IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Multiproduct Mathematical Model for Productive Company S.A. Fahad Department of Computer Science, College of Education Ibn- Al- Haitham, University of Baghdad Received in : 30, March , 2011 Accepted in : 30, May , 2011 Abstract This model is an extension to H.M .M.S and related developments models of a single product. These models will be converted to deal with Multiproduct for productive company. This model executed by computer programming technique to maximize profits. Key words : Multi product, Company, Factory , Maximize. Introduction There are many companies produce more than one product each manufactured by plant or firm each one is independent from the others. The management of the company needs such a system to make a control on production, inventory, work–force, sales and prices for all products and provide an optimal solution to maximize profit. Holt, Modigliani, Muth and Simon (H.M .M.S) developed a dynamic model to plan aggregate control of production, inventory and work – force for a single – item (product) which fully reported in their text [1] . It was developed under the assumption that the receipt of orders would be erratic and fluctuating and it would therefore need to eradicate any excessive movements in the rates of production , inventory and work – force , in order to cut down the costs of running a manufactory in mathematical terms and , to that end, H.M .M.S subdivided the total cost as follows : a- Regular payroll 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 total cost function to be minimized is the summation of the above costs . Multiproduct Model This model will depend on the original model of H.M .M.S [1] and [2] and also last developments on their model were done by the researchers [3] and [4] , all above models were dealing with a single product and will be converted to deal with multiproduct , as follows : Modify the equation (1,1) to eq. (1.4) to be : Regular payroll costs = C1i Wti + C13i …(2.1) Hiring and lay off = C2i (Wti -W(t-1)i - C11i) 2 …(2.2) Overtime and idle time costs = 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= production rate for product i in period t IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Iti = level of inventory for product i at the end of period t. Sti= shipment of product i in period t = Oti the order level for that month. C1i – C13i numerical constants which must be evaluated from historical costs for each product (see [2]) . Price variable have introduced to H.M .M.S model to influence on the ordering pattern (see [3]) to move heavy demand away from peak periods and smoothing Pt, It and Wt and reducing costs. For each product is written as follows: Oti = ai - bti p ti …(2.5) where Oti = forecasted order for product i ai = M aximum productive capacity for product i bti - the measure of change in demand per unit change in price for each product pti = variable price for product i ai = optimal value of labour productivity X initial level of work – force X possible 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 compensate for unknown components in the productive capacity and for any large forecasted demands in the interval t = 1 to t = 12 By substituting equation (2.6) in to equation (2.5) we obtain: Oti = C4i Woi X Ni X Vi - bti p ti …(2.7) By substituting equation (2.7) into equation (2.4) we obtain: Inventory connected costs = C7i [ Iti - C8i - C9i ( C4i Woi X Ni X Vi - bti pti)] 2 …(2.8) The variable price policy will bear the manufacturer the following cost: Opportunity cost = Qi . Pci – T ti t 1 p   (C4i Woi X Ni X Vi - bti p ti ) …(2.9) where Pci = the (constant) salling price for product i Qi = the total quantity. That would have been sold during the period t = 1 to t = T The total 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 pti)] 2 – pti (C4i Woi X Ni X Vi– bti p ti )+ Qi . pci ] …(2.10) Subject to the following restriction Iti = I(t-1)i + Pti – C4i Woi X Ni X Vi + bti pti …(2.11) By differentiating CT with respect 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) pti = 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. SCI. 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 substituting the decision variables Pti , Iti and pti 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-periods of simultaneous linear equations and imposing to end condition W10 = w11 = w12. By applying the gauss – Jordan method to the system above, we have got the optimal values of wti, ti = 1 to 14. Characteristics of the Model a- Optimize the production rats, inventory rats, work–force, sales quantity and prices to maximize profit for each month within a year until N years. b- The computer program was written which was referred to as the (pred 3) which designed to execute the model above. This p rogram will accept any number of products or factories, see [5]. c- Can run the program in any month within each year by giving the input variable II value represents the difference between t = 1 and the new period (month ) d- The parameter Vi in equation (2.7) have given many values and selected value caused a smallest variation in the work – force per each year and smoothing the decision variables (see equation (2.12), (2.13) and (2.14)) according to the principles of initiating H.M .M.S model. e- There are two subroutines in (pred 3) to forecast future demands, the first is moving average demand and the second is exponential 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 system 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 product where produced by different factories belong to one company , each factory independent from the other in term of costs and demand and this case is applicable in the international companies. h- The most difficult decision face the manager is firing man-power. The idle people are international problem. Such system like this model will reduce number of idle people. For example, in the table (4.8) for product (factory) 2 in the month 11 must fire one work–force but the manager can hire him in factory (product) 1 in the same month. See table (4-1). So changes work–place for the work–force helps to reduce the layoff workers. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Results Obtained from Program ( pred 3) This program has given many values for number of years and number of products and the execution was succeeded. But to reduce the number and areas the tables as output of this program for this research paper I chose (2) years and (2) products as input values to run the program. The value of Vi determine the productive capacity (see eg.(2.6)). This parameter has a positive relationship with maximum and minimum work-force, production rate and sales levels, and also with revenue and profit. 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 product 1 will be (0.9 , 1) and for product 2 ( 1,1 ) . The out put of program is as follows: a- The results of each product will be printed out according to the sequence of input data of products. b- Print out the input data for each product in the beginning of its results. c- Three tables for each year, first table for decision variables (Pt, It, Wt, pt) for each month and yearly total of Pt and It. See tables ((4-1), (4-3)) for product 1 and ((4-8), (4-10)) for product 2. Second table is for monthly basic costs and total of them in each month, and total each of them in a year. These 4 tables are not important to 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 year, see tables ((4-2), (4-4)) for product 1 and ((4–9), (4–11)) for products 2 d- Three tables for each product represent the yearly totals, first table to inventory, production and sales, see table (4–5) for product 1 and table (4–12) for product 2. Second table contains yearly total of each kind of cost and their summation for N years, see table ( 4 – 6 ) for product 1 and table (4–13) for product 2. Third table contains yearly total of revenue, other cost and profit for N years and their summation, see table (4-7) for product 1 and table (4–14) for product 2. e- Four tables for the company (all products or all factories) as final results of all products. The first table (4–15) contains the total of each basic cost for every product and their totals for the company. The second table (4–16) contains the total of revenue, other cost and profit for each product and their summation for the company. The third table contains the monthly total of each basic cost for all products as well as monthly summation. See table (4–17). The fourth table (4–18) contains the monthly total of revenue, other cost and profit for all products. Comparison with H.M.M.S Model: One of the main purposes 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 pred3 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 program is written in general to accept any number of products and any number of years. Execution time is 5 seconds for two years and two products. It consists 525 programming instructions and statements. 1- Definition for integer and real variables. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 2- Read No. of years, No. of products and forecasting selection variable. 3- Declaration for 57 dimensions. 4- Main loop for number of products. 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 forecasting 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- Main loop for number of years. 13- Loop for monthly computations. 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 system of equations by Gauss Jordan method to obtain Wt , t = 1 to 14 and select Wt , Wt+1 and Wt+2 .This step will be executed 12 times per each year. See [10] , [11] , [12] . 17- Compute C26 to C40 and compute Pt, It and pt. 18- Compute the basic costs then revenue, other cost and profit. 19- Accumulate monthly costs for all products. 20- Accumulate monthly revenue, other cost and profit for all products. 21- Compute check which represent equation (2.11) and must equal zero otherwise there is an error in mathematical operations of this model or in programming this model. 22- K = K + 1 , M = M + 1 then step 13 to compute another month . 23- If the reminder of K 0 12  step 24. 24- Print 3 tables in 4.C above for each year. 25- Go back to step 13. 26- In end of yearly loop print 3 tables as yearly totals for each product see 4.d. 27- Go back to step 4 to compute another product. 28- When finished from the last product p rint out 4 tables for the company. see 4.C. References 1. Holt, C.; Modigiliani, F.; Muth, J. and Simon, H.A. (1960), Planning, production, Inventories and Work–Force, prentice–Hall, Englewood Cliffs, N.J. 2. Holt, C.Modigiliani, F., and Simon, H. A., (1955), A linear Decision Rule for Production and Employment Scheduling, Management Science, 2(1). 3. Fahad, S.A., (2008), Mathematical Model for One Year Planning of a Manufactory , Ibn–AL- Haitham Journal for Pure and Applied Sciences, Baghdad University , 21(4). 4. Fahad, S.A. (2009), Month–to–Month Until N Years Prediction for Planning a Productive Firm, Ibn–AL- Haitham Journal for Pure and Applied Sciences, Baghdad University , 22(4). 5. Lynwood, A.; Johnson, Douglas, C. Montgomery, (1973), Operations Research in Production Planning, Scheduling and Inventory Control, Johnwiley & Sons, INC; New York, London. 6. Brown, R.G. (1967), Decision Rules for Inventory Management. Holt, Rinehart and Winston, New York. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 7. Montgomery, D.C., (1968), An Introduction to short – term Forecasting, Journal of Industrial Engineering, XIX (10). 8. Brown, R.G. (1959), Statistical Forecasting for Inventory Control, Mc Graw–Hill, New York. 9. Winters, P.R. (1960), Forecasting sales by Exponentially Weighted Moving Averages, Management Science, 6 (3). 10. Philips and Taylor, (1973), Theory and Applications 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 Moler, C. B. (1967), Computer Solution of Linear Algebraic Systems, Englewood, N. J. prentice – Hill. Product 1 Table: ( 4 - 1 ) Year 1 When 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. SCI. 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 Sales 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. Sales Y. Production 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. SCI. VOL.24 (3) 2011 Table :( 4 - 6 ) Yearly Total of Each Basic Cost (Regular Payroll, Hiring And Layoff, Overtime, Inventory Related, Opportunity Cost and their total for each year) Table ( 4 – 7 ) Product 2 Table: ( 4 – 8 ) Year 1 When 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 Sales 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.Incc 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. Other 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. SCI. 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 Sales 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 .sals 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. SCI. VOL.24 (3) 2011 Table: ( 4 – 13 ) Yearly Total of Each Basic Cost ( Regular Payroll , Hiring And Layoff , Overtime , Inventory Related , Opportunity 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. other 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 basic cost for product ( factory ) ( Regular Payroll , Hiring And Layoff , Overtime , Inventory Related , Opportunity Cost and their total for each product ) C.ToTc C.opc C.lncc 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 basic cost for the company ( all products ) Tote Opc Incc Otc Hlc Rpac 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. SCI. 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 Production 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 596854 348776.7 547295.78 807736.9 531 414 117 725 284 441 332 301 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