This paper first defines the profitability to be the probability of achieving a target profit under the optimal ordering policy, and introduces a new index (achievable capacity index; IA) which can briefly analyze the profitability for newsboy-type product with normally distributed demand. Note that since the level of profitability depends on the demand mean μ and the demand standard deviation σ if the related costs, selling price, and target profit are given, the index IA is a function of μ and σ. Then, we assess level performance which examines if the profitability meets designated requirement. The results can determine whether the product is still desirable to order/manufacture. However, μ and σ are always unknown, and the demand quantity is common to be imprecise, especially for new product. To tackle these problems, a constructive approach combining the vector of fuzzy numbers is introduced to establish the membership function of the fuzzy estimator of IA. Furthermore, a three-decision testing rule and step-by-step procedure are developed to assess level performance based on fuzzy critical values and fuzzy p-values.
The classical newsboy problem (single-period problem) deals with the purchasing inventory problem for short shelf-life products with the uncertainty of demand. For such problems, the managers should determine ordering quantity at the beginning of each period. Products cannot be sold in the next period and need additional cost (excess cost) to dispose it if the ordering quantity exceeds actual demand. Therefore, the determination of the ordering quantity is critical in the classical newsboy problem. Several extensions to the newsboy problem have been proposed and solved in the literature. Among those extensions are alternative objective functions such as minimizing the expected cost (Nahmias, 1993), maximizing the expected profit (Khouja, 1995), maximizing the expected utility (Ismail and Louderback, 1979 and Lau, 1980), and maximizing the probability of achieving a target profit (Ismail and Louderback, 1979, Khouja, 1996, Lau, 1980, Sankarasubramanian and Kumaraswamy, 1983 and Shih, 1979). In fact, these maximum and minimum values can be adopted to measure product’s capacity. For example, the maximum expected profit and maximum probability of achieving the target profit can measure product’s profitability.
In this paper, we consider the newsboy-type product with normally distributed demand and assume that the profitability is defined to be the probability of achieving a target profit under the optimal ordering policy. Furthermore, in order to simplify the calculation, we develop a new index, which has a simple form and can correspond to the profitability, and so-called “achievable capacity index (ACI)”, and be denoted by I A. Note that since the level of profitability depends on the demand mean μ and the demand standard deviation σ if the related costs, selling price, and target profit are given, the index I A is a function of μ and σ . Then, we assess level performance which examines if the profitability meets designated requirement. However, μ and σ are always unknown. To tackle this problem, one should collect the historical data of demand, and then implement the following hypothesis testing, H0:IA⩽CH0:IA⩽C versus H1:IA>CH1:IA>C, where C is a designated requirement. Critical value of the test must be calculated to determine the results. The results can determine whether the product is still desirable to order/manufacture. But in practice, especially for new product, the demand quantity is difficult to acquire due to lack of information and historical data. In this case, the demand quantity is approximately specified based on the experience. Some papers have dealt with this case by applying fuzzy theory. Petrovic, Petrovic, and Vujosevic (1996) first proposed a newsboy-type problem with discrete fuzzy demand. Dutta, Chakraborty, and Roy (2007) studied the newsboy problem with reordering opportunities under fuzzy demand. Zhen and Xiaoyu (2006) considered the multi-product newsboy problem with fuzzy demands under budget constraint. Kao and Hsu (2002) compared the area of fuzzy numbers to obtain the optimal order quantity. To the best of our knowledge, no researchers have investigated the fuzzy hypothesis testing for assessing level performance. In this study, we first use a new approach in fuzzy statistics to estimate the demand mean and variance parameters of normal distribution ( Buckley, 2004, Buckley, 2005a and Buckley, 2005b). Then, a general method combining the vector of fuzzy numbers of sample mean View the MathML sourcex¯˜, and sample variance View the MathML sources˜2 is proposed to derive the membership function of the fuzzy estimator of IA. Furthermore, a three-decision testing rule for assessing level performance according to two different criteria, critical value and fuzzy p-value are proposed. Based on the test, we develop a step-by-step procedure for managers to use so that decisions made in examining the profitability are more reliable. The rest of the paper is organized as follows. In the next section, we calculate the profitability, and develop a new index IA to correspond profitability. Section 3 discusses the statistical properties of estimation for IA based on crisp data. In Section 4, we present some basic definitions, notations of fuzzy sets and the α-cuts of fuzzy estimation for IA. Section 5 deals with implementing fuzzy hypothesis testing for assessing level performance. Following critical value and fuzzy p-value, decision rules and testing procedures are developed. In Section 6, a numerical example is discussed to illustrate the procedure of solving the problem. Some conclusions are given in the final section.
