An EPQ model of imperfect production processes with shortages and quality cost.
The primary operation strategies and goals of most manufacturing firms are to seek a high satisfaction to customer's demands and to become a low-cost producer. To achieve these goals, the company must be able to effectively utilize resources and minimize costs. The economic order quantity (EOQ) model was the first mathematical model introduced several decades ago by Harris (1913) to assist corporations in minimizing total inventory costs. It balances inventory holding and setup costs and derives the optimal order quantity. Regardless of its simplicity, the EOQ model is still applied industry-wide today. In the manufacturing sector, when items are produced internally instead of being obtained from an outside supplier, the economic production quantity (EPQ) model introduced by Warets (1994) is often employed to determine the optimal production lot size that minimizes overall production/inventory costs. It is also known as the finite production model because of its assumption that the production rate must be much larger than the demand rate. In developing classified EPQ models it has been assumed that the product quality and production process are perfect (that is all units produced are of good quality). In most of the existing literature, products are assumed to be no defective items when the production is considered. But, in real situation the defective items are popular in many kinds of products. Hence, the defective rate cannot be ignored in the production process. Defective items will be produced in each cycle of production in most practical situations. A considerable amount of research has been carried out to address the problems of imperfect quality EPQ models. Several scholars have investigated the effect of imperfect quality production on economic production models. Maity et al. (2009) presented an optimal control recovery production inventory system with shortages under inflation and discounting in fuzzy stochastic environment. The product defectiveness is random. The defective product also is treated as return product. Swapan Kumar Manna (2010) developed two deterministic economic production quantity (EPQ) models for Weibull-distribution deteriorating items with demand rate as ramp type function of time. It is assumed that the finite production rate is proportional to the time-dependent demand rate and the unit production cost is inversely proportional to the production rate. The EPQ model without shortages is studied first and with shortages is investigated next. Porteus (1986) formulated the relationship between process quality improvement and setup cost reduction and illustrated that the annual cost can be further reduced when a joint investment in both process quality improvement and setup reduction is optimally made. Zhang and Gerchak (1990) considered a joint lot sizing and inspection policy in order to develop the EOQ model where the number of defective items in each lot is random and defective units cannot be used and, thus, they must be replaced with non- defective ones. Cheng (1991) addressed an EOQ model with demand dependent unit cost and imperfect production processes and formulated and optimization problem as a geometric program to obtain a closed--form optimal solution. Lee et al. (1997) developed a model of batch quantity in a multi-stage production system considering various proportions of defective items produced in every stage while they ignored the rework situation. Recently, Ben-Daya et al. (2006) developed integrated inventory inspection models with and without replacement of nonconforming items discovered during inspection. Inspection policies include no inspection, sampling inspection, and 100% inspection. They proposed a solution procedure for determining the operating policies for inventory and inspection consisting of order quantity, sample size, and acceptance number. Sridevi et al. (2010) developed and analyzed an inventory model with the assumption that the rate of production is random and follos a Weibull distribution and the demand is a function of selling price. Nita H.Shah et al. (2010) developed an inventory model that jointly optimizes cost of manufacturer and retailer under buoyant market condition. Proposed model also considers imperfect production processes and partly backlogging is allowed only at the retailer's end. Gour Chandra Mahata et al. (2010) developed an EPQ model for deteriorating items in the fuzzy sense where delay in payments for both retailer and customer are permissible to reflect realistic situations. Singa Wang Chiu (2011) described a cost reduction (n+1) delivery policy into an imperfect EPQ model with repairable items and algebra approach used to find the economic lot size, long run average production inventory delivery cost. Ting J. Lee (2011) considered a cost reduction distribution policy for an integrated manufacturing system operating under quality assurance practice and also improves its replenishment lot-size solution in terms of lowering producer's stock holding cost. Abdur Rahim (2011) developed a mathematical model for determining an optimal production run length for a deteriorating production system with allowable shortages for products sold with a free minimal repair warranty period. The length of the warranty period is dependent upon the quality of the product. Srinivasa Rao (2011) concerned with a production inventory system with the assumption that the life time of the product is random and follows a Weibull distribution. The Weibull rate of decay includes increasing, decreasing and constant rates of decay. In this paper an Economic Production Quantity model with shortage has been developed by incorporating the defective items and quality cost. This paper is organized as follows. Section 2 is concerned with assumptions and notations; Section 3 is presented problem formulation. In Section 4, we present a sensitivity analysis for the developed model. In section 5, conclusions and topics for further research are presented.
Assumptions and Notations
The imperfect production process, due to process deterioration or other factors may randomly generate x percentage of defective items. In this paper, all defective items are assumed to be scrap items. It follows that the production rate of the scrap items d can be expressed as d = Px. Suppose a manufacturer produces a certain product and sells it in a market. All items are produced and sold in each cycle.
1. A single type of product in a single stage production system is considered.
2. Production rate is constant and greater than demand rate.
3. Proportion of defective is constant and only one type of defective is produced in each cycle.
4. All demands must be satisfied.
5. Backlogging permitted.
6. Proportion of scrap is less than the proportion of defectives.
7. Inspection cost is ignored since it is negligible with respect to other costs.
8. The other assumption in classical EPQ model.
The following notations are used in our analysis.
1. P--Production rate in units per unit time
2. D--Demand rate in units per unit time
3. B--shortage level
4. [Q.sub.1]--on hand inventory level
5. [Q.sup.*]--Optimal size of production run
6. [C.sub.0]--Setup cost
7. [C.sub.h]--Holding cost per unit/year
8. [C.sub.Q]--Cost of quality
9. [C.sub.d]--Unit scrap or defective cost per item of imperfect quality.
10. [C.sub.p]--Production Cost per unit
11. d--rate of defective items from end customers in units per unit time (d = PY)
12. x--proportion of defective items from regular production (x is between 0.01 to 0.09)
13. [t.sub.1]--The time during which the stock is building up at a constant rate of P-D-d units per unit time
14. [t.sub.2]--The time during which there is no production and inventory is decreasing at a constant demand rate D per unit time.
15. t--unit time in one cycle
16. TC--Total cost
An EPQ Model with Defective Items and Shortages permitted
The proposed inventory system operates as follows. The cycle starts at t=0 and the inventory accumulates at a rate of P-D-d up to time t= [t.sub.1] where the production stops. After that, the inventory level starts to decrease due to demand and defective at a rate of D up to time t = [t.sub.2] where shortages start to accumulate at a rate D up to time t = [t.sub.3]. Production restarts again at time t = [t.sub.4] and ends at time t =t with a rate P--D--d to recover the previous shortage in the period [t.sub.3] and satisfy demand in the period [t.sub.4].
The process is repeated. A real life production process due to process deteriorate or other factors may generate randomly x percent of defective items at a production rate d (d=Px). A line AK indicates the slope of P-D-d. Therefore, the net defectives produced during time [t.sub.1] is Qx say line JK indicates that amount. On hand inventory of defective items during production uptime [t.sub.1] is [dt.sub.1] = Px[t.sub.1] = Px(Q/P)= Qx. The behavior of the inventory levels is shown in figure-1.
[FIGURE 1 OMITTED]
Time [t.sub.1] needed to built up [Q.sub.1] units of items, therefore, [t.sub.1] = [Q.sub.1]/P - D - d (1)
The inventory level start to decrease due to demand at a rate D upto time
[t.sub.2], [therefore] [t.sub.2] = [Q.sub.1]/D (2)
Shortages starts to accumulates at a rate of B, the inventory level is zero at time [t.sub.2] but shortages accumulate at the rate of D upto time [t.sub.3].
Therefore, time [t.sub.3] needed to built up B units of items, therefore,
[t.sub.3] = B/D (3)
The production restarts again at time [t.sub.4] at a rate of P-D-d to recover both the previous shortages into period [t.sub.3] and to satisfy demand in the period
[t.sub.4]. [therefore] [t.sub.4] = B/P-D-d (4)
Let [Q.sub.1] be the maximum inventory available at the end of time [t.sub.1] that is expected to be consumed during the remaining period [t.sub.2] at the demand rate D.
Then [Q.sub.1] =(P-D-W) [t.sub.1] - B = (P - D - d) (Q/P) - B (5)
The inventory level during production cycle is (from equations (1) to (4))
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Calculation of Average Inventory
According to the definition, Q =DT, therefore, T = Q/D and Q = P[t.sub.1], therefore, [t.sub.1] = Q/P.
Therefore, average Inventory
= 1/2T [[Q.sub.1][t.sub.1] + [Q.sub.1][t.sub.2]] = [Q.sub.1]/2T[[Q.sub.1]/P - D -d + [Q.sub.1]/D] (from equations (1) and (2))
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Average inventory during shortage period (from equations (3) and (4))
= 1/T [1/2 B[t.sub.3] + 1/2 B[t.sub.4]] = B/2T [B/D + B/P - D -d] = [PB.sup.2]/2Q(P - D - d) (8)
1. When x=0 and B =0 then [bar.I] = Q/D which is standard inventory model.
2. When x =0, the [bar.I] = q(p - D)/2P which is standard inventory model. And x=0, then [bar.I] = Q(P - D)/2P
Total Cost: The total cost components consist of production cost, setup cost, holding cost, shortage cost, defective cost and cost of quality.
Production Cost = 1/T [QC.sub.p] = [DC.sub.p]/1-x (9)
Setup Cost = [C.sub.0]/T = [DC.sub.0]/Q(1 - x) (10)
Holding Cost = [QC.sub.h] (P - D - d)/2P + [C.sub.h] [PB.sup.2]/2Q(P - D - d) - [BC.sub.h] (11)
Shortage cost = [C.sub.S][PB.sup.2]/2Q(P - D - d) (12)
Defective cost = 1/T [dt.sub.1] [C.sub.d] = Dx[C.sub.d]/1 - x (13)
Cost of Quality = 1/T d[C.sub.Q][t.sub.1] = Dx[C.sub.Q]/1 - x (14)
Total Cost (TC (Q,B)) (from the equations (9) to (14)) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Partially differentiate with respect to Q,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
Therefore, [Q.sup.2] = (2PD[C.sub.0]/[C.sub.h] (P - D - d) + [P.sup.2] [B.sup.2] ([C.sub.h] + [C.sub.S])/[C.sub.h] [(P - D - d).sup.2] (15)
Partially differentiate w.r.t B,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Therefore, B = [QC.sub.h](P - D - d)/P([C.sub.h] + [C.sub.S]) (16)
Substitute in the above equation (16) in (15),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
Note: when x=0, then Q = [square root of (2PD[C.sub.0]([C.sub.h] + [C.sub.S])/[C.sub.h][C.sub.S](P - D))]
Example, Let the inventory system has the following parameter values P = 5000 units, D =4500 units, [C.sub.h] =10, [C.sub.p] = 100, [C.sub.0] =100, x = 0.01 to 0.09, [C.sub.d] = 5, [C.sub.S] =10, [C.sub.Q] =5
From the above, table, a study of rate of defective items (x) with optimal quantity([Q.sup.*]), cycle time (t) and it is concluded that when the rate of defective items increases then the optimal quantity, Production uptime ([t.sub.1]), Demand down time ([t.sub.2]), Total cycle time (t) increases but on hand inventory level ([Q.sub.1]) and shortage level B decreases.
From the above, a study of rate of defective items with Production cost, Ordering cost, Holding cost, shortage cost, Defective cost, quality cost and Total cost and it is concluded that when the rate of defective items increases then the Production cost, defective cost, quality cost and total cost increase but ordering cost and holding cost decreases.
Cycle time verification
To verify the mode, from equations (1) to (5) [t.sub.1] = 0.1421, [t.sub.2] = 0.0142, [t.sub.3] = 0.0142, [t.sub.4] = 0.1421and t = 0.3126 t = [t.sub.1] + [t.sub.2] + [t.sub.3] + [t.sub.4] = 0.1421 + 0.0142 + 0.0142 = 0.1421 = 0.3126, from equation (6), it is found that t = 0.3126 which proves the model.
An EPQ Model with defective items and Shortages not permitted s
The proposed inventory system operates as follows. The cycle starts at t=0 and the inventory accumulates at a rate of P-D-d upto time t=[t.sub.1] where the production stops. After that, the inventory level starts to decrease due to demand and defective at a rate of D upto time t = [t.sup.2] The process is repeated. The behavior of the inventory levels is shown in figure -2.
From the figure 2,
[Q.sub.1] =(P-D-W) [t.sub.1] = (P - D - d)(Q/P), [t.sub.1] = [Q.sub.1]/P - D - d (18)
;[t.sup.2] = [Q.sub.1]/D (19)
The inventory level during production cycle is (from equations (18) and (19))
t = [t.sub.1] + [t.sub.2] = [Q.sub.1]/P - D - d + [Q.sub.1]/D = [Q.sub.1](P - d)/D(P - D - d)
= [PQ.sub.1](1 - x)/D(P - D - d) = Q/D (1 - x) where d = Px (20)
[FIGURE 2 OMITTED]
Average Inventory (from equations (18) and (19))
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
Note: When x=0 then [bar.I] = Q(P - D)/2P which is standard inventory model
Total Cost: The total cost components consist of production cost, setup cost, holding cost, defective cost and cost of quality.
Production Cost = 1/T [QC.sub.p] = [DC.sub.p]/1 - x (22)
Setup Cost = [C.sub.0]/T = [DC.sub.0]/Q(1 - x) (23)
Holding Cost = [QC.sub.h](P - D - d)/2P (24)
Defective cost = 1/T [dt.sub.1] [C.sub.d] = Dx[C.sub.d]/1 - x (25)
Cost of Quality = 1/T d[C.sub.Q][t.sub.1] = Dx[C.sub.Q]/1 - x (26)
Total Cost (TC (Q,B)) = [DC.sub.p/1 - x + [DC.sub.0]/Q(1 - x) + [QC.sub.h](P - D - d)/2P + Dx[C.sub.d]/1 - x + Dx[C.sub.Q]/1 - x (27)
Partially differentiate with respect to Q,
[partial derivative]/[partial derivative]Q(TC) = 0 [right arrow] - [DC.sub.0]/[Q.sup.2] (1 - x) + [C.sub.h](P - D - d)/2P = 0
[[partial derivative].sup.2]/[partial derivative][Q.sup.2](TC) = 2[DC.sub.0]/[Q.sup.3](1 - x) > 0
Therefore, [Q.sup.2] = 2PD[C.sub.0]/[C.sub.h](P - D - d)(1 - x),
Therefore, Q = [square root of (2PD[C.sub.0]/[C.sub.h](P - D - d)(1 - x))] (28)
Note: When x=0, then Q = [square root of (2[PDC.sub.0]/[C.sub.h](P - D)] which is standard inventory model.
Example, Let us consider the cost parameters
P = 5000 units, D =4500 units, [C.sub.h] = 10, [C.sub.p] = 100, [C.sub.0] = 100, x = 0.01 to 0.09, [C.sub.d] = 5, [C.sub.Q] = 5
Optimum Solution (from equations (22) to (28))
Q = 1005.04, Q1 = 90.45, Production cost = 454545.45, Setup cost = 452.27; Holding cost = 452.27; Defective cost = 227.27; Quality cost =227.27; Total cost = 455904.53
[t.sub.1] = 0.2010, [t.sub.2] = 0.0201 and t = 0.2211
Cycle time verification from equations (18) and (19)
To verify the mode, from equations,
[t.sub.1] = 0.2010, [t.sub.2] = 0.0201 and t = 0.2211
t = [t.sub.1] + [t.sub.2] = 0.2010 + 0.0201 = 0.2211, from equation (20)it is found that t = 0.2211 which proves the model.
The total cost functions are the real solution in which the model parameters (quantity, proportion of defectives, and proportion of scrap) are assumed to be static values. It is reasonable to study the sensitivity, i.e., the effect of making changes in the model parameters over a given optimum solution. It is important to find the effects on different system performance measures, such as cost function, inventory system, etc. For this purpose, sensitivity analyses of various system parameters for this model of this research is given.
An EPQ model with Defective items and shortages permitted
From the above table, for the assigned values of parameters and demand parameters, the optimal values of cycle time (T), Optimal production Quantity (Q*), and total system cost have been determined. The values of above parameters are varied further to observe the trend in optimal policies and the results obtained are shown in Table-1. The increase in rate of defective items (x), the optimum values maximum inventory [Q.sub.1] and shortage stock has shown decreasing trend but it has been increasing trend in Production time ([t.sub.1]), cycle time (T), optimum quantity and Total cost and so on.
An EPQ model with Defective items and shortages not permitted
From the above table, for the assigned values of parameters and demand parameters, the optimal values of cycle time (T), Optimal production Quantity (Q*), and total system cost have been determined. The values of above parameters are varied further to observe the trend in optimal policies and the results obtained are shown in Table-1. The increase in rate of defective items (x), the optimum values maximum inventory [Q.sub.1] and shortage stock has shown decreasing trend but it has been increasing trend in Production time ([t.sub.3]), cycle time (T) and Total cost. The increase in rate of setup cost ([C.sub.0]), the optimum values of maximum inventory Q and shortage stock has shown increasing trend but there is decreasing trend in cycle time (T), Optimum quantity (Q) and total cost and so on.
In this paper, two inventory models have been developed for a single product manufacturing system with defective rate is considered as a variable of known proportions and each of the demand, production and as well as all cost parameters are known. Shortages are allowed. The objective is to minimize the overall total relevant inventory cost. An exact mathematical model and a solution procedure is established. A numerical example is provided to demonstrate its practical usage. Result validation is a necessary step in this research. For validation, the model was coded in Microsoft Visual Basic 6.0. The proposed model can assist the manufacturer and retailer in accurately determining the optimal quantity, cycle time and annual total cost. Moreover, the proposed inventory model can be used in inventory control of certain items such as food items, fashionable commodities, stationary stores and others. For further research, this model can be extended in several ways, for instance, time value of money, price discounts, quantity discounts, rework of defective items.
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C. Krishnamoorthi (1) and Dr. S. Panayappan (2)
(1) Research Scholar, Bharathiar University, Coimbatore-641 035, India (2) Principal, Chickanna Government Arts College, Tirupur, India E-mail: firstname.lastname@example.org
Table 1: Variation of Rate of Defective Items, Q, t and total Cost Rate of Q B [Q.sub.1] [t.sub.1] Defective items 0.01 1421.34 63.96 63.96 0.1421 0.02 1515.23 60.61 60.61 0.1515 0.03 1628.17 56.97 56.97 0.1628 0.04 1767.77 53.03 53.03 0.1768 0.05 1946.66 48.67 48.67 0.1947 0.06 2187.97 43.76 43.76 0.2188 0.07 2540.00 38.10 38.10 0.2540 0.08 3127.72 31.28 31.28 0.3128 0.09 4447.50 22.24 22.24 0.4447 Rate of [t.sub.2] [t.sub.3] [t.sub.4] t Defective items 0.01 0.0142 0.0142 0.1421 0.3126 0.02 0.0135 0.0135 0.1515 0.3300 0.03 0.0127 0.0126 0.1628 0.3509 0.04 0.0118 0.0118 0.1768 0.3771 0.05 0.0108 0.0108 0.1947 0.4107 0.06 0.0097 0.0097 0.2188 0.4570 0.07 0.0085 0.0085 0.2540 0.5249 0.08 0.0069 0.0069 0.3128 0.6394 0.09 0.0049 0.0049 0.4447 0.8994 Table 2: Variation of Rate of Defective Items with Inventory and Total Cost Rate of Production Ordering Holding Shortage Defective Cost Cost Cost cost Items 0.01 454545 319.80 159.90 159.90 0.02 459183 303.05 151.52 151.52 0.03 463917 284.93 142.46 142.46 0.04 468750 265.16 132.58 132.58 0.05 473684 243.33 121.67 121.67 0.06 478723 218.80 109.40 109.40 0.07 483870 190.50 95.25 95.25 0.08 489130 158.36 78.19 78.19 0.09 494505 111.19 55.59 55.59 Rate of Defective Cost of Total Cost Defective cost Quality Items 0.01 227.27 227.27 455639.60 0.02 459.18 459.18 460708.13 0.03 695.08 695.08 465879.14 0.04 937.50 937.50 471155.33 0.05 1184.21 1184.21 476539.29 0.06 1436.17 1436.17 482033.34 0.07 1693.55 1693.55 487639.06 0.08 1956.52 1956.52 493356.25 0.09 2225.27 2225.27 499178.42 Table 1: Effect of Demand and Defective parameters on Optimal Policies. Parameters Optimum Values [Q.sub.1] B [t.sub.1] x 0.01 63.96 63.96 0.1421 0.02 60.61 60.61 0.1515 0.03 56.99 56.99 0.1628 0.04 53.03 53.03 0.1768 0.05 48.67 48.67 0.1947 [C.sub.h] 8 60.30 75.38 0.1675 9 62.25 69.17 0.1537 10 63.96 63.96 0.1421 11 65.46 59.51 0.1323 12 66.80 55.67 0.1237 [C.sub.0] 80 57.21 57.21 0.1271 90 60.68 60.68 0.1348 100 63.96 63.96 0.1421 110 67.08 67.08 0.1491 120 70.06 70.06 0.1557 [C.sub.p] 80 63.96 63.96 0.1421 90 63.96 63.96 0.1421 100 63.96 63.96 0.1421 110 63.96 63.96 0.1421 120 63.96 63.96 0.1421 [C.sub.S] 8 75.38 60.30 0.1340 9 69.17 62.25 0.1383 10 63.96 63.96 0.1421 11 59.51 65.46 0.1455 12 55.67 66.80 0.1485 [C.sub.Q] 3 63.96 63.96 0.1421 4 63.96 63.96 0.1421 5 63.96 63.96 0.1421 6 63.96 63.96 0.1421 7 63.96 63.96 0.1421 Parameters Optimum Values T Q Total Cost x 0.3126 1421.34 455639.60 0.3300 1515.23 460708.13 0.3509 1628.18 465879.14 0.3771 1767.77 471155.33 0.4110 1946.66 475653.29 [C.sub.h] 0.3317 1507.56 455603.02 0.3213 1460.29 455622.54 0.3126 1421.34 455639.60 0.3055 1388.66 455654.65 0.2994 1360.83 455668.04 [C.sub.0] 0.2797 1271.28 455572.08 0.2966 1348.40 455606.78 0.3126 1421.34 455639.60 0.3280 1490.71 455670.82 0.3425 1557.00 455700.65 [C.sub.p] 0.3127 1421.34 364730.51 0.3127 1421.34 410185.06 0.3126 1421.34 455639.60 0.3127 1421.34 501094.15 0.3127 1421.34 546548.69 [C.sub.S] 0.3317 1507.56 455603.02 0.3213 1460.28 455622.54 0.3126 1421.34 455639.60 0.3055 1388.65 455654.65 0.2994 1360.83 455668.04 [C.sub.Q] 0.3126 1421.34 455548.69 0.3126 1421.34 455594.15 0.3126 1421.34 455639.60 0.3126 1421.34 455685.06 0.3126 1421.34 455730.51 Table 2: Effect of Demand and Defective parameters on Optimal Policies Parameters Optimum Values [Q.sub.1] [t.sub.1] [t.sub.1] x 0.01 90.45 0.2010 0.0201 0.02 85.71 0.2143 0.0190 0.03 80.59 0.2302 0.0179 0.04 75.00 0.2500 0.0167 0.05 68.82 0.2753 0.0153 [C.sub.h] 8 101.13 0.2247 0.0225 9 95.34 0.2119 0.0211 10 90.45 0.2010 0.0210 11 86.24 0.1916 0.0192 12 82.57 0.1835 0.0183 [C.sub.h] 80 80.90 0.1797 0.0180 90 85.81 0.1907 0.0190 100 90.45 0.2010 0.0201 110 94.87 0.2108 0.0211 120 99.09 0.2202 0.0220 [C.sub.p] 80 90.45 0.2010 0.0201 90 90.45 0.2010 0.0201 100 90.45 0.2010 0.0201 110 90.45 0.2010 0.0201 120 90.45 0.2010 0.0201 [C.sub.Q] 3 90.45 0.2010 0.0201 4 90.45 0.2010 0.0201 5 90.45 0.2010 0.0201 6 90.45 0.2010 0.0201 7 90.45 0.2010 0.0201 Parameters Optimum Values T Q Total Cost x 0.01 0.2211 1005.04 455904.53 0.02 0.2333 1071.43 460959.18 0.03 0.2481 1151.29 466115.18 0.04 0.2667 1250.00 471375.00 0.05 0.2906 1376.49 476740.88 [C.sub.h] 8 0.2472 1123.67 455809.04 9 0.2330 1059.40 455858.12 10 0.2211 1005.04 455904.53 11 0.2108 958.26 455948.68 12 0.2018 917.47 455990.86 [C.sub.h] 80 0.1977 898.93 455809.04 90 0.2097 953.46 455858.12 100 0.2211 1005.04 455904.53 110 0.2319 1054.09 455948.68 120 0.2422 1100.96 455990.87 [C.sub.p] 80 0.2211 1005.04 364995.44 90 0.2211 1005.04 410500.00 100 0.2211 1005.04 455904.53 110 0.2211 1005.04 501359.08 120 0.2211 1005.04 546813.62 [C.sub.Q] 3 0.2211 1005.04 455722.71 4 0.2211 1005.04 455813.62 5 0.2211 1005.04 455904.53 6 0.2211 1005.04 455995.44 7 0.2211 1005.04 456086.35
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|Author:||Krishnamoorthi, C.; Panayappan, S.|
|Publication:||International Journal of Computational and Applied Mathematics|
|Date:||Mar 1, 2012|
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