IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Multiproduct Mathematical Model for Productive Company S.A. Fahad Departme nt of Computer Scie nce, College of Education I bn- Al- Haitham, Unive rsity of Baghdad Received in : 30, March , 2011 Accepte d in : 30, May , 2011 Abstract This model is an extension to H.M .M .S and related developments models of a single p roduct. T hese models will be converted to deal with M ultip roduct for p roductive comp any. This model executed by comp uter programming technique to maximize p rofits. Key words : M ulti p roduct, Comp any, Factory , M aximize. Introduction There are many comp anies p roduce more than one p roduct each manufactured by p lant or firm each one is indep endent from the others. The management of t he comp any needs such a sy st em to make a control on p roduction, inventory , work–force, sales and p rices for all p roducts and provide an optimal solution to maximize p rofit. Holt, M odigliani, M uth and Simon (H.M .M .S) developed a dy namic model to p lan aggregate control of p roduction, inventory and work – force for a single – item (p roduct) which fully reported in their text [1] . It was develop ed under the assump tion that t he receip t of orders would be erratic and fluctuating and it would therefore need to eradicate any excessive movements in the rates of p roduction , inventory and work – force , in order to cut down t he costs of running a manufactory in mathematical terms and , to that end, H.M .M .S subdivided the tot al cost as follows : a- Regu lar p ayroll costs = 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 + C5 Pt - C6 Wt + C12 Pt Wt …(1.3) d – Inventory related costs = C7 [ It – ( C8 + C9 St ) ] 2 …(1.4) The tot al cost function to be minimized is t he summation of the above cost s . Multiproduct Model This model will depend on the original model of H.M .M .S [1] and [2] and also last developments on t heir model were done by the researchers [3] and [4] , all above models were dealing with a single p roduct and will be converted to deal with multip roduct , as follows : M odify the equation (1,1) to eq. (1.4) to be : Regular p ayroll costs = C1i Wti + C13i …(2.1) Hiring and lay off = C2i (Wti -W(t-1) i - C11i) 2 …(2.2) Overtime and idle time cost s = C3i ( Pti - C4i Wti ) 2 + C5i Pti - C6iWti + C12i Pti Wti …(2.3) Inventory related costs = C7i (Iti - ( C8i + C9i Sti)) 2 …(2.4) where Wti = level of work – force for product i in period t Pti= p roduction rate for p roduct i in period t IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Iti = level of inventory for p roduct i at t he end of p eriod t. Sti= shipment of p roduct i in period t = Oti the order level for that month. C1i – C13i numerical constants which must be evaluated from hist orical costs for each p roduct (see [2]) . Price variable have introduced to H.M .M .S model to influence on the ordering p att ern (see [3]) to move heavy demand away from peak p eriods and smoothing Pt, It and Wt and reducing costs. For each product is writt en as follows: Oti = ai - bti p ti …(2.5) where Oti = forecast ed order for p roduct i ai = M aximum productive capacity for p roduct i bti - the measure of change in demand p er unit change in p rice for each p roduct p ti = variable p rice for product i ai = op timal value of labour p roductivity X initial level of work – force X p ossible maximum shift ratio Xv i.e. ai= C4i Woi XNi X Vi …(2.6) where i number of shifts possible per day N number of shifts worked per day  Vi = a factor to comp ensate for unknown components in the p roductive capacity and for any large forecasted demands in the interval t = 1 to t = 12 By substitut ing equation (2.6) in to equation (2.5) we obtain: Oti = C4i Woi X Ni X Vi - bti p ti …(2.7) By substitut ing equation (2.7) into equation (2.4) we obtain: Inventory connected costs = C7i [ Iti - C8i - C9i ( C4i Woi X Ni X Vi - bti p ti)] 2 …(2.8) The variable p rice policy will bear the manufacturer the following cost: Op p ortunity cost = Qi . Pci – T ti t 1 p   (C4i Woi X Ni X Vi - bti p ti ) …(2.9) where Pci = the (const ant) salling p rice for product i Qi = the tot al quantity . That would have been sold during the period t = 1 to t = T The tot al cost function for all products is a summation of the equations (2.1), (2.2), (2.3), (2.8) and (2.9) n T T i 1 t 1 C     [( C1i - C6i ) Wti + C13i + C2i ( Wti - W(t-1) i - C11i ) 2 + C3i( Pti - C4i Wti ) 2 + C5i Pti + C12iPti Wti+C7i [Iti-C8i -C9i(C4i Woi X Ni X Vi – bti p ti)] 2 – p ti (C4i Woi X Ni X Vi– bti p ti )+ Qi . p ci ] …(2.10) Subject t o the following restriction Iti = I(t-1) i + Pti – C4i Woi X Ni X Vi + bti p ti …(2.11) By differentiating CT with resp ect to Wti , Iti and pti result a linear decision rules as follows : Pti = g1i – g2i W(t-1) i + g3i Wti – g2i W(t+1)i …(2.12) Iti = C26(t)i + C27(t)i W(t-1) i – C28(t)i Wti + C29(t)i W(t+1)i – C30(t)i W(t+2)i …(2.13) p ti = C36(t)i – C37(t)i W(t-1) i + C38(t)i Wti – C39(t)i W(t+1)i + C40(t)i W(t+2)i …(2.14) where IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 1i 6i 2i 1i 2i 3i 4i 3i 4i C C C g , g 2C C C C    andg3i = 2g21 + C4i C26(t)i = C8i + C9i ( C4i ( C4i Woi X Ni X Vi–C5i bti) – bti C10i)/ 2C4i C27(t)i = C2i (1+ C7i C9i bti (C9i + 1)) / (C4i C7i ) C28(t)i = C2i (3+ C7i C9i bti (3C9i + 2)) / (C4i C7i ) C29(t)i = C2i (3+ C7i C9i bti (3C9i + 1)) / (C4i C7i ) C30(t)i = C2i (1+ C7i C9i bti ) / (C4i C7i ) C36(t)i = (C4i Woi X Ni X Vi + bti (C10i + C4i C5i ))/(2C4i bti) C37(t)i = C2i (C9i + 1 ) / C4i C38(t)i = C2i ( 3C9i + 2 ) / C4i C39(t)i = C2i ( 3C9i + 1 ) / C4i C40(t)i = C2i C9i / C4i By substitut ing the decision variables Pti , Iti and p ti above in equation ( 2.11 ) obtain for t > 1 C27(t)i W(t-2) i -C41(t)iW(t-1) i +C42(t)i Wti - C43(t)iW(t+1)i + C44(t)iW(t+2)i = C4i Woi X Ni X Vi – C45(t)i …(2.15) And for t = 1 C47(1) i W1i - C48(1) i W2i + C49(1) i W3i = C4i Woi X Ni X Vi – Ioi + C46(1) i Woi - C50(1) i …(2.16) From equations (2.15) and (2.16) we have got 12-p eriods of simultaneous linear equations and imposing to end condition W10 = w11 = w12. By app lying the gauss – Jordan method t o the sy st em above, we have got the optimal values of wti, ti = 1 to 14. Characteristics of the Model a- Op timize the p roduction rats, inventory rats, work–force, sales quantity and p rices to maximize p rofit for each month within a year until N y ears. b- The computer p rogram was written which was referred to as the (p red 3) which designed to execute the model above. This p rogram will accep t any number of p roducts or factories, see [5]. c- Can run the p rogram in any month within each y ear by giving the input variable II value represents the difference between t = 1 and the new p eriod (month ) d- The p arameter Vi in equation (2.7) have given many values and selected value caused a smallest variation in the work – force p er each y ear and smoothing the decision variables (see equation (2.12), (2.13) and (2.14)) according to the p rinciples of initiating H.M .M .S model. e- There are two subrout ines in (p red 3) to forecast future demands, the first is moving average demand and the second is exp onential weighted average see [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] . f- To obtain values for the decision variables for one month we would need 12 monthly values of forecasted demand and execute the sy st em of equations (2.15) and (2.16) for each t, and select Wti , W( t+1)i , w(t+2)i, see equations ((2.12) (2.13)(2.14)). g- This model can be used for a single p roduct where p roduced by different factories belong to one comp any , each factory indep endent from the other in term of costs and demand and this case is ap p licable in the international comp anies. h- The most difficult decision face the manager is firing man-p ower. The idle p eople are international p roblem. Such sy st em like this model will reduce number of idle p eople. For examp le, in the table (4.8) for p roduct (factory ) 2 in the month 11 must fire one work–force but the manager can hire him in factory (p roduct) 1 in the same month. See table (4-1). So changes work–p lace for the work–force help s t o reduce the lay off workers. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Results Obtained from Program ( pred 3) This p rogram has given many values for number of y ears and number of p roducts and the execution was succeeded. But to reduce the number and areas the tables as outp ut of this p rogram for this research paper I chose (2) y ears and (2) p roducts as input values to run the p rogram. The value of Vi determine the p roductive cap acity (see eg.(2.6)). This p arameter has a p ositive relationship with maximum and minimum work-force, p roduction rate and sales levels, and also with revenue and p rofit. So it is easy to get better results than another models by increasing the value of Vi. But it is not fair to do so. For this reason value of Vi will be chosen according to 3.d above the set of Vi for p roduct 1 will be (0.9 , 1) and for p roduct 2 ( 1,1 ) . The out p ut of p rogram is as follows: a- The results of each p roduct will be printed out according to the sequence of input data of p roducts. b- Print out the inp ut data for each product in the beginning of its results. c- Three tables for each y ear, first table for decision variables (Pt, It, Wt, p t) for each month and yearly total of Pt and It. See tables ((4-1), (4-3)) for p roduct 1 and ((4-8), (4-10)) for p roduct 2. Second table is for monthly basic costs and total of them in each month, and tot al each of them in a y ear. These 4 tables are not imp ortant t o be listed in this research while the table of the yearly cost in d below is a good breviary. The third table contains the sales, revenue, other cost and profit for each month and their total for each y ear, see tables ((4-2), (4-4)) for p roduct 1 and ((4–9), (4–11)) for p roducts 2 d- Three tables for each p roduct represent the y early totals, first table to inventory , p roduction and sales, see table (4–5) for p roduct 1 and table (4–12) for product 2. Second table contains y early total of each kind of cost and their summation for N y ears, see table ( 4 – 6 ) for p roduct 1 and table (4–13) for product 2. Third table contains y early total of revenue, other cost and p rofit for N y ears and their summation, see table (4-7) for product 1 and table (4–14) for product 2. e- Four t ables for the comp any (all products or all factories) as final results of all p roducts. The first table (4–15) contains the total of each basic cost for every p roduct and their totals for t he company. The second table (4–16) contains the total of revenue, other cost and p rofit for each p roduct and their summation for the comp any. The third table contains the monthly total of each basic cost for all p roducts as well as monthly summation. See table (4–17). The fourth t able (4–18) contains the monthly total of revenue, other cost and profit for all p roducts. Comparison with H.M.M.S Model: One of the main p urp oses of these models concerned is to smooth out the raw time–series representing fluctuations in work- force, production, inventory and sales levels. In table (4–19) blew shows the maximum and minimum and variation for the decision variables of H.M .M .S and pred 3. It is clearly that variation in p red3 is considerably less than H.M .M .S model and this smoothing is effective in increasing the profit and reducing costs. Main Steps of pred. 3 Program This p rogram is written in general to accep t any number of p roducts and any number of y ears. Execution time is 5 seconds for two y ears and two p roducts. It consists 525 p rogramming instructions and st atements. 1- Definition for integer and real variables. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 2- Read No. of y ears, No. of p roducts and forecast ing selection variable. 3- Declaration for 57 dimensions. 4- M ain loop for number of p roducts. 5- Read C1 to C43, W0, Io and II. 6- Read historical demand. 7- Compute G1 to G5 and C10 to C18 8- K = 1, M = 12 . 9- Print C1 to C13 and initial values. 10- Selection the method of forecast ing 1 moving average forecasting subroutine FORCA 2 exp onential weighted average 3 forecasted sales equal to actual demands     11- Read Pc SHN and N. 12- M ain loop for number of y ears. 13- Loop for monthly comp utations. 14- Compute productive capacity and bt. 15- Compute C41 to C45 and C46 to C50. 16- Build up the matrix by using equations (2–15) and (2–16). solve the sy st em of equations by Gauss Jordan method to obtain Wt , t = 1 to 14 and select Wt , Wt+1 and Wt+2 .This st ep will be executed 12 times p er each y ear. See [10] , [11] , [12] . 17- Compute C26 to C40 and compute Pt, It and pt. 18- Compute the basic costs t hen revenue, other cost and profit. 19- Accumulate monthly costs for all p roducts. 20- Accumulate monthly revenue, other cost and profit for all p roducts. 21- Compute check which rep resent equation (2.11) and must equal zero otherwise there is an error in mathematical op erations of this model or in p rogramming this model. 22- K = K + 1 , M = M + 1 then step 13 to comp ute another month . 23- If the reminder of K 0 12  st ep 24. 24- Print 3 tables in 4.C above for each y ear. 25- Go back to step 13. 26- In end of yearly loop p rint 3 t ables as y early totals for each p roduct see 4.d. 27- Go back to step 4 to comp ute another product. 28- When finished from the last p roduct p rint out 4 tables for the company. see 4.C. Re ferences 1. Holt, C.; M odigiliani, F.; M uth, J. and Simon, H.A. (1960), Planning, p roduction, Inventories and Work–Force, prentice–Hall, Englewood Cliffs, N.J. 2. Holt, C.M odigiliani, F., and Simon, H. A., (1955), A linear Decision Rule for Production and Emp loyment Scheduling, M anagement Science, 2(1). 3. 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). 4. Fahad, S.A. (2009), M onth–to–M onth Unt il N Years Prediction for Planning a Productive Firm, Ibn–AL- Haitham Journal for Pure and Ap p lied Sciences, Baghdad University , 22(4). 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. Brown, R.G. (1967), Decision Rules for Inventory M anagement. Holt, Rinehart and Winst on, New York. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 7. M ontgomery , D.C., (1968), An Introduction to short – term Forecast ing, Journal of Indust rial Engineering, XIX (10). 8. Brown, R.G. (1959), Statist ical Forecast ing for Inventory Control, M c Graw–Hill, New York. 9. Winters, P.R. (1960), Forecast ing sales by Exp onentially Weighted M oving Averages, M anagement Science, 6 (3). 10. Philip s and Taylor, (1973), Theory and Ap p lications of Numerial 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. p rentice – Hill. Product 1 Table: ( 4 - 1 ) Year 1 Whe n V = 0.9 95.64 81 314 443 2 96.09 81 319 440 3 96.60 81 319 438 4 94.90 80 320 437 5 94.93 80 319 436 6 94.43 80 319 435 7 92.75 80 321 434 8 94.18 80 320 435 9 94.37 80 318 435 10 93.62 81 318 436 11 91.86 81 320 437 12 3806.6 5256.34 Tot. Product 1 Table :( 4 – 2 ) year 1 V = 0.9 Check Profit Other cost Revenue sales Month 0 18776.94 2873.44 39973.62 414 1 0 20654.84 2817.10 41075.4 429 2 0 21261.13 2796.73 41867.89 436 3 0 21546.68 2786.25 42333.13 438 4 0 21357.3 2776.15 41322.93 435 5 0 21434.87 2770.13 41379.92 436 6 0 21424.47 2764.67 41099.93 435 7 0 21277.02 2759.88 40062.01 432 8 0 21453.31 2763.28 41023.98 436 9 0 21491.12 2767.5 41206.88 437 10 0 21449.65 2771.6 40884.56 437 11 0 21314.12 2777.43 39935.58 435 12 253441.5 33425.04 492165.8 5200 Tot. Prices Work Force Inventory Production Month 96.53 81 301 452 1 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Product 1 Table : ( 4 – 3 ) Year 2 V = 1 Prices workforce Inventory Production Month 91.97 83 318 456 13 93.48 84 318 464 14 94.94 86 317 471 15 95.97 67 318 478 16 96.38 88 322 485 17 97.79 90 332 493 18 114.18 91 328 506 19 124.08 92 321 516 20 125.62 93 318 522 21 129.36 94 312 526 22 123.16 95 312 526 23 115.49 95 318 525 24 3834.71 5965.57 Tot. Product 1 Table :( 4 – 4 ) Year 2 V = 1 Check Profit Other cost Revenue S ales Month 0 23365.42 2898.80 42040.88 457 13 0 23314.33 2948.27 43385.64 464 14 0 23415.02 2994.15 44737.17 471 15 0 23545.81 3037.43 45789.96 477 16 0 23502.12 3081.24 46315.7 481 17 0 23280.44 3135.14 47205.07 483 18 0 29194.28 3215.03 58246.94 510 19 0 34802.34 3278.55 64822.94 522 20 0 35616.1 3317.59 65871.08 524 21 0 38662.92 3342.54 68732.15 531 22 0 34610.28 3345.35 64868.23 527 23 0 30103.43 3340.96 59968.2 519 24 343412.5 37935.04 651984 5967 Tot. Yearly Total For Product 1 Table ( 4 - 5) Y. S ales Y. Pro duction Y. inventory Year 5200 5256.34 3806.60 1 5967 5965.57 3834.71 2 11167 11221.9 7641.31 Tot. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Table :( 4 - 6 ) Yearly Total of Each Basi c Cost (Regular Payroll, Hiring And Layoff, Overtime , Inventory Related, Opportuni ty Cost and their total for each year) Table ( 4 – 7 ) Product 2 Table: ( 4 – 8 ) Year 1 Whe n V = 1 Prices Work-Force Inventory Production Year Month 67.74 61 1055 965 1 60.49 61 1103 964 2 66.73 61 1107 968 3 71.71 62 1095 972 4 66.09 62 1114 971 5 77.57 62 1088 972 6 67.39 61 1096 964 7 65.33 61 1100 956 8 65.65 60 1093 949 9 60.52 60 1100 940 10 60.95 59 1099 934 11 59.74 59 1099 929 12 13149.6 11484 Tot. Product 2 Table: ( 4 – 9 ) Year 1 V = 1 Check Profit Other cost Revenue S ales Month 0 8803.79 6139.54 54906.19 811 1 0 16302.23 6131.35 55398.61 916 2 0 21696.19 6155.92 64380.03 965 3 0 26414.2 6178.32 70495. 95 983 4 0 20235 6176.08 62946.85 952 5 0 32968.37 6178.14 77356.66 997 6 0 21465.23 6130.85 64436.26 956 7 0 20467.57 6080.84 62217.14 952 8 0 21157.27 6032.80 62740.37 956 9 0 18316.63 5978.37 56452.17 933 10 0 18807.27 5938.37 57022.35 936 11 0 18394.71 5906.46 55458.17 928 12 245028.5 73027 743810.8 11284.84 Tot. Y. Totc Y. Opc Y.Inc c Y.OTC Y. HLC Y. RPAC Year 205299.3 - 27009.5 34.33 376.40 24.76 231873.3 1 270636.4 7806.80 29.97 3081.91 1168.52 258549.2 2 475935.7 - 19202.7 64.30 3458.31 1193.28 490422.6 Tot. Y. Profit Y. Othe r cost Y. Re venue Year 253441.5 33425.04 492165.8 1 343412.5 37935.04 651984 2 596854 71360.08 1144150 Tot. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Product 2 Table: ( 4 – 10 ) Year 2 V = 1 Prices Work Force Inventory Production Month 59.35 59 1098 926 13 58.06 59 1099 925 14 58.18 59 1096 927 15 56.36 59 1101 931 16 56.88 60 1116 941 17 65.94 61 1120 956 18 79.28 61 1099 970 19 77.76 62 1090 975 20 72.38 62 1085 972 21 62.76 61 1103 965 22 65.52 61 1104 961 23 67.80 61 1095 956 24 13206.25 11403.44 Tot. Product 2 Table: ( 4 – 11 ) Year 2 V = 1 Check Profit Other cost Revenue S ales Month 0 18441.12 5887.189 55045.98 927 13 0 18251.19 5880.17 53613.3 923 14 0 18524.47 5891.97 54074.24 929 15 0 18320.85 5921.99 52190.63 926 16 0 17820.99 5982.39 52657.44 926 17 0 20650.28 6077.57 62780.7 952 18 0 33997.01 6168.14 78589.18 991 19 0 31541.62 6199.37 76520.95 984 20 0 26272.06 6181.34 70696.24 977 21 0 18790.4 6136.57 59415.46 947 22 0 20990.19 6108.59 62892.4 960 23 0 22739.99 6079.16 65411.82 965 24 266340.2 72514.46 743888.3 11407 Tot. Yearly total for product 2 Table :( 4 – 12 ) Y .sal s Y .production Y . inventory Year 11284.84 11484.04 13149.6 1 11407.44 11403.44 13206.25 2 22692.28 22887.48 26355.85 Tot IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Table: ( 4 – 13 ) Yearly Total of Each Basi c Cost ( Regular Payroll , Hiring And Layoff , Overtime , Inventory Related , Opportuni ty Cost and their total for each year ) Y.TOTC Y.OPC Y.incc Y.OTC Y.HLC Y.RPac Year 425755.3 59921.03 250.61 801.51 191.86 364590.3 1 405033.7 41921.11 108.16 610.57 138. 362210.8 2 830789 101842.14 358.77 1412.08 374.86 726801.1 Tot Table: ( 4 – 14 ) Y. profit Y. othe r cost Y. revenue Year 245028.5 73027.03 743810.8 1 266340.2 72514.46 743888.3 2 511368.7 145541.49 1487699.1 Tot Final results for company ( all products ) Table :( 4 – 15 )Total each basi c cost for product ( factory ) ( Regul ar Payroll , Hiring And Layoff , Overtime , Inventory Related , Opportuni ty Cost and their total for each product ) C.ToTc C.opc C.l ncc C.oTc C.HLC C.RPAC Product 475935.7 -19202.7 64.30 3458.31 1193.28 490422.6 1 830789 101842.11 358.77 1412.08 374.86 726801.1 2 1306724.7 82639.43 423.07 4870.39 1568.14 1217223.7 Tot Table: ( 4 – 16 )Total of revenue , other cost and profit for each product and for the company C. profit C. other cost C. revenue Product 596854 71360.08 1144150 1 511368.6 145541.5 1487699 2 1108222.6 216901.6 2631849 Tot IBN AL- HAITHAM J. 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VOL.24 ( 3) 2011 Table: ( 4 – 17 )Monthly total of each basi c cost for the company ( all products ) Tote Opc Incc Otc Hlc Rpa c Month 58286.08 7400.53 233.74 557.87 91.08 49982.86 1 50567.59 462.26 3.61 45.91 4.66 50051.15 2 54337.96 4155.09 4.47 53.16 6.28 50118.95 3 55903.62 5655.34 2.6 74.88 4.83 50165.97 4 53725.26 3514.94 19.38 58.86 1.78 50130.29 5 55385.08 5200.0 13.74 106.26 1.57 50063.5 6 53750.98 3829.87 1.52 65.95 13.42 49840.22 7 51693.85 2027.3 0.06 52.80 18.2 49595.47 8 52357.7 2909.61 4.83 52.48 18.87 49371.91 9 49105.43 -109.07 0.23 30.39 21.91 49161.98 10 48940.01 -134.52 0.67 31.71 17.65 49024.49 11 47001.03 -1999.84 0.08 27.63 16.37 48956.8 12 46494.33 -3111.33 0.82 196.4 164.72 49243.72 13 46604.98 -3288.11 0.52 167.2 146.82 49578.55 14 47985.81 -2272.75 2.09 131.95 136.04 49988.47 15 47154.51 -3550.47 0.52 81.45 135.12 50487.89 16 48586.4 -2764.64 27.13 56.5 150.83 51116.58 17 56842.32 4590.24 52.67 152.23 178.21 51868.98 18 64261.65 11050.96 5.03 464.22 163.16 52578.28 19 65522 11759.66 10.8 620.08 102.54 53128.92 20 65180.23 11262.12 22.10 609.38 68.41 53217.32 21 61215.17 7385.73 5.77 533.75 52.84 53237.07 22 62706.22 9028.44 7.14 406.24 31.35 53233.05 23 63116.48 9638.04 2.64 273.08 21.47 53181.24 24 IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Table :( 4 – 18 ) Monthly total of revenue , other cost and profit for the company ( all products ) Profit Other cost Revenue Month 27580.75 9012.97 94879.81 1 36957.07 8949.34 96474.01 2 42957.32 8952.64 106247.9 3 47960.88 8964.57 112829.1 4 41592.29 8952.23 104269.8 5 54403.23 8948.27 118736.6 6 42889.7 8895.52 105536.2 7 41744.59 8840.72 102279.2 8 42610.58 8796.07 103764.4 9 39807.75 8745.87 97659.05 10 40256.91 8709.98 97906.9 11 39708.83 8683.89 95393.75 12 41806.54 8785.99 97086.85 13 41565.52 8828.44 96998.93 14 41939.49 8886.12 98811.41 15 41866.66 8959.43 97980.59 16 41323.11 9063.62 98973.13 17 43930.72 9212.72 109985.8 18 63191.29 9383.17 136836.1 19 66343.96 9477.92 141343.9 20 61888.16 9498.93 136567.3 21 57453.32 9479.11 128147.6 22 55600.47 9453.94 127760.6 23 52843.42 9420.12 125380 24 Table :(4-19) Comparison pred .3 with H.M.M.S Model in terms of smoothing of Wt , Pt , It and sales as well as cost and profit Profit Total-Cost Sales Inventory Produ ction Work-Force P red 3 H .M .M .S P red 3 H .M .M .S P red 3 H .M .M .S P red 3 H .M .M .S P red 3 H .M .M .S p ed 3 H .M .M .S 5 9 6 8 5 4 3 4 8 7 7 6 .7 5 4 7 2 9 5 .7 8 8 0 7 7 3 6 .9 531 414 117 725 284 441 332 301 31 455 216 239 526 434 92 661 360 301 95 67 28 109 65 44 Max. Min. Var. 2011) 3( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد لشركة إنتاجیة نموذج ریاضي لمنتوجات متعددة فهد سمیر عبد الوهاب ابن الهیثم ، جامعة بغداد –قسم الحاسبات ، كلیة التربیة 2011، اذار،30:استلم البحث في 2011 ،یار، ا30: قبل البحث في الخالصة H.Mنموذج انموذج امتداد إلى ھذا اال .M .S تج واحـد ذه النمـاذج سـتغیر . والنماذج المطورة لھ التـي تتعامـل مـع مـن ـھ . نموذج نفذ بتقنیة برمجة الحاسوب لتعظیم األرباحھذا اال. لتتعامل مع منتجات متعددة لشركة إنتاجیة .متعدد ، شركة ، مصنع ، تعظیم: الكلمات المفتاحیة