Assessing Profitability of a Newsboy-Type Product with Normally Distributed Demand Based on Multiple Samples

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(1)This article was downloaded by: [National Chiao Tung University 國立交通大學] On: 24 April 2014, At: 07:24 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK. Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsta20. Assessing Profitability of a Newsboy-Type Product with Normally Distributed Demand Based on Multiple Samples a. b. Rung-Hung Su , Dong-Yuh Yang & W. L. Pearn a. c. Electronics Division, SEDA Chemical Product Co., LTD. , Taipei , Taiwan. b. Institute of Information and Decision Sciences , National Taipei College of Business , Taipei , Taiwan c. Department of Industrial Engineering and Management , National Chiao Tung University , Hsinchu , Taiwan Accepted author version posted online: 14 Apr 2013.Published online: 16 Jul 2013.. To cite this article: Rung-Hung Su , Dong-Yuh Yang & W. L. Pearn (2013) Assessing Profitability of a Newsboy-Type Product with Normally Distributed Demand Based on Multiple Samples, Communications in Statistics - Theory and Methods, 42:18, 3401-3419, DOI: 10.1080/03610926.2011.624245 To link to this article: http://dx.doi.org/10.1080/03610926.2011.624245. PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions.

(2) Communications in Statistics—Theory and Methods, 42: 3401–3419, 2013 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0926 print/1532-415X online DOI: 10.1080/03610926.2011.624245. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. Assessing Profitability of a Newsboy-Type Product with Normally Distributed Demand Based on Multiple Samples RUNG-HUNG SU1 , DONG-YUH YANG2 , AND W. L. PEARN3 1. Electronics Division, SEDA Chemical Product Co., LTD., Taipei, Taiwan 2 Institute of Information and Decision Sciences, National Taipei College of Business, Taipei, Taiwan 3 Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu, Taiwan This article develops a new index “Achievable Capacity Index,” IA , which can accurately measure the profitability of newsboy-type product with normally distributed demand. An unbiased and effective estimator of IA is derived to estimate actual IA as the parameters of distribution are unknown. Practically, the market information regarding demand is obtained from multiple samples rather than single sample. Therefore, we estimate and test IA based on multiple samples. A hypothesis testing for examining whether the profitability meets designated requirement is presented. Critical values of the test are calculated to determine the evaluation results. Finally, a real case on the sales of donuts is presented to illustrate the applicability of our approach. Keywords Achievable capacity index; Estimating and testing; Multiple samples; Newsboy; Normal demand. Mathematics Subject Classification Primary 62F03; Secondary 62P30.. 1. Introduction In the traditional newsboy problem, it usually focused on short shelf-life products such as daily newsarticles, monthly/weekly magazines, milks, seasonal products, fresh food, and many others. Since the surplus products are subject to storage for a short period of time, one ought to pay additional costs to dispose these items. If the unsatisfied demand is lost, the opportunity cost may be occurred. Received April 1, 2011; Accepted September 9, 2011 Address correspondence to Dong-Yuh Yang, Institute of Information and Decision Sciences, National Taipei College of Business, Taipei 100, Taiwan; E-mail: yangdy@webmail. ntcb.edu.tw. 3401.

(3) Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. 3402. Su et al.. Generally, the demand presented in the newsboy problem is unknown and assumed to be a random variable with a known probability distribution. Consequently, the determination of the ordering quantity (or manufacturing quantity) is critical for achieving certain objective function in the newsboy problem. There is an excellent survey of the literature on the various objective functions such as minimizing the expected cost (Nahmias, 1993), maximizing the expected profit (Khouja, 1995), maximizing the expected utility (Ismail and Louderback, 1979; Lau, 1997), and maximizing the probability of achieving a target profit (Ismail and Louderback, 1979; Shih, 1979; Lau, 1980; Sankarasubramanian and Kumaraswamy, 1983). However, so far, existing researches never care about the value of the maximum expected profit and the probability of achieving a target profit. These values can be expressed the product’s profitability. Whenever the demand is uncertain, several researchers always considered that the demand is random and follows common distributions with known parameters. For example, the normal is preferred when the demand per cycle is relatively large, while the Poisson is better for low-demand items because it is discrete. Lau (1997) has pointed that some seasonal or fashion products which have very high demand uncertainties may be more suitably modeled by the exponential distribution. In practical work, the parameter(s) of demand distribution is/are unknown and depend(s) on the estimation technique. Berk et al. (2007) used the frequentist and the Bayesian approaches for demand estimation. Also, most of the research focused on distribution-free newsboy problem, where the form of the demand distribution is unknown but only the mean and variance are specified. It was the pioneer work of Scarf (1958), in which the minimax approach applied to minimize the maximum cost resulting from the worst possible demand distribution. This approach can derive a simple closed-form expression for the ordering quantity that maximizes expected profit. Moon and Choi (1995) studied a distribution-free newsboy problem with balking, in which customers are allowed to balk when inventory level is low. Ouyang and Wu (1998) presented an inventory model with mixture of backorders and lost sales, which relaxes the assumption about the normal distribution of lead-time demand. Ouyang and Chang (2002) modified the continuous review inventory models involving variable lead time with a mixture of backorders and lost sales. They utilized the minimax distribution-free procedure for finding the optimal inventory strategy in the fuzzy sense where information about the lead time demand distribution is partial. Alfares and Elmorra (2005) extended the analysis of the distribution-free newsboy problem to the case when shortage cost is taken into consideration. Mostard et al. (2005) derived a simple closedform formula to determine the order quantity for the distribution-free newsboy inventory problem with returns. It was shown in Mostard et al. (2005) that the distribution-free order rule performs well when the coefficient of variation cv is at most 0.5, but is far from optimal when the cv is large. Recently, Kevork (2010) developed appropriate estimators for the optimal ordering quantity and the maximum expected profit when demand is normally distributed. They investigated the statistical properties for both small and large samples analytically and through Monte Carlo simulation. The product profitability evaluation is a practical problem frequently occurred in the inventory control. In many real-world inventory systems, the new products are unceasingly introduced. In order to hold the competitive advantage, the managers not only determine the optimal ordering/manufacturing quantity but also.

(4) Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. Assessing Profitability of a Newsboy-Type Product. 3403. measure the old product’s characteristic for evaluating whether the old product is worthy of being ordered/manufactured. If the old product’s characteristic is not satisfied, it would be curtailed or substituted for new product due to spatial or capacity constraints. In this article, we consider a newsboy-type product with random demand D, and study the profitability evaluation problem which deals with examining whether the product’s profitability meets designated requirement. Note that the product’s profitability is defined as the probability of achieving the target profit under the optimal ordering policy. Since the form of the profitability ought to be complex, it is hard to effectively find the statistical estimation of profitability when the parameter(s) of demand distribution is/are unknown. This motivated us to develop a simple index combined with product’s profitability. A new index is called “Achievable Capacity Index.” To the best of our knowledge, the index depends on demand mean ED and demand standard deviation VarD if the selling price and the related costs are given. Therefore, if the normal demand is considered, the achievable capacity index, IA , is a function of and . Under the assumptions that and are unknown, we ought to collect past demand data, and develop an unbiased and effective estimator of IA to estimate actual IA . For the demand data, the demand is the sum of the sales volume and the unsatisfied demand. It seems as if the unsatisfied demand is unable to observed or record. Practically, in order to understand the actual demand for controlling inventory and diminishing the lost sale opportunity cost, the retailers not only care about the sales volume but also try to record unsatisfied demand. Some products would appear to fit these conditions such as high-profit products and new products. Another kind of possibility is that the product is purchased by using the order. At this time, the order can be referred to demand. On the other hand, the majority of the results related to the distributional properties of the estimators were obtained based on the assumption of having a single sample. However, from a practical perspective, several stores have observed a weekly-based (or daily-based) demand records for monitoring profitable status such as fast food restaurants, dairy industries, chemical industries, and so on. Therefore, in these particular environments, the demand data is collected from multiple samples rather than single sample. In order to tackle the profitability evaluation problem, we implement the statistical hypothesis testing methodology based on multiple samples. Critical values of the test are calculated to determine the evaluation results. Furthermore, for practitioners’ convenience, we provide a simple procedure to use in making decision on whether the profitability meets designated requirement. The rest of the article is organized as follows. In the next section, we study the profitability measure for the newsboy-type product with normally distributed demand and devise the achievable capacity index IA . More noteworthy is that IA has simple form and can accurately measure the profitability. In Sec. 3, we derive an unbiased estimator IA to estimate actual IA based on multiple samples. The distribution of IA is also discussed. In Sec. 4, the critical value of the test is calculated to determine the evaluation results. Section 5 presents an example for donuts to illustrate the practicality of the approach to data collected from a donut store for profitability evaluation. In the final section, concluding remarks are given.. 2. Profitability Measurement In this section, we consider a newsboy-type product. The demand D follows a normal distribution, N 2 , and satisfies that the coefficient of variation cv is.

(5) 3404. Su et al.. below 0.3 for neglecting the negative tail, i.e., fD < 0 = −/ = −1/cv < −1/03 ≈ 0. However, if cv ≥ 03, the truncated normal distribution is more suitable for modeling the demand instead of normal distribution. In addition, the profitability is defined as the probability of achieving the target profit k > 0 under the optimal ordering policy, in which the target profit is set according to the product property and the sales experience.. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. 2.1. Achievable Capacity Index If the selling price and the related cost (shortage, excess, and purchasing/ manufacturing costs per unit) are given, the optimal ordering quantity and the level of profitability depend on the demand mean ED and the demand standard deviation VarD. We develop a new index, which is a function of ED and VarD to express the product’s profitability, and so-called “Achievable Capacity Index (ACI).” For the normal demand, the achievable capacity index IA is defined as follows: ED − T −T IA = = VarD where p the selling price per unit, p > 0; c the purchasing/manufacturing cost per unit, p > c > 0; and T the target demand which is the minimal demand required for achieving profit, i.e., T = k/p − c > 0. The numerator of IA provides the difference between demand mean and target demand. The denominator gives demand standard deviation. Obviously, it is desirable to have an IA as large as possible. 2.2. Interrelationship between Profitability and IA Based on the literature Sankarasubramanian and Kumaraswamy (1983), the profit Z depends on the demand D and the ordering quantity Q, which are formulated as follows: pD − cd Q − D − cQ = cp + ce D − ce Q 0 ≤ D ≤ Q Z= pQ − cs D − Q − cQ = −cs D + cp + cs Q D > Q where cp the net profit per unit i.e., cp = p − c > 0 cd the disposal cost for a surplus product, cd > 0 ce the excess cost per unit i.e., ce = cd + c > 0. and. cs the shortage cost per unit, cs > 0 Note that if the surplus products can be salvaged, the value of cd is negative and redefine into salvage price. It is well known that in order to possibly achieve the.

(6) Assessing Profitability of a Newsboy-Type Product. 3405. target profit, the ordering quantity must be greater than target demand, i.e., Q ≥ T . For any Q ≥ T , Z is strictly increasing in D ∈ 0 Q and strictly decreasing in D ∈ Q , and has a maximum at point D = Q. The maximum value of Z is equal and higher than k, i.e., Z = pD − cQ = cp D = cp Q ≥ cp T = k. The target profit will be realized when D is equal to either LALQ or UALQ, so the target profit will be achieved in D ∈ LALQ UALQ , where. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. LALQ =. cp + cs Q − k ce Q + k and UALQ = cp + c e cs. are the lower and upper achievable limits, respectively, and both are the functions of Q. Under the assumption that the demand is normally distributed, the probability of achieving the target profit is . UALQ − Pr Z ≥ k = . . . LALQ − − . (1). where · is the cumulative distribution function of the standard normal distribution. Before calculating the profitability, we first find the optimal ordering quantity that maximizes Pr Z ≥ k. We take the first-order of Pr Z ≥ k with respect to Q, and obtain 2 cp + cs − 1 UALQ− 2 dPr Z ≥ k ce 1 − 21 LALQ− 2 e − e =√ dQ cs cp + c e 2 It is well known that the necessary condition for Q to be optimal must satisfy the equation dPrZ ≥ k/dQ = 0, which implies =. UALQ + LALQ

(7) 2 − UALQ − LALQ 2. (2). where

(8) = ln 1 + cp A/cs ce and A = cp + ce + cs . For Q ≥ T , we solve Eq. (2), then obtain the unique optimal ordering quantity cs cp + ce cp − k Q∗ = T + + cp cp A + 2ce cs . . cs cp + ce cp − k cp cp A + 2ce cs . 2 +. 2cs2 cp + ce 2

(9) 2 > T cp Acp A + 2ce cs (3). In addition, the sufficient condition is given by. cp + cs d2 Pr Z ≥ k 1 UALQ∗ − = − exp − √ ∗ dQ2 2 2 3 cs2 cp + ce Q=Q UALQ∗ − LALQ∗ cp A + 2ce cs × 2. cp A

(10) 2 + < 0 UALQ∗ − LALQ∗ .

(11) 3406. Su et al.. We can conclude that the stationary point Q∗ is a global maximum. By using Eq. (2) and substituting Eq. (3) into Eq. (1), the profitability, AC, can be obtained as follows:.

(12)

(13)

(14)

(15) − −G + AC = G + 2G 2G where. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. UALQ∗ − LALQ∗ G= =M 2. = MIA + M 2 IA2 + M

(16) > 0. . −T . . +. M2. −T . 2 + M

(17). and M=. cp A > 0 2cp A + 2ce cs . One can easily sees that AC is a function of IA . Taking the first-order derivative of ACIA with respect to IA , we have 1 dACIA MG

(18)

(19) 2

(20) e = − 1 e− 2 G+ 2G > 0 e

(21) + 1 + 2 2 dIA 2G 2 MIA + M

(22) Consequently, ACIA is a strictly increasing function of IA . Therefore, we can express the product’s profitability according to the value of IA , and the value of IA is as large as possible.. 3. Estimating IA Based on Multiple Samples The historical data of the demand ought to be collected in order to estimate the actual IA due to unknown and . For multiple samples of m groups each of size n is given as xi1 xi2 xin , where i = 1 2 m, let x¯ i = nj=1 xij /n and si2 = nj=1 xij − x¯ i 2 /n − 1 be the ith sample mean and sample standard deviation, respectively. We first consider the natural estimator IA which is obtained ¯¯ = m ¯ i /m and sp = by replacing the and by their unbiased estimators x i=1 x 2 1/2 i.e., m i=1 si /m. x¯¯ − T IA = sp Furthermore, the natural estimator IA can be written as ¯. x¯ − √ √ + /−T x¯¯ − T 1 / mn mn =√ IA = × 2 /2 mn−1s sp mn p mn−1 √ 1 1 Z + mnIA Z = √ =√ × × A W W mn mn mn−1. . mn−1. √ √ 2 where ZA = Z + mnIA ∼ N mnIA 1, Z ∼ N0 1, W = mn − 1sp2 /2 ∼ mn−1 . Since ZA and W are independent, the estimator IA is distributed as mn−1/2 tmn−1 ,.

(23) Assessing Profitability of a Newsboy-Type Product. 3407. where tmn−1 is a non central t distribution with mn − 1 degree of freedom and the non-centrality parameter = mn1/2 IA . Since mn − 1/2 1/2 mn − 1 − 1/2. E IA = × IA = IA mn − 1/2. the natural estimator IA is biased. To tackle this problem, we add a correction factor as follows:. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. b=. 2/mn − 1 1/2 mn − 1/2. mn − 1 − 1/2. Then we can obtain unbiased estimator b IA , which is denoted by IA . Since IA is based 2 ¯ solely on the complete and sufficient statistics x¯ sp , it leads to the conclusion that the estimator IA is the uniformly minimum variance unbiased estimator (UMVUE) IA = R is of IA based on multiple samples. The probability density function of derived as follows (for more details, see Appendix A):

(24) mn−1. √ 2 . 2 2mn mn−1 2 vr 1 − bI A

(25) dv vmn−1 exp − + mn − 1v2 fR r = √ 2 b2 /mn 0 b mn−1 2 − < r < Figure 1 plots the probability density function of R, IA = 10, 1.5, 2.0, n = 3, 4, 5, and m = 10, 25, 40 (from bottom to top in plots). From Fig. 1, we can see that (1) for fixed sample sizes m and n, the variance of IA = R increases as IA increases; (2) IA = R decreases as m increases; and (3) for a for a fixed n and IA , the variance of IA = R decreases as n increases. fixed m and IA , the variance of 3.1. Discussion For the case with unequal sample sizes, the natural estimator of IA can straightforwardly be expressed as:. x¯¯ − T IA = s p. . where x¯¯ = m ¯ i /N is the grand mean of the overall sample, N = m i=1 ni x i=1 ni is the 2 m number of observation in the total sample, and s p = i=1 ni − 1si2 /N − m is the pooled sample variance. The estimator IA can be rewritten as ¯. x¯ − −T. √ + √ x¯¯ − T 1 / N / N. = IA = ×. √ s p N −ms 2p /2 N N −m √ 1 1 Z + N IA Z. = √ × =√ × A W. W. N N N −m. √. √. N −m. where Z A = Z + N IA ∼ N N IA 1, W = N − ms 2p /2 ∼ N2 −m . Since Z A and IA is distributed as N−1/2 tN −m , where tN −m W are independent, the estimator .

(26) Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. 3408. Su et al.. Figure 1. Probability density function plots of r for n = 3, 4, 5 and m = 10, 25, 40 (from bottom to top in plots).. is a non central t distribution with N − m degree of freedom and the non-centrality parameter = N1/2 IA . Similarly, we also obtain the unbiased estimator IA , IA = b . 1/2 IA . where b = 2/N − m N − m/2 / N − m − 1/2 is the correction factor of . 4. Testing IA Based on Multiple Samples In order to judge whether the product’s profitability meets the designated requirement, the achievable capacity index IA is adopted to be a criterion. We consider the following hypothesis testing: H0 IA ≤ C vs. H1 IA > C where C is the designated requirement of IA . The critical value is used for making decision in profitability performance testing with designated Type I error (i.e., the chance of incorrectly judging IA ≤ C as IA > C). Since IA is distributed as bmn−1/2 tmn−1 , the critical value, c0 , is determined by: = Pr IA ≥ c0 IA = C √ btmn−1 mnc0 = Pr IA = C ≥ c0 IA = C = Pr tmn−1 ≥ √ b mn.

(27) Assessing Profitability of a Newsboy-Type Product. 3409. Thus, we have c0 =. btmn−1 √ mn. where c0 = tmn−1 is the upper quantile of a non central t distribution with mn − 1 degrees of freedom satisfying Pr tmn−1 ≥ tmn−1 = . If the. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. Table 1 Critical values c0 for = 005, 0.025, 0.01 based on multiple samples with n = 315, m = 10240, and C = 100220 = 005. C = 10. C = 12. C = 14. n. n. n. m. 3. 4. 5. 3. 4. 5. 3. 4. 5. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40. 1.445 1.422 1.402 1.385 1.369 1.355 1.343 1.332 1.321 1.312 1.304 1.296 1.288 1.281 1.275 1.269 1.264 1.258 1.253 1.249 1.244 1.240 1.236 1.232 1.228 1.225 1.222 1.218 1.215 1.212 1.210. 1.367 1.348 1.332 1.318 1.305 1.294 1.284 1.275 1.267 1.259 1.252 1.246 1.240 1.234 1.229 1.224 1.219 1.215 1.211 1.207 1.204 1.200 1.197 1.194 1.191 1.188 1.185 1.182 1.180 1.177 1.175. 1.319 1.303 1.289 1.277 1.267 1.257 1.248 1.240 1.233 1.227 1.221 1.215 1.210 1.205 1.200 1.196 1.192 1.188 1.185 1.182 1.178 1.175 1.172 1.170 1.167 1.165 1.162 1.160 1.158 1.156 1.154. 1.690 1.664 1.642 1.623 1.606 1.590 1.577 1.564 1.553 1.543 1.533 1.524 1.516 1.509 1.502 1.495 1.489 1.483 1.478 1.472 1.468 1.463 1.458 1.454 1.450 1.446 1.443 1.439 1.436 1.433 1.429. 1.601 1.580 1.563 1.547 1.533 1.521 1.510 1.500 1.491 1.483 1.475 1.468 1.461 1.455 1.449 1.444 1.439 1.434 1.430 1.426 1.422 1.418 1.414 1.411 1.407 1.404 1.401 1.398 1.396 1.393 1.391. 1.548 1.530 1.515 1.502 1.490 1.479 1.470 1.461 1.453 1.446 1.440 1.434 1.428 1.423 1.418 1.413 1.409 1.405 1.401 1.397 1.394 1.390 1.387 1.384 1.381 1.379 1.376 1.374 1.371 1.369 1.367. 1.938 1.910 1.885 1.864 1.845 1.828 1.813 1.799 1.787 1.775 1.765 1.755 1.746 1.738 1.730 1.723 1.716 1.710 1.704 1.698 1.693 1.688 1.683 1.678 1.674 1.669 1.665 1.662 1.658 1.654 1.651. 1.838 1.815 1.796 1.778 1.763 1.750 1.738 1.727 1.717 1.708 1.700 1.692 1.685 1.678 1.672 1.666 1.660 1.655 1.650 1.646 1.641 1.637 1.633 1.629 1.626 1.622 1.619 1.616 1.613 1.610 1.607. 1.778 1.759 1.742 1.728 1.715 1.704 1.693 1.684 1.675 1.668 1.660 1.654 1.647 1.642 1.636 1.631 1.627 1.622 1.618 1.614 1.610 1.607 1.603 1.600 1.597 1.594 1.591 1.588 1.586 1.583 1.581. (continued).

(28) 3410. Su et al. Table 1 Continued. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. = 005. C = 16. C = 18. C = 20. n. n. n. m. 3. 4. 5. 3. 4. 5. 3. 4. 5. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40. 2.188 2.157 2.130 2.106 2.086 2.067 2.051 2.036 2.022 2.010 1.998 1.988 1.978 1.969 1.960 1.952 1.945 1.938 1.931 1.925 1.919 1.914 1.908 1.903 1.898 1.894 1.889 1.885 1.881 1.877 1.873. 2.076 2.052 2.030 2.012 1.995 1.981 1.967 1.955 1.945 1.935 1.925 1.917 1.909 1.902 1.895 1.889 1.883 1.877 1.872 1.867 1.862 1.857 1.853 1.849 1.845 1.841 1.838 1.834 1.831 1.828 1.825. 2.011 1.990 1.972 1.956 1.942 1.929 1.918 1.908 1.899 1.890 1.882 1.875 1.868 1.862 1.856 1.851 1.845 1.841 1.836 1.832 1.828 1.824 1.820 1.817 1.813 1.810 1.807 1.804 1.801 1.798 1.796. 2.441 2.406 2.377 2.351 2.328 2.308 2.290 2.274 2.259 2.245 2.233 2.221 2.211 2.201 2.191 2.183 2.175 2.167 2.160 2.153 2.147 2.141 2.135 2.129 2.124 2.119 2.114 2.110 2.105 2.101 2.097. 2.316 2.290 2.266 2.246 2.228 2.212 2.198 2.185 2.173 2.162 2.153 2.143 2.135 2.127 2.120 2.113 2.106 2.100 2.094 2.089 2.084 2.079 2.074 2.070 2.065 2.061 2.057 2.054 2.050 2.047 2.044. 2.244 2.222 2.202 2.185 2.170 2.156 2.144 2.133 2.123 2.114 2.105 2.097 2.090 2.083 2.077 2.071 2.065 2.060 2.055 2.050 2.046 2.042 2.038 2.034 2.030 2.027 2.023 2.020 2.017 2.014 2.011. 2.694 2.657 2.625 2.597 2.572 2.550 2.530 2.513 2.497 2.482 2.468 2.456 2.444 2.434 2.424 2.414 2.405 2.397 2.389 2.382 2.375 2.368 2.362 2.356 2.350 2.345 2.340 2.335 2.330 2.326 2.321. 2.558 2.529 2.504 2.482 2.462 2.445 2.430 2.416 2.403 2.391 2.380 2.371 2.361 2.353 2.345 2.337 2.330 2.324 2.317 2.312 2.306 2.301 2.296 2.291 2.286 2.282 2.278 2.274 2.270 2.266 2.263. 2.479 2.454 2.433 2.415 2.398 2.384 2.371 2.359 2.348 2.338 2.329 2.320 2.312 2.305 2.298 2.292 2.286 2.280 2.275 2.270 2.265 2.260 2.256 2.252 2.248 2.244 2.241 2.237 2.234 2.231 2.228. (continued). observed value of the statistic IA = w is higher than the critical value, the null hypothesis is rejected. We then conclude that the profitability is better than designated requirement with 1 − × 100% confidence level. Note that the p-value can be also adopted for making decisions in this testing, which presents the actual risk of misjudging IA ≤ C as IA > C, i.e., p − value = Pr IA ≥ w IA = C.

(29) Assessing Profitability of a Newsboy-Type Product. 3411. Table 1 Continued. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. = 0025. C = 10. C = 12. C = 14. n. n. n. m. 3. 4. 5. 3. 4. 5. 3. 4. 5. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40. 1.559 1.528 1.502 1.479 1.459 1.441 1.425 1.410 1.397 1.385 1.374 1.364 1.355 1.346 1.338 1.330 1.323 1.317 1.310 1.304 1.299 1.293 1.288 1.284 1.279 1.275 1.270 1.266 1.263 1.259 1.255. 1.455 1.431 1.410 1.392 1.376 1.362 1.349 1.337 1.327 1.317 1.308 1.300 1.293 1.286 1.279 1.273 1.267 1.262 1.257 1.252 1.248 1.243 1.239 1.235 1.231 1.228 1.225 1.221 1.218 1.215 1.212. 1.393 1.373 1.355 1.340 1.326 1.314 1.303 1.293 1.284 1.276 1.269 1.262 1.255 1.249 1.244 1.238 1.233 1.229 1.224 1.220 1.216 1.213 1.209 1.206 1.202 1.199 1.196 1.194 1.191 1.188 1.186. 1.817 1.783 1.753 1.728 1.705 1.685 1.668 1.652 1.637 1.624 1.612 1.600 1.590 1.580 1.571 1.563 1.555 1.548 1.541 1.534 1.528 1.522 1.516 1.511 1.506 1.501 1.497 1.492 1.488 1.484 1.480. 1.698 1.672 1.649 1.629 1.611 1.595 1.581 1.569 1.557 1.547 1.537 1.528 1.520 1.512 1.505 1.498 1.492 1.486 1.480 1.475 1.470 1.465 1.461 1.457 1.452 1.449 1.445 1.441 1.438 1.435 1.431. 1.629 1.607 1.587 1.570 1.555 1.542 1.530 1.519 1.510 1.501 1.492 1.485 1.478 1.471 1.465 1.459 1.454 1.449 1.444 1.439 1.435 1.431 1.427 1.423 1.420 1.417 1.413 1.410 1.407 1.404 1.402. 2.079 2.041 2.008 1.980 1.955 1.933 1.913 1.896 1.879 1.865 1.851 1.839 1.828 1.817 1.807 1.798 1.789 1.781 1.773 1.766 1.759 1.753 1.747 1.741 1.735 1.730 1.725 1.720 1.715 1.711 1.707. 1.945 1.915 1.890 1.868 1.849 1.832 1.816 1.802 1.790 1.778 1.768 1.758 1.749 1.740 1.732 1.725 1.718 1.712 1.705 1.700 1.694 1.689 1.684 1.679 1.675 1.671 1.667 1.663 1.659 1.655 1.652. 1.868 1.843 1.822 1.803 1.787 1.772 1.759 1.748 1.737 1.727 1.718 1.710 1.702 1.695 1.688 1.682 1.676 1.670 1.665 1.660 1.655 1.651 1.647 1.643 1.639 1.635 1.632 1.628 1.625 1.622 1.619. (continued). . btmn−1 ≥ w IA = C √ mn √ w mn = Pr tmn−1 ≥ IA = C b. = Pr. (4). If p-value< , the null hypothesis is rejected. We conclude that the profitability is better than designated requirement with the actual Type I error p-value (rather than.

(30) 3412. Su et al. Table 1 Continued. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. = 0025. C = 16. C = 18. C = 20. n. n. n. m. 3. 4. 5. 3. 4. 5. 3. 4. 5. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40. 2.344 2.301 2.266 2.234 2.207 2.183 2.161 2.142 2.124 2.108 2.093 2.080 2.067 2.055 2.044 2.034 2.025 2.016 2.008 2.000 1.992 1.985 1.978 1.972 1.966 1.960 1.954 1.949 1.944 1.939 1.934. 2.194 2.162 2.134 2.110 2.089 2.070 2.053 2.038 2.024 2.011 2.000 1.989 1.979 1.970 1.961 1.953 1.946 1.939 1.932 1.926 1.920 1.914 1.909 1.904 1.899 1.894 1.890 1.885 1.881 1.877 1.874. 2.108 2.081 2.058 2.038 2.020 2.004 1.990 1.977 1.965 1.955 1.945 1.936 1.927 1.920 1.912 1.905 1.899 1.893 1.887 1.882 1.877 1.872 1.867 1.863 1.859 1.855 1.851 1.847 1.844 1.841 1.837. 2.610 2.564 2.525 2.491 2.461 2.434 2.411 2.389 2.370 2.353 2.336 2.322 2.308 2.295 2.283 2.272 2.262 2.252 2.243 2.234 2.226 2.219 2.211 2.204 2.198 2.191 2.185 2.179 2.174 2.169 2.163. 2.445 2.409 2.379 2.353 2.330 2.310 2.291 2.275 2.260 2.246 2.233 2.222 2.211 2.201 2.192 2.183 2.175 2.167 2.160 2.153 2.146 2.140 2.134 2.129 2.124 2.119 2.114 2.109 2.105 2.100 2.096. 2.350 2.321 2.296 2.274 2.255 2.238 2.222 2.208 2.195 2.184 2.173 2.163 2.154 2.146 2.138 2.130 2.123 2.117 2.111 2.105 2.099 2.094 2.089 2.084 2.080 2.076 2.071 2.067 2.064 2.060 2.057. 2.879 2.829 2.786 2.749 2.716 2.687 2.662 2.639 2.618 2.598 2.581 2.565 2.550 2.536 2.523 2.511 2.500 2.490 2.480 2.470 2.461 2.453 2.445 2.437 2.430 2.423 2.417 2.411 2.405 2.399 2.393. 2.697 2.659 2.626 2.598 2.573 2.551 2.531 2.513 2.496 2.482 2.468 2.455 2.444 2.433 2.423 2.413 2.404 2.396 2.388 2.381 2.374 2.367 2.361 2.355 2.349 2.344 2.339 2.334 2.329 2.324 2.320. 2.594 2.562 2.535 2.511 2.490 2.472 2.455 2.440 2.426 2.414 2.402 2.392 2.382 2.373 2.364 2.356 2.349 2.341 2.335 2.328 2.323 2.317 2.312 2.306 2.302 2.297 2.292 2.288 2.284 2.280 2.276. (continued). . Table 1 displays the critical values for = 005 0025 001 based on multiple samples n = 315, m = 10240, and IA = 100220. Next, we also calculate the risk. Once the sample size and the risk are defined, the power function PowerIA may be expressed by: PowerIA = Pr IA ≥ c0 IA > C.

(31) Assessing Profitability of a Newsboy-Type Product. 3413. Table 1 Continued. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. = 001. C = 10. C = 12. C = 14. n. n. n. m. 3. 4. 5. 3. 4. 5. 3. 4. 5. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40. 1.705 1.663 1.628 1.598 1.571 1.548 1.527 1.508 1.491 1.476 1.462 1.449 1.437 1.425 1.415 1.405 1.396 1.388 1.380 1.372 1.365 1.359 1.352 1.346 1.340 1.335 1.330 1.325 1.320 1.315 1.311. 1.564 1.533 1.506 1.483 1.463 1.444 1.428 1.414 1.400 1.388 1.377 1.367 1.357 1.348 1.340 1.333 1.325 1.319 1.312 1.306 1.301 1.295 1.290 1.285 1.281 1.276 1.272 1.268 1.264 1.260 1.257. 1.484 1.458 1.436 1.416 1.399 1.384 1.370 1.358 1.347 1.336 1.327 1.318 1.310 1.303 1.296 1.289 1.283 1.277 1.272 1.267 1.262 1.257 1.253 1.249 1.245 1.241 1.237 1.234 1.230 1.227 1.224. 1.980 1.934 1.894 1.860 1.831 1.805 1.781 1.760 1.742 1.724 1.709 1.694 1.681 1.668 1.657 1.646 1.636 1.627 1.618 1.610 1.602 1.594 1.587 1.581 1.574 1.568 1.562 1.557 1.551 1.546 1.541. 1.820 1.785 1.756 1.730 1.707 1.687 1.669 1.653 1.638 1.625 1.613 1.601 1.591 1.581 1.572 1.564 1.556 1.548 1.541 1.535 1.529 1.523 1.517 1.512 1.507 1.502 1.497 1.493 1.488 1.484 1.480. 1.730 1.701 1.676 1.655 1.636 1.619 1.604 1.590 1.578 1.567 1.556 1.547 1.538 1.530 1.522 1.515 1.508 1.502 1.496 1.490 1.485 1.480 1.475 1.470 1.466 1.462 1.458 1.454 1.451 1.447 1.444. 2.260 2.208 2.165 2.127 2.094 2.065 2.039 2.016 1.995 1.976 1.959 1.943 1.928 1.914 1.902 1.890 1.879 1.868 1.859 1.849 1.841 1.833 1.825 1.817 1.810 1.804 1.797 1.791 1.785 1.780 1.774. 2.079 2.041 2.008 1.980 1.955 1.933 1.913 1.895 1.879 1.864 1.851 1.838 1.827 1.816 1.806 1.797 1.788 1.780 1.772 1.765 1.758 1.752 1.746 1.740 1.734 1.729 1.724 1.719 1.714 1.710 1.706. 1.978 1.947 1.919 1.896 1.875 1.857 1.840 1.825 1.812 1.799 1.788 1.777 1.768 1.759 1.750 1.742 1.735 1.728 1.722 1.716 1.710 1.704 1.699 1.694 1.689 1.685 1.680 1.676 1.672 1.669 1.665. (continued). . btmn−1 ≥ c0 IA > C √ mn √ c0 mn = Pr tmn−1 ≥ IA > C b = Pr. The power of the test for C = 10 14 18 vs. various values of IA , n = 3 4 5, m = 101040, and = 005 is showed in Figure 2. It is seen that the larger the sample size, the larger the power of test, and consequently, the smaller the risk..

(32) 3414. Su et al. Table 1 Continued. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. = 001. C = 16. C = 18. C = 20. n. n. n. m. 3. 4. 5. 3. 4. 5. 3. 4. 5. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40. 2.543 2.486 2.438 2.396 2.360 2.328 2.300 2.274 2.251 2.230 2.211 2.194 2.177 2.162 2.148 2.135 2.123 2.112 2.101 2.091 2.082 2.073 2.064 2.056 2.048 2.041 2.034 2.027 2.021 2.015 2.009. 2.341 2.299 2.263 2.232 2.205 2.180 2.159 2.139 2.122 2.105 2.091 2.077 2.065 2.053 2.042 2.032 2.023 2.014 2.005 1.997 1.990 1.983 1.976 1.970 1.964 1.958 1.952 1.947 1.942 1.937 1.932. 2.229 2.195 2.165 2.139 2.116 2.096 2.078 2.062 2.047 2.034 2.021 2.010 1.999 1.989 1.980 1.972 1.964 1.956 1.949 1.942 1.936 1.930 1.924 1.919 1.914 1.909 1.904 1.900 1.895 1.891 1.887. 2.829 2.767 2.714 2.668 2.629 2.594 2.562 2.534 2.509 2.486 2.465 2.446 2.429 2.412 2.397 2.383 2.369 2.357 2.345 2.334 2.324 2.314 2.305 2.296 2.287 2.279 2.272 2.264 2.257 2.251 2.244. 2.606 2.560 2.520 2.486 2.456 2.430 2.406 2.385 2.366 2.348 2.332 2.318 2.304 2.291 2.279 2.268 2.258 2.248 2.239 2.231 2.223 2.215 2.208 2.201 2.194 2.188 2.182 2.176 2.170 2.165 2.160. 2.482 2.444 2.412 2.384 2.359 2.337 2.318 2.300 2.284 2.269 2.256 2.244 2.232 2.221 2.212 2.202 2.194 2.185 2.178 2.170 2.164 2.157 2.151 2.145 2.139 2.134 2.129 2.124 2.119 2.115 2.111. 3.117 3.049 2.991 2.942 2.899 2.860 2.827 2.796 2.769 2.744 2.721 2.700 2.681 2.663 2.647 2.631 2.617 2.603 2.591 2.579 2.567 2.557 2.547 2.537 2.528 2.519 2.511 2.503 2.495 2.488 2.481. 2.872 2.822 2.779 2.742 2.710 2.681 2.656 2.633 2.612 2.593 2.575 2.559 2.544 2.531 2.518 2.506 2.495 2.484 2.474 2.465 2.456 2.448 2.440 2.433 2.425 2.419 2.412 2.406 2.400 2.394 2.389. 2.737 2.696 2.661 2.630 2.604 2.580 2.559 2.540 2.522 2.506 2.492 2.478 2.466 2.455 2.444 2.434 2.424 2.416 2.407 2.399 2.392 2.385 2.378 2.372 2.366 2.360 2.355 2.349 2.344 2.339 2.335. 4.1. Profitability Evaluation Procedure In the following, we develop a simple step-by-step procedure for the practitioners to use for judging whether the profitability meets the designated requirement. Step 1. Determine the value of the designated requirement C, -risk, and sample size m n. Step 2. Calculate the value of the estimator, IA , form the given sample..

(33) Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. Assessing Profitability of a Newsboy-Type Product. Figure 2. 30, 40.. 3415. Power curves for C = 10, 1.4, 1.8, with sample sizes n = 3, 4, 5 and m = 10, 20,. Step 3. Find the corresponding critical value, IA , based on , C, m, and n form the Table 1. Also, we calculate the p-value from the Eq. (4) based on C, m, and n. Step 4. Conclude that the profitability meets the designated requirement if IA > c0 (or p-value < ). Otherwise, the profitability does not meet the designated requirement.. 5. Application Example We consider a dessert store, which provides delicious donuts made fresh daily in Taipei, Taiwan. This store is a Japanese-owned incarnation of a donut franchise formerly out of America. Fifty varieties of donuts are offered, one half of them are American style and another half of them are Japanese style. All of the donuts range from NT$20–35. Besides, each donut comes with a label indicating its level of sweetness. However, these donuts only have approximate 12-h shelf-life due to texture deterioration. In order to provide the best texture, this store prepares the donut each day and disposes the overdue donuts after closing store. If the.

(34) 3416. Su et al.. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. Table 2 The profitability for p c cd cs k = 25 10 1 3 2500 and IA = 000001309 IA. 0.00. 0.01. 0.02. 0.03. 0.04. 0.05. 0.06. 0.07. 0.08. 0.09. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0. 0.4249 0.4617 0.4987 0.5357 0.5723 0.6082 0.6431 0.6768 0.7090 0.7395 0.7683 0.7951 0.8200 0.8428 0.8636 0.8824 0.8992 0.9142 0.9275 0.9391 0.9492 0.9579 0.9653 0.9716 0.9769 0.9814 0.9851 0.9881 0.9906 0.9926 0.9943. 0.4285 0.4654 0.5024 0.5394 0.5759 0.6117 0.6465 0.6800 0.7121 0.7425 0.7710 0.7977 0.8223 0.8449 0.8655 0.8841 0.9008 0.9156 0.9287 0.9401 0.9501 0.9587 0.9660 0.9722 0.9774 0.9818 0.9854 0.9884 0.9908 0.9928 0.9944. 0.4322 0.4691 0.5061 0.5431 0.5795 0.6152 0.6499 0.6833 0.7152 0.7454 0.7738 0.8002 0.8247 0.8471 0.8675 0.8859 0.9024 0.9170 0.9299 0.9412 0.9510 0.9594 0.9666 0.9728 0.9779 0.9822 0.9857 0.9887 0.9911 0.9930 0.9945. 0.4359 0.4728 0.5099 0.5467 0.5831 0.6188 0.6533 0.6866 0.7183 0.7483 0.7765 0.8028 0.8270 0.8492 0.8694 0.8876 0.9039 0.9184 0.9311 0.9423 0.9519 0.9602 0.9673 0.9733 0.9784 0.9826 0.9861 0.9889 0.9913 0.9932 0.9947. 0.4395 0.4765 0.5136 0.5504 0.5868 0.6223 0.6567 0.6898 0.7214 0.7512 0.7792 0.8053 0.8293 0.8513 0.8713 0.8893 0.9054 0.9197 0.9323 0.9433 0.9528 0.9610 0.9680 0.9739 0.9788 0.9830 0.9864 0.9892 0.9915 0.9933 0.9948. 0.4432 0.4802 0.5173 0.5541 0.5903 0.6258 0.6601 0.6931 0.7245 0.7541 0.7819 0.8078 0.8316 0.8534 0.8732 0.8910 0.9070 0.9211 0.9335 0.9443 0.9537 0.9617 0.9686 0.9744 0.9793 0.9833 0.9867 0.9894 0.9917 0.9935 0.9949. 0.4469 0.4839 0.5210 0.5577 0.5939 0.6293 0.6634 0.6963 0.7275 0.7570 0.7846 0.8103 0.8339 0.8555 0.8751 0.8927 0.9084 0.9224 0.9346 0.9453 0.9545 0.9625 0.9692 0.9749 0.9797 0.9837 0.9870 0.9897 0.9919 0.9937 0.9951. 0.4506 0.4876 0.5246 0.5614 0.5975 0.6327 0.6668 0.6995 0.7305 0.7598 0.7873 0.8127 0.8361 0.8575 0.8769 0.8944 0.9099 0.9237 0.9358 0.9463 0.9554 0.9632 0.9698 0.9754 0.9801 0.9841 0.9873 0.9899 0.9921 0.9938 0.9952. 0.4543 0.4913 0.5283 0.5650 0.6011 0.6362 0.6701 0.7026 0.7335 0.7627 0.7899 0.8151 0.8384 0.8596 0.8788 0.8960 0.9114 0.9250 0.9369 0.9473 0.9562 0.9639 0.9704 0.9759 0.9806 0.9844 0.9876 0.9902 0.9923 0.9940 0.9953. 0.4580 0.4950 0.5320 0.5687 0.6046 0.6396 0.6734 0.7058 0.7365 0.7655 0.7925 0.8176 0.8406 0.8616 0.8806 0.8976 0.9128 0.9262 0.9380 0.9482 0.9570 0.9646 0.9710 0.9764 0.9810 0.9848 0.9879 0.9904 0.9925 0.9941 0.9954. manufacturing quantity can not satisfy the demand, then the manager must pay the lost sale opportunity cost. Therefore, the donut exactly belongs to the newsboytype product. Now, the manager would like to know whether the profitability of the designated donut is higher than some level. If it is incapable, the manager is going to plan a sale promotion. The selling price of the donut is NT$25 per unit, the manufacturing cost is NT$10 per unit, and the target profit is NT$2500. In addition, the lost sale opportunity cost is NT$3 per unit. The disposal cost for overdue donut is NT$1 per unit. Table 2 displays the profitability for p c cd cs k = 25 10 1 3 2500 and IA = 000001309. For the demand data, because of Saturday and Sunday are always have high demand. In order to avoid these extreme values, we only consider the demand on Monday-Friday. Note that the unsatisfied demand is record. Twenty samples of size five (i.e., 20-weeks demand).

(35) Assessing Profitability of a Newsboy-Type Product. 3417. Table 3 The 5-sample data each of 20 observations Demand units/day. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. Group (Week) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20. Observations in sample of size five MON. TUE. WED. THU. FRI. 185 221 208 224 202 189 219 188 189 215 178 183 221 172 191 187 176 199 187 192. 169 220 213 195 218 198 196 215 206 225 173 214 194 217 199 223 205 184 183 178. 189 191 217 208 208 212 190 188 194 198 186 244 187 205 183 183 211 235 206 210. 201 180 212 214 197 204 229 191 191 191 224 212 194 216 196 219 216 186 212 180. 192 203 196 224 189 225 198 185 186 212 212 217 174 214 179 198 198 184 203 195. are displayed in Table 3. Due to the store’s propertied restriction, the prices, costs, and sample data were modified. If the designated requirement of the IA value is C = 18, we implement the hypothesis testing: H0 IA ≤ 18 vs. H1 IA > 18. We first use the Kolmogorov-Smirnov test for the sample data from Table 3 to confirm if the data is normally distributed. A test result in p-value> 0.05, which means that data is normally distributed. For the data displayed in Table 3, we calculate the overall sample mean, pooled sample variance, and sample estimator, and obtain that IA = 21753. If the Type I error -risk set to 0.05, the x¯¯ = 20048, sp2 = 23710, and critical value is 2.1050 form Table 1. Since IA = 21753 > 21050 = c0 , we conclude that the profitability meets the designated requirement, than it is unnecessary to plan a sale promotion. For calculating the p-value, we obtain p-value=0.0244<0.05. Therefore, it suggests the same evaluation result.. 6. Conclusions In this article, we developed a new index, achievable capacity index, IA , which has a simple-form to measure the profitability of the newsboy-type product with normally distributed demand. In practical application, the demand data is collected from multiple samples rather than single sample. Hence, we considered an unbiased and effective estimator of IA based on multiple samples. The evaluation testing of IA is investigated, i.e., H0 IA ≤ C vs. H1 IA > C, where C is the designated requirement.

(36) 3418. Su et al.. of IA . The critical value of the test is calculated to determine evaluation result under the preset risk (Type I error). The implementation of the existing statistical theory for the profitability of Newsboy-type product makes it possible to apply the complicated theoretical results to the actual productions. For convenience, we also provided a simple step-by-step procedure for the practitioners to use in making decisions. Finally, a real-world example on the sales of donuts is presented to illustrate the practicality of the exact approach.. Downloaded by [National Chiao Tung University ] at 07:24 24 April 2014. Appendix A We first define R = bx¯¯ − T/sp = Y/V , where Y = bx¯¯ − T/ and V = sp /. It is easy to see that if the demand is normally distributed, D ∼ N 2 , we have Y ∼ Nb − T/ b2 /mn. Since mn − 1sp2 /2 follows the chi-squared distribution with mn − 1 degree of freedom, we then have V 2 ∼ mn − 1/2 2/mn − 1. By using the technique of change-of-variable, the probability density function of V is derived as follows: 2vmn−1−1 mn − 1 2 fV v = exp − v

(37) mn−1

(38) 2 2 2 mn−1 2 mn−1 Because Y and V are independent continuous random variables, the probability density function of R can be obtained by the Jacobian approach, i.e., fR r =. . 0. fY vrfV v v dv.

(39) mn−1. √ 2 . 2mn mn−1 2 1 vr − bIA 2 mn−1 2

(40) = dv v exp − + mn − 1v √ 2 b2 /mn 0 b mn−1 2. References Alfares, H. K., Elmorra, H. H. (2005). The distribution-free newsboy problem: Extensions to the shortage penalty case. Int. J. Prod. Econ. 93–94:465–477. Berk, E., Gurler, U., Levine, R. A. (2007). Bayesian demand updating in the lost sales newsvendor problem: a two-moment approximation. Eur. J. Oper. Res. 182:256–281. Ismail, B., Louderback, J. (1979). Optimizing and satisfying in stochastic cost-volume profit analysis. Decis. Sci. 10:205–217. Kevork, I. S. (2010). Estimating the optimal order quantity and the maximum expected profit for single-period inventory decisions. Omega 38:218–227. Khouja, M. (1995). The newsboy problem under progressive multiple discounts. Eur. J. Oper. Res. 84:458–466. Lau, H. (1997). Simple formulas for the expected costs in the newsboy problem: An educational note. Eur. J. Oper. Res. 100:557–561. Lau, H. S. (1980). The newsboy problem under alternative optimization objectives. J. Oper. Res. Soc. 31:525–535. Moon, I., Choi, S. (1995). The distribution free newsboy problem with balking. J. Oper. Res. Soc. 46:537–542. Mostard, J., Koster, R., Teunter, R. (2005). The distribution-free newsboy problem with resalable returns. Int. J. Prod. Econ. 97:329–342..

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