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We consider an extension of the standard newsvendor problem by allowing for multiple classes of customers. The product is first sold to customers with the highest priority, and the remaining units (if any) are sold at a discounted price to customers in decreasing order of priority until all classes of customers have been served, limited only by the available stock. Unsold items, if any, have a salvage value. The demands of different priority customers are independent random variables with known probability distributions. The problem is to find the purchase quantity that maximizes the expected profit. We show that this problem actually reduces to the standard newsvendor problem with the demand distribution being a mixture of the input demand distributions. Since this mixture of distributions is typically hard to handle analytically, we propose a simple general heuristic which can be implemented using different types of distributions. Some of these implementations produce near optimal solutions. We tested these implementations for the case of two demand classes of customers and found that they outperform previously published heuristics in almost all instances. We suggest applications for this model in the Chinese pharmaceutical industry, apparel industry, and perishable goods among others. We also propose an extension involving shortage cost.

Arrow et al. [

There have been many extensions to the newsvendor problem [^{th} class of customers before the remaining units (if any) are sold at a discounted price ^{th }class of customers. Thus, the newsvendor sells newspapers starting at the highest price location, and will move to the next highest price location after meeting the realized demand at the current location. For example, the price charged in the morning is higher than that in the afternoon. The model we examine is common, for example, in the apparel industry and retailing of perishable goods where discounts are used to sell excess inventory; see [

The main theoretical result of this paper is that the newsvendor problem with multiple demand classes characterized by decreasing prices actually reduces to the standard newsvendor problem with the demand distribution being a mixture of the input distributions, the selling price

The paper is organized as follows. In Section 2, we provide some additional motivations related to the main problem addressed in this paper, and in Section 3 we formulate the problem and show its reduction to the standard newsvendor problem. In Section 4, we describe our proposed heuristics; their performance is empirically examined in Section 5. Section 6 indicates some extensions to the problem, while Section 7 presents some concluding remarks.

First, we discuss the motivation generated by the practices we gleaned through our recent survey of the Chinese Pharmaceutical industry in Central China. Generally, the customers of a pharmaceutical distributor in China are classified into two types: Hospitals (including clinics) and retail pharmaceutical franchisees (Pharmacies). In China, hospitals and pharmacies are separate and both play important roles in the Pharmaceutical Supply Chain [

This model is also applicable for annual order placements by US manufactures to their Asian contract manufactures for apparels and fashion items, as well as for the general purchase of perishable items by the organizations which have multiple classes of customers with fixed known prices.

The general newsvendor problem with decreasing priority demand classes of customers can be formulated as follows. A product with a unit cost of c is sold to n classes of customers in a sequential order. It is sold first to the first class of costumers at a price of^{th} class of customers and

Therefore, letting

Since for any random variable Y whose CDF G has a finite mean,

where

The first derivative of

and the second derivative is obviously non-positive. Therefore, the expected profit

the optimal purchase quantity q^{*} is a solution to the equation^{*} satisfies

Letting now

Since

the newsvendor problem with n decreasing priority demand classes of customers actually reduces to the standard newsvendor problem with the demand CDF G, the unit selling price

Şen and Zhang [

H1. Define the standard newsvendor problem with the demand

H2. Solve separately n standard newsvendor problems with the demands

up the obtained purchase quantities. Thus,

We have shown above that the problem under study can be reduced to a specific standard newsvendor prob-

lem, and the optimal purchase quantity is

CDF G cannot be assumed to have a closed analytical form, and hence its inverse

mined. Also, the equation

Therefore, the use of heuristics is fully justified. Although G is hard to handle, its moments (around zero) are

easily determined. Clearly, if

G are:

and

We propose a general heuristic named H3, which can be implemented by replacing the mixture CDF G by a more tractable CDF

we developed four implementations of H3, named H3N, H3L, H3G and H3W, in which

For a given heuristic H that yields the purchase quantity

For every conducted experiment, we computed the average and maximum relative percentage errors ARPE (H) and MRPE (H).

We have limited our experiments to the case n = 2, though they are easily extendable to any value of n. The demands

All numerical computations were performed using MS Excel. The optimal purchase quantity

to which (3) is reduced for n = 2. Excel Solver was also used to determine

case of exponentially distributed demands. For normally distributed demands, we employed Excel function NORM.INV to find both

Whenever the expected profit,

(assume n = 2 in (1)) could not be analytically determined, we applied the Simpson method for approximating

the integrals

We reconsidered the 240 instances defined in Şen and Zhang [

It should be added here that the presented errors are strongly biased upward by some rather unrealistic instances for which

We found that in general the heuristics H3N and H3G with normal and gamma distributions for

Heuristics | ARPE | MRPE |
---|---|---|

H1 | 22.91% | 100.00% |

H2 | 2.91% | 36.84% |

H3N | 2.00% | 28.65% |

H3G | 1.71% | 29.89% |

H3L | 2.03% | 38.96% |

H3W | 3.48% | 49.48% |

In all of our simulation experiments, it is assumed that there are only two classes of customers and_{2} and c, we generated two additional data sets. For Dataset 2, the average values of

In order to have comparable results, we used the same 100 mean demands

Let be the mean relative percentage error induced by heuristic H for a particular simulation experiment conducted on 100 problem instances. We are interested in testing the following null and alternative hypotheses:

Since all of the heuristics are applied on the same instances, we have a dependent (matched) sampling. We tried to apply the ANOVA test for a randomized block design. Unfortunately, we were unable to verify the needed normality assumptions in any of the simulation experiments. Consequently, we turned to the non-para- metric Friedman test followed by the non-parametric HSD (honestly significance difference) Tukey’s test. For each simulation experiment, the null hypothesis stated above was rejected by Friedman’s test with a p-value of virtually zero in all of the 21 experiments. Therefore, below we present only the results of Tukey’s multiple comparison test conducted at a 5% significance level. For example, the notation

First, we consider normally distributed demands _{1} and X_{2} were assumed to have the same coefficient of variation_{1} and μ_{2}, the standard deviations were

Next, we consider uniformly distributed demands X_{1} and X_{2}with means μ_{1} and μ_{2},and the same range 2d.

Since the coefficient of variation for the uniform distribution on the interval _{1} and X_{2} is

Heuristic | CV = 1/2 | CV = 1/3 | CV = 1/4 | |||
---|---|---|---|---|---|---|

ARPE | MRPE | ARPE | MRPE | ARPE | MRPE | |

H1 | 0.94% | 8.67% | 0.93% | 6.62% | 0.79% | 6.08% |

H2 | 1.41% | 21.86% | 0.59% | 2.60% | 0.42% | 1.28% |

H3N | 0.08% | 1.16% | 0.11% | 0.95% | 0.29% | 2.62% |

H3G | 0.23% | 1.36% | 0.36% | 2.89% | 0.63% | 5.02% |

Tukey test | {H3N,H3G} ≺ {H1,H2} | {H3N,H3G} ≺ {H2} ≺ {H1} | {H3N,H2} ≺ {H1,H3G} |

Heuristic | CV = 1/2 | CV = 1/3 | CV = 1/4 | ||||
---|---|---|---|---|---|---|---|

ARPE | MRPE | ARPE | MRPE | ARPE | MRPE | ||

H1 | 1.19% | 7.27% | 1.60% | 8.27% | 1.48% | 6.41% | |

H2 | 5.25% | 49.87% | 0.91% | 5.88% | 0.39% | 1.94% | |

H3N | 0.21% | 5.06% | 0.12% | 1.72% | 0.43% | 2.20% | |

H3G | 0.22% | 1.41% | 0.33% | 2.00% | 0.80% | 3.68% | |

Tukey test | {H1,H3N,H3G} ≺ {H2} | {H3N,H3G} ≺ {H2} ≺ {H1} | {H2,H3N} ≺ {H3G} ≺ {H1} | ||||

Heuristic | CV = 1/2 | CV = 1/3 | CV = 1/4 | |||
---|---|---|---|---|---|---|

ARPE | MRPE | ARPE | MRPE | ARPE | MRPE | |

H1 | 2.65% | 9.09% | 3.29% | 9.21% | 3.03% | 7.65% |

H2 | 7.83% | 72.43% | 1.23% | 18.53% | 0.52% | 5.71% |

H3N | 0.42% | 4.54% | 0.20% | 1.87% | 0.32% | 1.59% |

H3G | 0.21% | 2.51% | 0.22% | 2.02% | 0.53% | 3.89% |

Tukey test | {H1,H3N,H3G} ≺ {H2} | {H3N,H3G} ≺ {H2} ≺ {H1} | {H2,H3N,H3G} {H1} |

are presented in Tables 5-7. It may be noted that H3N dominates all other heuristics for all datasets and CV values reported. H3G falls in the best performing heuristics group in all but three cases.

Finally, we consider exponentially distributed demands X_{1} and X_{2} with means

In the standard newsvendor problem, no shortage cost is assumed if the purchased quantity is less than the demand. Although this cost might be difficult to define in practice, the authors of [^{th} class, the expected profit is:

Heuristic | CV = 1/2 | CV = 1/3 | CV = 1/4 | |||
---|---|---|---|---|---|---|

ARPE | MRPE | ARPE | MRPE | ARPE | MRPE | |

H1 | 1.00% | 7.44% | 1.01% | 7.24% | 0.84% | 6.78% |

H2 | 2.24% | 14.49% | 1.05% | 3.34% | 0.74% | 2.09% |

H3N | 0.07% | 0.51% | 0.16% | 2.05% | 0.37% | 4.17% |

H3G | 0.27% | 2.02% | 0.40% | 4.30% | 0.71% | 6.87% |

Tukeytest | {H3N,H3G} ≺ {H1} ≺ {H2} | {H3N,H3G} ≺ {H1,H2} | {H3N} ≺ {H1,H2,H3G} |

Heuristic | CV = 1/2 | CV = 1/3 | CV = 1/4 | |||
---|---|---|---|---|---|---|

ARPE | MRPE | ARPE | MRPE | ARPE | MRPE | |

H1 | 0.93% | 5.81% | 1.59% | 7.07% | 1.54% | 6.01% |

H2 | 6.66% | 20.03% | 1.80% | 7.35% | 0.85% | 4.37% |

H3N | 0.10% | 0.41% | 0.14% | 1.91% | 0.50% | 5.62% |

H3G | 0.22% | 0.79% | 0.33% | 2.40% | 0.85% | 5.70% |

Tukeytest | {H1,H3N,H3G} ≺ {H2} | {H3N,H3G} ≺ {H1,H2} | {H2,H3N,H3G} ≺ {H1} |

Heuristic | CV = 1/2 | CV = 1/3 | CV = 1/4 | |||
---|---|---|---|---|---|---|

ARPE | MRPE | ARPE | MRPE | ARPE | MRPE | |

H1 | 2.42% | 8.38% | 3.50% | 9.16% | 3.21% | 7.27% |

H2 | 5.11% | 23.44% | 1.29% | 7.35% | 0.81% | 3.55% |

H3N | 0.15% | 0.92% | 0.23% | 1.12% | 0.47% | 3.26% |

H3G | 0.27% | 0.82% | 0.24% | 2.65% | 0.64% | 4.80% |

Tukeytest | {H3N,H3G} ≺ {H1} {H2} | {H3N,H3G} ≺ {H2} ≺ {H1} | {H2,H3N,H3G} ≺ {H1} |

Heuristic | Dataset 1 | Dataset 2 | Dataset 3 | |||
---|---|---|---|---|---|---|

ARPE | MRPE | ARPE | MRPE | ARPE | MRPE | |

H1 | 0.27% | 5.16% | 0.25% | 4.15% | 0.56% | 5.17% |

H2 | 4.32% | 32.19% | 16.38% | 41.45% | 13.42% | 47.46% |

H3N | 2.35% | 10.40% | 3.53% | 10.83% | 4.63% | 15.64% |

H3G | 0.01% | 0.22% | 0.01% | 0.18% | 0.01% | 0.13% |

Tukeytest | {H1,H3G} ≺ {H3N} ≺ {H2} | {H1,H3G} ≺ {H3N} ≺ {H2} | {H1,H3G} ≺ {H3N} ≺ {H2} |

Consequently, the optimal purchase quantity

where

creasing priority demand classes and the shortage penalties reduces to the standard newsvendor problem with the demand CDF G, the selling price

We assumed so far that the demands

deviation

The quantities

We reconsidered the Şen and Zhang [

The two heuristics along with the two from previous work [

The reduction of the newsvendor problem with decreasing priority demand classes to the standard newsvendor problem revealed in this paper, remains valid when penalties are imposed for not meeting the demands. This reduction is also very useful in the case of incomplete probabilistic information about the demand distributions. In particular, we showed extensions of Scarf’s ordering rule when the means and standard deviations of random demands are the only parameters available. Additional studies are needed to consider different assumptions concerning the demands whose distributions cannot be fully specified. For example, the work of [

We would like to thank Prof. Jerzy Kamburowski for his significant contributions to Sections 3 and 6.

BoLi,PillaiboothamgudiSundararaghavan,UdayanNandkeolyar, (2015) A Newsvendor with Priority Classes and Shortage Cost. American Journal of Operations Research,05,337-346. doi: 10.4236/ajor.2015.55027