Optimal Ordering Policy for an Economic Order Quantity Model with Inspection Errors and Inspection Improvement Investment

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1 International Journal of Information and Management Sciences 25 (2014), Optimal Ordering Policy for an Economic Order Quantity Model with Inspection Errors and Inspection Improvement Investment Liang-Yuh Ouyang, Chia-Hsien Su, Chia-Huei Ho and Chih-Te Yang Tamkang University, Tungnan University, Ming Chuan University and Chien Hsin University of Science and Technology Abstract The rise of consumer rights has caused businesses to focus increasingly on product quality. The inability of businesses to identify defective items before selling them results in higher return costs, decreased sales revenue, damaged reputations, and decreased competitiveness. This study examines the economic order quantity(eoq) model in which the retailer discovers defective goods among received products. Although retailers conduct quality inspections, the inspection process is imperfect. We assume that Type I and Type II inspection errors occur during product quality inspection and that the market demand rate is sensitive to Type II inspection errors. To improve inspection, the retailer invests capital to decrease Type II inspection errors. This study investigates the optimal order quantity and the power of the test to maximize total profit per unit time. Mathematical analysis is used to show the optimal solution exists. An algorithm is then developed to calculate the optimal solution. Finally, numerical examples demonstrate the solution process and sensitivity analysis with respect to major parameters is carried out. Keywords: Inventory, defective products, inspection error, power of the test, capital investment. 1. Introduction In 1913, Ford Whitman Harris presented the original economic order quantity (EOQ) model (Harris [9]). Inventory management has since attracted researcher attention. The traditional EOQ model assumes that products produced by manufacturers are good quality and ignores production concerns such as poor production equipment, flawed raw materials, and human operation errors. Waters[33] suggested it was unrealistic to assume that a production system produced only good products. Porteus [22] argued that deficient production processes produce defective products. He assumed that manufacturers invest capital to enhance production process controls and determine optimal production policies. Lee and Rosenblatt [17] developed an economic manufacturing quantity

2 318 LIANG-YUH OUYANG, CHIA-HSIEN SU, CHIA-HUEI HO AND CHIH-TE YANG (EMQ) model with production cycle length and equipment maintenance intervals as decision variables. Lee and Rosenblatt [18] then incorporated precautionary maintenance and recovery costs into an economic production quantity (EPQ) model with imperfect production processes. Zhang and Gerchak [38] focused on an EOQ model to determine the optimal order quantity and inspection strategy when flawed products are randomly produced. Groenevelt et al. [8] determined optimal production quantities by considering equipment damage and related maintenance in an EPQ model. Kim and Hong [15] discovered the optimal production cycle length by calculating equipment failure time using three different functions: a fixed parameter, a linear increasing function, and an exponentially distributed function. Wu et al. [34] assumed that manufacturers offer free repairs on flawed products and investigated the effects of quality guarantee strategies on optimal production quantity. Results in this regard can be found in Yassine et al. [35], Rezaei and Salimi [24], Chang [3], and Ouyang and Chang [19]. These studies showed that flawed products affect optimal production or ordering strategies, but they did not discuss the treatment of defective items. To ensure quality products and maintain consumer goodwill, enterprises typically conduct product quality restriction before sales and establish handling procedures for defective products. Salameh and Jaber [27] considered an EOQ model that accounted for defective products and product inspection on all goods, in which defective products are sold at lower prices. Cárdenas-Barrón [1] corrected Salameh and Jaber [27] equation and numerical results. Chan et al. [2] incorporated product inspection on all goods into an EPQ model. They treated imperfect and defective goods by selling them at lower prices, reworking them, or rejecting them. Chiu [5] developed an EPQ model that allowed for inventory shortages and assumed that defective products are randomly produced, some of which are reworked or discarded, thereby determining the optimal production quantity and inventory shortage level. Papachristos and Konstantaras [21] based their study on the work of Salameh and Jaber [27] and concluded that defective products should only be sold when all quality products in a batch are sold. Chiu [4] assumed that defective products are produced randomly and can either be discarded or repaired. The repair process can also result in discarding certain goods. Other studies that have investigated defective goods handling strategies include Kulkarni [16], Sarker et al. [28], Jaber et al. [12], Roy et al. [25], El Saadany and Jaber [6], Sana [26], Su [30], and Glock and Jaber [7]. These studies focused on determining optimal production or order quantity by using various defective product handling strategies. They assumed that no errors occur in the inspection process, and that products deemed good or defective after inspection are actually good or defective products, respectively. However, the inspection process involves human error, equipment obsolescence, and technical constraints, which results in inspection errors. Quality restriction occasionally treats quality products as defective products (Type I inspection error) and defective products as good ones (Type II inspection error). Research has recently incorporated these inspection errors into inventory models to investigate the effect of inspection errors on production quantity and cost. Yoo et al. [36] created an EPQ model with imperfect production quality and inspection

3 OPTIMAL ORDERING POLICY FOR AN ECONOMIC ORDER QUANTITY MODEL 319 Table 1: Comparison between this study and related research. Article Model type Inspection process Demand rate Capital investment Yoo et al. [36] EPQ Type I and II errors constant no Wang et al. [32] EPQ Type I and II errors constant no Khan et al. [13] EOQ Type I and II errors constant no Yoo et al. [37] EPQ Type I and II errors constant yes Hsu and Hsu [11] EOQ Type I and II errors constant no This study EOQ Type I and II errors a function of the power of test yes errors. The model considered defective sales returns and reverse logistics from customers when Type II inspection errors occur, and reworking and salvaging processes when Type I inspection errors occur. Wang et al. [32] proposed an EPQ model that allowed for Type I and Type II inspection errors for input materials. They showed that a partial inspection policy is more effective than a complete inspection policy or none at all. Khan et al. [14] extended the work of Salameh and Jaber [27] and developed an EOQ model with imperfect quality and inspection errors. They concluded that an inspection process performed by people instead of an automated screening system requires considering misclassification errors. Yoo et al. [37] established an EPQ model with imperfect production and inspection quality investments. The expected total profit function was decomposed into non-quality-related terms and quality-related terms. The objective of their model was to determine the optimal production and inspection policy to maximize total profit and minimize total quality cost. Recently, Hsu and Hsu [11] developed an EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns. Raouf et al. [23], Sheu et al. [29], and Wang [31] also conducted related studies. Defective products that are overlooked during the inspection process and then sold to customers result in the loss of uninspected defective items and decreased customer repurchase intention that creates a negative effect on customer demand rate. Businesses should filter out defective products before sales to protect their reputations and maintain competitiveness. Therefore, improving the quality of the inspection process is essential. From the retailer perspective, if a Type II error occurs, the retailer sells defective products to customers and suffers substantial costs. Hence, it is more crucial to reduce the proportion of Type II errors than Type I errors. We consider an EOQ model for defective goods in incoming purchases. Although the retailer implements quality inspection, the inspection results contain Type I and Type II errors. To decrease the proportion of Type II inspection errors, the retailer invests capital in increasing the power of the test. We establish a retailer total profit function per unit time and use a mathematical analysis approach to determine the optimal order quantity and the power of the test to maximize total profit function per unit time. Subsequently, we design an algorithm to determine optimal solutions and use numerical examples to explain the solution process and the sensitivity analysis results. Table 1 presents a comparison between this study and previous research regarding imperfect inspection processes.

4 320 LIANG-YUH OUYANG, CHIA-HSIEN SU, CHIA-HUEI HO AND CHIH-TE YANG 2. Notations and Assumptions The following notations and assumptions are used throughout this paper. Notations: D demand rate (units/time unit) A ordering cost per order ($/order) F freight cost per shipment ($/shipment) f fixed freight cost per shipment ($/shipment) γ freight cost per unit ($/unit) h 1 holding cost per non-defective item per unit time ($/unit/time unit) h 2 holding cost per defective item per unit time, h 2 h 1 ($/unit/time unit) v unit purchasing price ($/unit) p unit selling price, p > v ($/unit) s unit inspecting cost ($/unit) η unit handling cost of the defective item (including warranty cost, reverse logistics from customers back to the retailer, and goodwill loss) ($/unit) k unit sale price of the defective item in secondary market, k < v ($/unit) λ defective rate per shipment, λ [0,1) x inspection rate (units/time unit) α the proportion of Type I inspection error, i.e., the proportion in which the non-defective products are misclassified as defective ones, α (0, 1) and is given β 0 the proportion of Type II inspection error, i.e., the proportion in which the defective products are misclassified as non-defective ones, β 0 (0,1) and is given β the proportion of Type II inspection error after capital investment, β (0,β 0 ]; and hence, 1 β is the power of the test (decision variable) ρ opportunity cost of investment capital per dollar per unit time to reduce the proportion of Type II inspection error Q order quantity per order (decision variable) (units/order) T the length of the replenishment cycle (decision variable) (time unit) Assumptions: (1) Replenishments are instantaneous, and the lead time is zero. (2) Shortages are not allowed to occur. (3) In business dealings, the supplier often offers freight discounts to encourage retailers to order larger quantities. Hence, the freight charged per unit by the supplier has

5 OPTIMAL ORDERING POLICY FOR AN ECONOMIC ORDER QUANTITY MODEL 321 the following quantity schedule: γ 1, if q 1 Q < q 2 ; γ 2, if q 2 Q < q 3 ; γ =.. γ ε, if q ε Q < q ε+1 ;.. γ m, if q m Q < q m+1 ; where 0 < q 1 < q 2 < < q m < q m+1 = are the freight quantity boundary values in which the freight rate-break occurs; γ = γ ε is the unit freight cost applicable to shipping quantity Q that falls within the interval q ε to q ε+1 (ε = 1,2,...,m); and γ 1 > γ 2 > > γ m > 0. (4) The market demand rate for the product is a decreasing and convex function of the proportion of Type II inspection errors, β. The relationship formula is D(β) = D 0 e δ(1 β), where 1 β represents the power of the test, δ 0 is a constant representing a trend factor, and D 0 > 0 represents the market demand rate with zero trend factor (i.e., δ = 0). For simplicity, we use D(β) and D interchangeably. (5) The retailer capital investment, I(β), to improve product inspection quality (i.e., to reduce the proportion of Type II inspection errors from β 0 to β) is produced by a logarithmic function of β, I(β) = θln(β 0 /β) for 0 < β β 0, where θ = 1/ω and ω is the percentage decrease in β 0 per dollar increase in investment I(β). Porteus [22], Keller and Noori [13], Hong and Hayya [10], and Ouyang et al. [20] used this investment function. (6) The proportions of Type I and II inspection errors, α and β 0, are obtained using previous screening process data. For simplicity, we assume that α and β 0 are given constants. 3. Model Formulation The description of the problem is as follows. The retailer orders Q units every order with an ordering cost A. Arriving lot Q contains certain defective items with a defective rate of λ. When the retailer receives the lot, an all-items inspection process is conducted on the lot, with a fixed cost of s per unit. The inspection rate is assumed to be a constant, x. As a result of inspection process errors, Type I and Type II inspection errors occur at a proportion of α and β, respectively. In other words, α(1 λ)q units of non-defective products are misclassified as defective items. In addition, βλq units of defective products are overlooked and misclassified as non-defective items. Therefore, after inspection, there are [α(1 λ) + (1 β)λ]q defective items. These items are immediately sold in a single batch in a secondary market at lower price. On the other hand, there are [(1 α)(1 λ) + βλ]q items sold to customers. However, customers

6 322 LIANG-YUH OUYANG, CHIA-HSIEN SU, CHIA-HUEI HO AND CHIH-TE YANG Table 2: The results of stock inspection. inspection result actual quality non-defective defective total non-defective (1 α)(1 λ)q βλq (Type II inspection error) [(1 α)(1 λ)+βλ]q defective α(1 λ)q (Type I inspection error) (1 β)λq [α(1 λ)+(1 β)λ]q total (1 λ)q λq Q Figure 1: Inventory system of the retailer. sequentially return βλq units because of Type II inspection error and then receive the purchasing price refunds from the retailer. These returned defective items are stored and then sold in a single batch in a secondary market at the end of each cycle with a decreased price. Table 2 shows the results of the retailers product inspection process, and Figure 1 shows the inventory system. Based on the previous description and assumptions, the retailer s total profit per replenishment cycle includes the following elements: (a) Sales revenue (SR): For each shipment, (1 α)(1 λ) + βλ]q product units are classified as non-defective items and subsequently sold to customers with unit selling price p. However, customers return βλq units with purchasing price refunds because of Type II inspection errors. Thus, the sales revenue for the real non-defective items is p(1 α)(1 λ)q. In addition, [α(1 λ)+(1 β)λ]q product units are classified as defective items. These items with the returned βλq units are sold in a secondary market at markdown price k per unit. That is, sales revenue of defective items is k[α(1 λ) + (1 β)λ + βλ]q. Consequently, the sales revenue in a replenishment cycle is SR = p(1 α)(1 λ)q+k[α(1 λ)+λ]q. (b) Loss of uninspected defective items (LS): Because of Type II inspection errors, βλq units of defective products are overlooked during inspection and sold to customers,

7 OPTIMAL ORDERING POLICY FOR AN ECONOMIC ORDER QUANTITY MODEL 323 incurring defective item handling costs (including warranty cost, reverse logistics from customers to the retailer, and loss of goodwill) of ηβλq. Therefore, the loss related to uninspected defective returns is LS = ηβλq. (c) Ordering cost (OC): The ordering cost in a replenishment cycle is OC = A. (d) Freight cost (FC): For each delivery quantity Q, the freight cost in a replenishment cycle is FC = F(Q) = f +γq. (e) Inspection cost (IC): With unit inspection cost s, the inspection cost in a replenishment cycle is IC = sq. (f) Holding cost (HC): The cost of holding non-defective items per replenishment cycle is calculated using: h 1 {[(1 α)(1 λ)+βλ]qt +[α(1 λ)+(1 β)λ]q 2 /x}/2, where(1/2)[α(1 λ)+(1 β)λ]q 2 /x, isthecumulative inventory quantity of defective items before they are identified. The cost of holding defective items per replenishment cycle is calculated using: h 2 {(1/2)[α(1 λ)+(1 β)λ]q(q/x)+(1/2)βλqt}, where (1/2)βλQT is the cumulative inventory quantity of returned defective items from the market (see Figure 2). Therefore, the total holding cost in a replenishment cycle is: HC =(1/2){h 1 [(1 α)(1 λ)+βλ]+h 2 βλ}qt +[Q 2 /(2x)](h 1 +h 2 )[α(1 λ)+(1 β)λ]. (g) Purchasing cost (PC): The retailer orders Q units with unit purchasing price v, and hence, the purchasing cost in a replenishment cycle is PC = vq. (h) Opportunity cost of inspection power improvement investment (OP): With unit capital opportunity cost ρ per unit time, the investment cost per replenishment cycle is OP = ρθln(β 0 /β)t. The quantity of products that the retailer sells per cycle is [(1 α)(1 λ) +βλ]q which is equal to market demand DT, that is, DT = [(1 α)(1 λ)+βλ]q. Therefore, the cycle time for each replenishment is T = [(1 α)(1 λ)+βλ]q/d. For convenience, let φ (1 α)(1 λ) and K α(1 λ) + λ; then, the total profit per unit time is expressed as: Π(Q,S) =(SR OC FC IC HC PC LS OP)/T { { = pφq+kkq A+f+γQ+sQ+vQ+ηβλQ+ 1 2 QT[h 1(φ+βλ)+h 2 βλ] + Q2 ( 2x (h β0 ) } } / 1 +h 2 )(K βλ)+ρθln T T β

8 324 LIANG-YUH OUYANG, CHIA-HSIEN SU, CHIA-HUEI HO AND CHIH-TE YANG Figure 2: The retailer s inventory level behavior. = D { pφ+kk γ s A+f φ+βλ [h 1(φ+βλ)+h 2 βλ]q 2 Q (h 1 +h 2 )(K βλ)q 2x ( β0 ρθln β } v ηβλ ). (3.1) 4. Theoretical Results To calculate the optimal order quantity Q and the power of the test 1 β that maximizes total profit per unit time, the following steps are followed. The necessary conditions to maximize the total profit per unit time Π(Q,β) in (3.1) are Π(Q,β)/ Q = 0 and Π(Q,β)/ β = 0. That is, Π(Q, β) Q and Π(Q,β) β = (A+f)D (φ+βλ)q 2 1 [ 2 =0, h 1 (φ+βλ)+h 2 βλ+ D(h 1 +h 2 )(K βλ) (φ+βλ)x ] (4.1) = D {[ λ ][ φ+βλ φ+βλ +δ pφ kk+γ+s+ A+f Q + (h 1 +h 2 )(K βλ)q 2x ] [ (h1 +h 2 )Q ]} +v +ηβλ +λ η (h 1 +h 2 )λq + ρθ = 0, (4.2) 2x 2 β where D = D(β) = D 0 e δ(1 β). Because of the high-power expression of the exponential demand rate function, it is difficult to obtain the closed-form solutions of Q and β and to prove the concavity property of total profit per unit time Π(Q,β) in (3.1) with a Hessian matrix. Hence, the problem is solved using the following search procedure.

9 OPTIMAL ORDERING POLICY FOR AN ECONOMIC ORDER QUANTITY MODEL 325 For any given β, the second-order partial derivative of Π(Q,β) with respect to Q is 2 Π(Q,β) Q 2 = 2D(A+f) < 0. (4.3) (φ+βλ)q3 Therefore, Π(Q,β) is a concave function of Q for fixed β, and a unique value of Q exists (denoted by Q β ) such that Π(Q β,β) is the maximum. Solving (4.1), Q β is obtained as: 2xD(A+f) Q β = D(h 1 +h 2 )(K βλ)+x(φ+βλ)[h 1 (φ+βλ)+h 2 βλ]. (4.4) Q β in (4.4) is independent of the freight cost per unit, γ and is well-defined because K βλ = α(1 λ) +λ(1 β) > 0. Substituting Q = Q β and D = D(β) = D 0 e δ(1 β) into (3.1) produces: Π(β)=Π(Q β,β) = D 0e δ(1 β) φ+βλ { pφ+kk γ s A+f (h 1 +h 2 )(K βλ)q β Q β 2x ( β0 ρθln β [h 1(φ+βλ)+h 2 βλ]q β 2 } v ηβλ ). (4.5) 5. Solution Procedures Π(β) in (4.5) is a continuous function on interval (0,β 0 ]. The following algorithm is designed to obtain the optimal solution (Q,β ) for the proposed model. Algorithm. Step 1: For given β 0, divide the interval (0,β 0 ] into n subintervals of equal length. Let β i = iβ 0 /n, i = 1,2,...,n, where n is sufficiently large. Step 2: For each β i, i = 1,2,...,n, calculate Q βi from (4.4), and then find Π(Q βi,β i ) from (4.5). (i) If q m Q βi < q m+1, then the optimal total profit per unit time Π(Q β i,β i ) = Π(Q βi,β i ). (ii) If q k Q βi < q k+1 for some k (1 k m 1), then the optimal total profit per unit time Π(Q β i,β i ) = Max{Π(Q βi,β i ), Π(q j,β i ), j = k+1,k +2,...,m}. Step 3: Find Max i=1,2,...,n Π(Q β i,β i ). Let Π(Q,β ) = Max i=1,2,...,n Π(Q β i,β i ). Hence, (Q,β ) is the optimal solution. Once the optimal solution (Q,β ) is obtained, the optimal replenishment cycle length per order for the retailer, T = [(1 α)(1 λ)+βλ]q /[D(β )], and the retailer s optimal capital investment for improving the quality of product inspection, I(β ) = θln(β 0 /β ), are calculated.

10 326 LIANG-YUH OUYANG, CHIA-HSIEN SU, CHIA-HUEI HO AND CHIH-TE YANG Table 3: The freight rate schedule. Q (units/ship) γ ($/unit) Example 1 Example 2 1 Q < Q < Q < Q Table 4: Comparative solutions with/without inspection improvement in Examples 1 and 2. β Q D(β ) Π(Q,β ) Example Without investment Example Without investment Numerical Example and Sensitivity Analysis This section presents numerical examples to demonstrate the applicability of the proposed model. The parameter effect of the model is analyzed using an extensive sensitivity analysis. Consider an inventory system with the following data: D 0 = 4,000units/year, A =$100/order, p =$15/unit, v =$8/unit, k =$2/unit, η =$3/unit, λ = 1%, s =$0.5/unit, x = 15,000 units/year, h 1 =$0.5/unit/year, h 2 =$0.25/unit/year, α = 0.05, β 0 = 0.06, ρ = 0.1, δ = 0.3, and ω = The fixed transportation cost f is $500/shipment and the freight rate schedule is shown in Table 3. Example 1. The decision process is performed by setting n = 500 and using the algorithm in Section 5 to obtain the optimal solution. The results are as follows: the optimal ordering quantity is units (i.e., Q = Q β48 = units), and the corresponding market demand is D(β 48 ) = D 0 e δ(1 β 48) = units/year. The optimal power of the test is (i.e., (1 β ) = = ), the corresponding freight rate is γ = γ 3 = 0.8 and the maximum total profit per unit time is Π(Q,β ) = Π(Q β48,β 48 ) =$ /year. Example 2. In this example, the gap between the freight rate and order quantity is increased to 0.1 (see Table 3). Using the algorithm in the previous section and setting n = 500producesanoptimalmarketdemandofD(β 46 ) = units/yearandapower of the test of (1 β ) = However, the retailer must increase the order quantity to Q = q 4 = 7200 units to obtain a lower corresponding freight rate of γ = γ 4 = 0.6. The corresponding maximum total profit per unit time is Π(Q,β ) = Π(q 4,β 46 ) = $ /year. To observe the effects of improved inspection power, we substitute β = β 0 = 0.06 into (4.4) to obtain the optimal solutions without capital investment in improving inspection errors. The results are shown in Table 4.

11 OPTIMAL ORDERING POLICY FOR AN ECONOMIC ORDER QUANTITY MODEL 327 Table 5: Effects of changing various parameters in Example 3. D 0 β 0 parameters β Q D(β ) Π(Q,β ) (units/order) (units/year) ($/year) Example 1 β 48 = Q β48 = % β 65 = Q β65 = % β 38 = q 4 = % β 99 = Q β99 = % β 31 = q 4 = % β 64 = Q β58 = % β 39 = Q β35 = % β 96 = Q β87 = % β 32 = Q β29 = % β 103 = q 4 = δ -25% β 68 = Q β68 = % β 36 = Q β36 = % β 28 = Q β28 = % β 24 = Q β24 = ρ -25% β 36 = Q β36 = % β 60 = Q β60 = % β 72 = Q β72 = % β 49 = Q β49 = λ -25% β 49 = Q β49 = % β 48 = Q β48 = % β 47 = Q β47 = % β 96 = Q β96 = ω -25% β 64 = Q β64 = % β 38 = Q β38 = % β 32 = Q β32 = Table 4 shows that it is beneficial for the retailer to invest capital for improving the proportion of Type II errors. This results in effectively promoting annual market demand and total profit. Example 3. This example involves sensitivity analysis to investigate the effects of crucial parameters on the optimal policy. The data used are the same as in Example 1. The sensitivity analysis is conducted by changing each parameter individually by an appropriate value while all other parameters remain unchanged. The results are shown in Table 5. The results in Table 5 lead to the following managerial insights: (1) The retailer determines the optimal order quantity by trading off the freight rate against additional holding cost. Because the optimal order quantity is determined based on the maximum total profit, this simultaneously determines the optimal freight rate. (2) When D 0, δ, or ω increases, the optimal proportion of Type II inspection errors (β ) decreases and market demand (D(β )) increases, causing an increase in total profit Π(Q,B ). The results show that an increasing market demand parameter or

12 328 LIANG-YUH OUYANG, CHIA-HSIEN SU, CHIA-HUEI HO AND CHIH-TE YANG improved proportion of Type II inspection error increases demand rate and hence increases total profit. (3) Regarding the original proportion of Type II inspection errors, although β 0 exerts a slight effect on the optimal solutions (Q,β ), total profit per unit time decreases with an increased value of β 0. (4) When the unit capital opportunity cost for improving production inspection quality, ρ, increases, the value of β increases but the values of Q, D(β ), and Π(Q,β ) decrease. This implies that the retailer reduces capital investment to improve inspection error, and the total profit decreases as the unit capital opportunity cost increases. (5) Thedefective rate, λ, exerts slight effects on the values of β and D(β ). Theoptimal order quantity increases while total profit decreases when the value of λ increases. 7. Conclusion and Future Research We considered an EOQ model for defective goods in incoming purchases in which the retailer implements quality inspection and assumes that inspection results contain Type I and Type II errors. The market demand rate is also an increasing function of the power of the test. To increase customer demand, or increase the power of the test, the retailer invests in improving product inspection to reduce the proportion of substandard goods reaching consumers. The study established the total profit function per unit time for the retailer, maximizing the total profit function, and used mathematical analysis to calculate the retailers optimal order quantity and the power of the test. An algorithm was designed to calculate the optimal solution. We presented numerical examples to explain the solution process, and conducted a sensitivity analysis to examine the effects of the primary parameters on the solution. The proposed model can be extended and examined in several aspects. The proposed inventory model can be manipulated to consider stochastic rather than deterministic inspection errors, defective rate, or market demand. Another possible extension is to consider freight capacity restriction in the model. Acknowledgements The authors would like to express their appreciation to the anonymous referees for the valuable and helpful suggestions. This research was supported by National Science Council, Republic of China under Grant NSC E References [1] Cárdenas-Barrón, L. E. (2000). Observation on: Economic production quantity model for items with imperfect quality, International Journal of Production Economics, Vol.64, [2] Chan, W. M., Ibrahim, R. N. and Lochert, P. B. (2003). A new EPQ model: integrating lower pricing, rework and reject situations, Production Planning and Control, Vol.14, [3] Chang, H. C. (2013). An economic production quantity model with consolidating shipments of imperfect quality items: A note, International Journal of Production Economics, Vol.144,

13 OPTIMAL ORDERING POLICY FOR AN ECONOMIC ORDER QUANTITY MODEL 329 [4] Chiu, S. W. (2007). Optimal replenishment policy for imperfect quality EMQ model with rework and backlogging, Applied Stochastic Models in Business and Industry, Vol.23, [5] Chiu, Y. P. (2003). Determining the optimal lot size for the finite production model with random defective rate, the rework process, and backlogging, Engineering Optimization, Vol.35, [6] El Saadany, A. M. A. and Jaber, M. Y. (2010). A production/remanufacturing inventory model with price and quality dependent return rate, Computers & Industrial Engineering, Vol.58, [7] Glock, C. H. and Jaber, M. Y. (2013). A multi-stage production-inventory model with learning and forgetting effects, rework and scrap, Computers & Industrial Engineering, Vol.64, [8] Groenevelt, H., Pintelon, L. and Seidmann, A. (1992). Production lot sizing with machine breakdowns, Management Science, Vol.38, [9] Harris, F. W. (1913). How many parts to make at once, Factory, The Magazine of Management, Vol.10, , 152. [10] Hong, J. D. and Hayya, J. C. (1995). Joint investment in quality improvement and setup reduction, Computers & Operations Research, Vol.22, [11] Hsu, J. T. and Hsu, L. F. (2013). An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns, International Journal of Production Economics, Vol.143, [12] Jaber, M. Y., Bonney, M. and Moualek, I. (2009). An economic order quantity model for an imperfect production process with entropy cost, International Journal of Production Economics, Vol.118, [13] Keller, G. and Noori, H. (1988). Impact of investing in quality improvement on the lot size model, Omega International Journal of Management Sciences, Vol.16, [14] Khan, M., Jaber, M. Y. and Bonney, M. (2011). An economic order quantity (EOQ) for items with imperfect quality and inspection errors, International Journal of Production Economics, Vol.133, [15] Kim, C. H. and Hong, Y. (1999). An optimal production run length in deteriorating production processes, International Journal of Production Economics, Vol.58, [16] Kulkarni, S. S. (2008). Loss-based quality costs and inventory planning: General models and insights, European Journal of Operational Research, Vol.188, [17] Lee, H. L. and Rosenblatt, M. J. (1987). Simultaneous determination of production cycle and inspection schedules in a production system, Management Science, Vol.33, [18] Lee, H. L. and Rosenblatt, M. J. (1989). A production and maintenance planning model with restoration cost dependent on detection delay, IIE Transactions, Vol.21, [19] Ouyang, L. Y. and Chang, C. T. (2013). Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging, International Journal of Production Economics, Vol.144, [20] Ouyang, L. Y., Wu, K. S. and Ho, C. H. (2007). An integrated vendor-buyer inventory model with quality improvement and lead time reduction, International Journal of Production Economics, Vol.108, [21] Papachristos, S. and Konstantaras, I. (2006). Economic ordering quantity models for items with imperfect quality, International Journal of Production Economics, Vol.100, [22] Porteus, E. L. (1986). Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research, Vol.34, [23] Raouf, A., Jain, J. K. and Sathe, P. T. (1983). A cost-minimization model for multicharacteristic component inspection, IIE Transactions, Vol.15, [24] Rezaei, J. and Salimi, N. (2012). Economic order quantity and purchasing price for items with imperfect quality when inspection shifts from buyer to supplier, International Journal of Production Economics, Vol.137, [25] Roy, A., Maity, K., Kar, S. and Maiti, M. (2009). A production-inventory model with remanufacturing for defective and usable items in fuzzy-environment, Computers & Industrial Engineering, Vol.56, [26] Sana, S. S. (2010). A production-inventory model in an imperfect production process, European Journal of Operational Research, Vol.200, [27] Salameh, M. K. and Jaber, M. Y. (2000). Economic production quantity model for items with imperfect quality, International Journal of Production Economics, Vol.64,

14 330 LIANG-YUH OUYANG, CHIA-HSIEN SU, CHIA-HUEI HO AND CHIH-TE YANG [28] Sarker, B. R., Jamal, A. M. M. and Mondal, S. (2008). Optimal batch sizing in a multi-stage production system with rework consideration, European Journal of Operational Research, Vol.184, [29] Sheu, S. H., Chen, Y. C., Wang, W. Y. and Shin, N. H. (2003). Economic optimization of off-line inspection with inspection errors, Journal of the Operational Research Society, Vol.54, [30] Su, C. H. (2012). Optimal replenishment policy for an integrated inventory system with defective items and allowable shortage under trade credit, International Journal of Production Economics, Vol.139, [31] Wang, C. H. (2007). Economic off-line quality control strategy with two types of inspection errors, European Journal of Operational Research, Vol.179, [32] Wang, C. H., Dohi, T. and Tsai, W.C. (2010). Coordinated procurement/inspection and production model under inspection errors, Computers & Industrial Engineering, Vol.59, [33] Waters, C. D. J. (1994). Inventory Control and Management, Chichester, Wiley. [34] Wu, C. C., Chou, C. Y.and Huang, C. (2009). Optimal price, warranty length and production rate for free replacement policy in the static demand market, Omega International Journal of Management Sciences, Vol.37, [35] Yassine, A., Maddah, B. and Salameh, M. (2012). Disaggregation and consolidation of imperfect quality shipments in an extended EPQ model, International Journal of Production Economics, Vol.135, [36] Yoo, S. H., Kim, D. S. and Park, M. S. (2009). Economic production quantity model with imperfectquality items, two-way imperfect inspection and sales return, International Journal of Production Economics, Vol.121, [37] Yoo, S. H., Kim, D. S. and Park, M. S. (2012). Lot sizing and quality investment with quality cost analyses for imperfect production and inspection processes with commercial return, International Journal of Production Economics, Vol.140, [38] Zhang, X. and Gerchak, Y. (1990). Joint lot sizing and inspection policy in an EOQ model with random yield, IIE Transactions, Vol.22, Department of Management Sciences, Tamkang University, New Taipei City 251, Taiwan. liangyuh@mail.tku.edu.tw Major area(s): Inventory management, supply chain management, mathematical programming Department of Hospitality Management, Tungnan University, New Taipei City 222, Taiwan. peony@mail.tnu.edu.tw Major area(s): Inventory management, supply chain management, mathematical programming. Department of Business Administration, Ming Chuan University, Taipei 111, Taiwan. chho@mail.mcu.edu.tw Major area(s): Inventory management, supply chain management, mathematical programming. Department of Industrial Management, Chien Hsin University of Science and Technology, Taoyuan 320, Taiwan. ctyang@uch.edu.tw Major area(s): Inventory management, supply chain management, mathematical programming. (Received March 2014; accepted September 2014)