period t + 1 and p rovide it with new values for I0 and W0. In this case the variable II represents the difference b etween t = 1 and the new p eriod (month) for example the new p eriod = 7 then II = 7 – 1 = 6. This variable will be an inp ut variable, in the normal case wi ll be equal to zero, see table (3-3- 1). The p rogram was written which was referred to as the (Pred.1) which was designed to execute the three cases (3-1, 3-2 and 3-3) above. Details of t his p rogram are given in section 5 below. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 If we give the two p rograms Pred. and Pred.1 the same input data the results will be comp atible in the first p eriod (month) only . 4.Results and Comparisons with H.M.M .S. Model: To execute the p rogram Pred.1, we need data of H.M .M.S. p aint factory which are available in [1] and [2]. But the factor v in equation (2.2) is not availab le, [2] sp ecified the relationship between this factor and the decision variables as well as with cost s and p rofit. He p roved that the increase in value of v y ields to the increase in the revenue and p rofits. So it is easy to let the methods used in this research bett er than the results of H.M .M.S. but it is not fair to do so. One of the main purp oses of H.M.M .S., p red. and this model is t o smooth out the time-series representing fluctuations in work-force, p roduction, inventory levels. Work-force smoothing y ields to smooth out the other decision variables according to the formulation of equations (2.7), (2.8), (2.9). Therefore the value of v wi ll b e chosen after many running of the p rogram until we get the best smooth for work-force and y ear after y ear until N y ears. Thus t he p referred set of v for five y ears are (0.9,0.9,1,1,1). The outp ut of p rogram is as follows: a. Three tables for each year, first table for decision v ariables (Pt,It,Wt,p t) for each month and y early total of Pt and I t. Second table is for monthly basic costs and then total for each month and total for each of them in a year. This table is not imp ortant to be list ed in this research while the second table in b below is a good breviary for the cost s. T he third table contains the sales, revenue, other cost and profit for each month and their total for a year. These tables will be repeated each y ear until N y ears, see tables from (4-1) to (4-10) below. b. Final results in the end of N y ears will b e three tables, first table is listing the y early total of inventory , p roduction and sales and their summation in the N y ears. Second table is list ing the y early total of each kind of cost and their summation in the end of N y ears. Third table is listing the yearly total of revenu e, other cost and profit and their sum mation in the end of N y ears, see tables (4-11) to (4-13). The other cost = p roduction rate  OC where OC = the other cost p er unit of p roduction. c. Comparison with H.M.M .S. M odel. Table (4-14) shows the maximu m and min imum for work-force, p roduction rate, inventory rate and sales and also the variation for both models. It is clearly that variation in our model is considerably less than H.M .M .S. model. And this smooting is effective in increasin g the profit and reducing costs. In the real life, the decision maker will choose valu e for v factor to sp cify his p roductive copacity according to his exp erience and knowledge in the market, (5,p .400) say , Predictions require skill, experi ence, and judgment, not all time series can be successfully predicted. 5.Main Steps of the Program The p rogram (Pred.1) is written in general to accep t any number of y ears by changing the input variable I R and provde the p rogram monthly historical demand M SL = (I R + 1)  12. Execution time is 13 seconds for 5 y ears p lanning and consist of 435 p rogramming instructions and statements. 1. variables declaration. 2. Read I R, Alp ha, crival, forca. 3. M SL = (I R + 1)  12. 4. Declaration of dimensions. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 5. Read I0, W0 and II. 6. From I = 1 to M SL read SL(I). 7. comp ute G1, G2 to G6 and common terms. 8. K= 1, M = 12 and IYLO = 0. 9. p rint c1 to c13, I0 and W0. 10. Test for forecasting method to be used = 1 12month moving average forecasting subrout ine forca = 2 e xp onential weighted average subrout ine = 3 forecosted sales equal to actual demands see [6], [7], [8] and [9]. 11. Read PC and SHN(N in equation (2.2)). 12. Year ly loop IY = 1 to I R. 13. Read v. 14. M onthly comp utations, bt must be > 0 from equation (2.2). 15. N = 14 and Jmax = N + 1 to book area in the memory for the matrix to build up simultaneous lin ear equations according to equations (2.10) and (2.11) and app ly ing the Gauss-Jordan method to get Wt, t = 1 to 14 and we select Wt, Wt + 1 and Wt + 2 which are required in equations (2.7), (2.8) and (2.9). This step will be executed 60 times for 5 y ears, see [10], [11], and [12]. 16. From I = K + II to M – 11 + II comp ute c26 – c40 and for I > 1 + II comp ute Pt, It and pt else for I = 1 + II compute Pt, It and pt. 17. From I = K +II t o m – 11 + II comp ute costs for t = 1 and t > 1. 18. From I = K + II to M – 11 + II comp ute sales, other cost, revenue and p rofit. Also comp ute check from equ ation (2.6) which must equal to z ero otherwise there is an error in mathematical op erations of this model or in p rogramming of the model. 19. If the reminder of 12  = z ero step 20 Ot herwise K = K + 1 M = M + 1 Go back to step 14 20. p rint out monthly results within each year KS = K – 11 + II M S = M – 11 From I = KS to M S p rint out the three tables exp lained in 4.a above. 21. K = K + 1 M = M + 1 II = 0 Go back to step 12 to comp ute another year. 22. From IY = 1 to IR p rint table of yearly total for Pt, It and St. And t he same for cost s and another for revenue, other cost and profit. Re ferences 1. Holt, C.; M odigiliani, F.; M uth, J. and Simon, H.A. (1960), Planning, Production, Inventories and Work-Force Prentice-Hall, Englewood Cliffs, N.J. 2. Fahad, S.A., (2008). M athematical M odel for One Year Planning of a M anufactory , Ibn- Al-Haitham Journal for Pure and Ap p lied Sciences, Baghdad University , 21(4). IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 3. Harvey, M .Wagner, (1975), Princip les of Op erations Research with Ap p lications to M anagerial Decisions, Prentice-Hall, International, Inc., London, 2 nd Edition. 4. Coutie, G.A.; Davies, O.L.; and Hossell, C.H. (1964), M athematical and Statist ical Techniques for Indust ry , M onograph, No.2, Short Term Forecast ing, Oliver &Boyd, Edinburgh. 5. Lynwood, A.;Johnson, Douglas, C.M ontgomery , (1973), Op erations Research in Production Planning, Scheduling and Inventory Control, Johnwiley & Sons, INC; New York, London. 6. Lewis, C.D. (1970), Scientific Inventory Control, Butt erworth Press. 7. Pindy ck, R.S. and Rubinfeld, D.L., (1976). Econometric M odels and Economic Forecast s, M c GRAW-Hill Book Company, 4 th . Edi.. 8. Brown, R.G. (1959), Statist ical Forecast ing for Inventory Control, M c Graw-Hill, New York. 9. Trigg, D.W. (1964), M onitoring a Forecating Sy st em, Op . Res. Qt ly, 15. 10. Philip s and Taylor, (1973), Theory and Ap p lications of Numerical Analysis, Academic Press, London and New York. 11. Fox, L. (1964), An Introduction to Numerical Linear Algebra, New York:Oxford University Press. 12. Forsuthe, G. and M oler, C.B. (1967), Computer Solution of Linear Algebraic Sy st ems, Englewood, N.J., Prentice-Hill. Table:( 3-3-1)Decision Variables when II= 6 and v = 0.9 and the same thing for cost table and profit table while the following years woul d be as calendar years C1 = 340.0, C2 = 64.3, C3 = 0.20, C4 = 5.67, C5 = 51.2, C6 = 281.0, C7 = 0.0825, C8 = 320.0, C9 = C11 = C12 = 0 W0 = 81 men, I0 = 275 unit s, P c = 100. Month Production Inventory Work-Force Prices 7 453 296 83 94.85 8 453 307 85 91.55 9 456 311 86 92.17 10 459 312 87 91.87 11 461 313 88 90.84 12 463 315 88 88.91 2751.6 1847.7 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Year 1 Table :(4-1)whe n v = 0.9 Table:( 4-2) when v = 0.9 Month Production Inventory Work-Force Prices 1 452. 301 81 96.5 2 443. 314 81 95.6 3 440 319 81 96.1 4 438 319 81 96.6 5 437 320 80 94.9 6 436 319 80 94.9 7 435 319 80 94.4 8 434 321 80 92.7 9 435 320 80 94.2 10 435 318 80 94.4 11 436 318 81 93.6 12 437 320 81 91.9 Tot. 5256 3807 Month S ales Revenue Oth.cost Profit Check 1 414 39973.6 2873.435 18776.96 0 2 429 41075.4 2817.999 20654.84 0 3 436 41867.89 2796.725 21261.13 0 4 438 42333.13 2786.251 21546.68 0 5 435 41322.93 2776.146 21357.3 0 6 436 41379.92 2770.126 21434.87 0 7 435 41099.93 2764.673 21424.47 0 8 432 40062.01 2759.876 21277.02 0 9 436 41023.98 2763.275 21423.31 0 10 437 41206.88 2767.50 21491.12 0 11 437 40884.56 2771.607 21449.65 0 12 435 39935.58 2777.427 21314.12 0 Tot. 5200 492165.8 33425.04 253441.5 0 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Year 2 Table :(4-3) when v = 0.9 Table :(4-4) when v = 0.9 Month Production Inventory Work-Force Prices 13 439 321 81 92.9 14 443 321 82 94.9 15 446 320 83 97.1 16 451 321 83 98.6 17 456 324 84 99.4 18 462 334 85 101.3 19 474 330 86 122.4 20 483 322 87 135.9 21 488 319 88 138.2 22 491 313 88 143.7 23 491 312 89 135.2 24 490 318 89 124.9 Tot. 5615 3856 Month S ales Revenue Oth.cost Profit Check 13 438 40690.25 2792.539 21359.43 0 14 443 42032.51 2814.231 21516.23 0 15 447 43379.31 2839.153 21768.89 0 16 450 44415.98 2866.012 21987.49 0 17 452 44896.22 2896.898 21953.22 0 18 452 45826.44 2940.874 21961.52 0 19 479 58572.75 3013.166 21504.37 0 20 490 66637.63 3071.028 40126.72 0 21 491 67926.71 3105.166 41447.71 0 22 498 71515.66 3125.387 45976.2 0 23 492 66518.3 3123.501 39683.23 0 24 484 60402.02 3114.966 32769.11 0 Tot. 5616 652813.8 35702.92 362054.1 0 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Year 3 Table : (4-5)when v = 1 Table : (4-6) when v = 1 Month Production Inventory Work-Force Prices 25 509 320 91 118.4 26 516 319 92 123.7 27 520 319 93 125.7 28 524 317 93 128.6 29 525 315 94 127.8 30 524 317 94 122.6 31 524 313 94 125.6 32 521 311 94 118.6 33 516 312 94 110.3 34 512 314 94 104.9 35 509 318 94 100.8 36 508 323 94 98.6 Tot. 6208 3798 Month S ales Revenue Oth.cost Profit Check 25 507 60034.6 3235.868 30126.08 0 26 517 63931.66 3278.793 33668.24 0 27 521 65442.37 3309.639 35077.97 0 28 526 67627.47 3331.249 37360.88 0 29 527 67364.36 3339.015 37100.73 0 30 522 63999.07 3334.758 33667.88 0 31 529 66436.77 3331.434 36348.96 0 32 523 61970.68 3310.569 32129.32 0 33 515 56748.36 3281.667 28072.88 0 34 510 53520.22 3256.184 26179.1 0 35 506 50955.2 3237.153 24955.89 0 36 503 49538.37 3229.369 24388.34 0 Tot. 6203 727569.1 39475.7 379076.3 0 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Year 4 Table: (4-7) when v = 1 Table:(4-8) when v = 1 Month Production Inventory Work-Force Prices 37 509 322 94 102.8 38 511 323 94 104. 39 513 326 94 105.1 40 516 326 94 110.1 41 519 324 94 115.8 42 521 317 94 121.5 43 520 318 94 114.4 44 519 314 94 115.6 45 516 318 94 107.8 46 515 319 94 107.2 47 514 319 94 107.6 48 513 324 94 103.9 Tot. 6186 3848 Month S ales Revenue Oth.cost Profit Check 37 510 52418.7 3238.215 25750.64 0 38 510 53085.52 3247.866 26025.76 0 39 510 53589.44 3260.601 26182.69 0 40 516 56902.75 3281.733 28256.71 0 41 522 60372.86 3302.816 30857.41 0 42 529 64207.63 3316.141 34361.96 0 43 519 59354.84 3307.677 29900.3 0 44 523 60429.01 3300.306 30984.11 0 45 512 55260.47 3281.509 27081.95 0 46 513 54977.12 3271.919 27012.84 0 47 514 55325.92 3266.683 27316.17 0 48 508 52729.62 3260.28 25649.91 0 Tot. 6185 678653.9 39335.75 339380.5 0 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Year 5 Table :( 4-9)when v = 1 Table :(4-10) when v = 1 Month Production Inventory Work-Force Prices 49 514 323 94 108.6 50 515 319 94 112.1 51 514 320 94 109.2 52 514 312 94 109.1 53 513 318 94 108.9 54 511 320 93 106.3 55 511 320 93 107. 56 510 318 93 107.5 57 508 319 93 104.5 58 507 323 93 102.6 59 508 320 93 107.8 60 507 320 93 105.5 Tot. 6132 3841 Month S ales Revenue Oth.cost Profit Check 49 515 55889.1 3268.135 27671.69 0 50 519 58168.63 3274.49 29322.01 0 51 514 56083.59 3269.85 27721.54 0 52 514 56068.57 3265.72 27750.21 0 53 514 55920.2 3260.1 27698.5 0 54 509 54136.96 3251.245 26542.92 0 55 511 54668.93 3246.956 26933.41 0 56 512 55012.54 3241.982 27213.82 0 57 507 53011.87 3231.969 25986.69 0 58 503 51601.76 3225.632 25201.28 0 59 512 55159.48 3230.654 27379.03 0 60 507 53462.98 3225.837 2625.11 0 Tot. 6135 659184.6 38992.58 325672.2 0 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Table :(4-11)Yearly Total of Inventory, Production and S ales Table: (4-12)Yearly Total of Each Basi c Cost (Regular Payroll, Hiring and Layoff, Overtime . Inventory Related and Opportuni ty Cost) Table: ( 4-13)Yearly Total of Revenue, O the r Cost and Profit Table :(4-14)Comparison with H.M.M.S . for 5 Years Year Y.In v. Y.Pro d. Y. S al. 1 3807 5256 5200 2 3856 5615 5616 3 3798 6208 6203 4 3848 6186 6185 5 3841 6132 6135 Tot. 19150 29397 29339 Year Y.RPAC. Y.HLC. Y.OTC. Y.INCC Y.OPC. Y.TOTC. 1 231873.3 24.76 376.40 34.33 - 27009.5 205299.3 2 246227.5 386.58 1429.44 36.23 6976.99 255056.7 3 268562.9 319.91 4259.14 24.21 35850.24 309017.1 4 270916.2 8.48 742.17 13.80 28257 299937.7 5 268669.9 10.11 229.04 2.37 25608.43 294519.8 Tot. 1286250 749.86 7036.2 110.94 69683.87 1363831 Year Y.Re v. Y.OTHC. Y. Prof. 1 492165.8 33425.04 253441.5 2 652813.8 35702.92 362054.1 3 727569.1 39475.7 379076.3 4 678653.9 39335.75 339380.5 5 659184.6 38992.58 325672.2 Tot. 3210387 186932 1659624.6 Work-for ce Produ ction Inventory Sales Total Cost Profit H.M. M.S. Pred .1 H.M. M.S. Pred .1 H.M. M.S. Pred .1 H.M. M.S. Pred .1 H.M.M .S. Pred.1 H.M.M.S. Pred.1 Max . 111 94 736 525 492 334 841 529 2,290,8 50 1,550,763 1,039, 573 1,659, 624 Min. 66 80 359 434 117 301 289 414 ar. 45 14 377 91 375 33 552 115