Robust Optimization for the Newsvendor Problem with Discrete Demand

A robust newsvendor model with discrete demand is initiatively studied, and the steps to obtain the optimal ordering decision are provided. The study shows that the optimal ordering decision with discrete demand is very different from that with continuous demand. Besides, the total number of demand points has almost no effect on the performance of both ordering decisions. Furthermore, for an ultralow-profit product, the ordering decision with discrete demand performs significantly better than that with continuous demand.


Introduction
The newsvendor problem has been investigated as a basic problem in stochastic inventory management since the eighteenth century in the economic literature, and it has been universally employed to analyze supply chains with fashionable and perishable products.Since the 50s of the last century, newsvendor problem has been extensively studied in operations research and extended to model various problems in real life.The simplest and most elegant version of the newsvendor problem is an optimal inventory problem in which a newsvendor needs to decide how much newspaper to order for the future demand, where the future demand is uncertain and follows a stationary distribution.This classical newsvendor problem has been extended in many different ways.See Khouja [1] and Qin et al. [2] for a review of various newsvendor-related models.See Choi [3] for the state-of-theart findings on both theoretical and applied research on the newsvendor problem.
The studies on the newsvendor problem mentioned above focus mainly on scenarios in which the underlying distribution of the random demand is precisely known.However, it is often very hard or impossible to figure out the demand distribution in practice, especially in fast-changing markets.
Since the decision is made on the basis of the assumed distribution, it might not hold for other distributions.Therefore, instead of assuming a stationary distribution, it is more reasonable to assume that the distribution belongs to the set of all distributions satisfying the known parameters.These parameters may come from estimates based on past realizations or some prediction by industry experts.The robust approach optimizes the worst-case objective (e.g., expected profit or regret or risk) over the parametric family.Most research that uses the robust approach in newsvendor models describes the distribution set by some known mean and variance.In these researches, Scarf [4] can be regarded as a pioneer who derived a closed-form formula for the optimal ordering rule that maximizes the expected profit against the worst possible distribution of the demand with the given mean and variance.He also pointed out that the worst distribution of the demand has positive mass at two points.Gallego and Moon [5] provided a simpler proof of Scarf 's formula and extended his ideas.In the last 25 years, since the appearance of Gallego and Moon [5], there have been numerous results in the literature related to extensions of Scarf 's ordering rule to different settings and applications; see, for example, maximizing the worst-case expected profit as a target [6][7][8][9][10][11][12][13][14][15][16][17][18], minimax regret as a target [10,[19][20][21], and optimizing the worst-case conditional value-at-risk as a target [22][23][24][25][26][27][28].
The existing research has revealed that the incompleteness of demand distributional information will have substantial impact on the newsvendor's decision.In general, these works are based on the assumption that the demand is a continuous random variable.Nevertheless, the actual demand for many products is discrete.The discrete goods can be bought and sold only in integral units, such as newspapers, cars, and machines.When the discrete goods are high-cost or low-profit products, their demand could not be approximately regarded as continuous demand.So far, there is only a little research aimed at studying the newsvendor problem with limited distributional information of discrete stochastic demand.Yu et al. [29] presented a general framework for considering discrete demand in robust newsvendor models.Carrizosa et al. [30] explored the robust newsvendor problem where demand is modeled as a time series which follows an autoregressive process.The advantage of robust autoregressive approach is that there is not much information needed, and it can be predicted by using self-variable series.But this method is also limited: (1) Autocorrelation coefficient is the key.If the autocorrelation coefficient is less than 0.5, it is inappropriate to use it; otherwise the prediction result is extremely inaccurate.(2) Autoregression can only be used to predict the economic phenomena related to their own earlier period, that is, the economic phenomena which are greatly influenced by their own historical factors, such as the amount of mining and the production of various natural resources and so on.In view of the limitations of the autoregressive method, we still only use robust approach and do not assume demand as a time series.On the basis of Yu et al. [29], we made a more systematic, more specific, and more in-depth study in robust newsvendor problem with discrete demand.
Specifically, the purpose of this paper is threefold.The first is to provide the solving steps to obtain the optimal ordering decision for the robust newsvendor model with discrete demand.The existing solution with continuous demand can merely be regarded as an approximate solution with discrete demand.The second is to compare the performance of our solution with discrete demand with the existing solution with continuous demand in different situations.The third is to study the effect of profit margin on these two kinds of solutions.
The rest of the paper is organized as follows.In Section 2, we review the existing results of classical/robust newsvendor problem.In Section 3, we formulate a robust newsvendor model with discrete demand and provide the approach to solve it.In Section 4, we conduct numerical experiments to calculate the optimal order quantity with discrete demand and compare it with that with continuous demand.In Section 5, we draw our conclusions.The proof of main result is given in Appendix.

Existing Related Results
The related notation are as follows: : the item's unit cost : the item's unit selling price : the item's unit salvage value, if an item is left unsold at the end of the sales period : the item's order quantity : the item's stochastic demand : the distribution function of the item's demand  : the item's expected demand over the sales period : standard deviation of the item's demand Γ(,   [31] assumes that the stochastic demand follows a known distribution .Its model is where   () fl ( − )min(, ) − ( − ) is the newsvendor's profit function.
Lemma 1 (continuous demand).If  is a known continuous distribution, then the optimal order quantity  *  maximizing In the degenerate case where In all other cases, the optimal order quantity is unique.

Robust Newsvendor Model with Continuous Demand.
The robust newsvendor problem assumes that the newsvendor only knows the mean  and the variance  2 of the demand distribution .The newsvendor model with free distribution of continuous demand is max where is the newsvendor's profit function.
Scarf [4] solved the robust newsvendor model with continuous demand in steps.Firstly, he solved the minimization problem min Then he solved the maximization problem max

Lemma 3.
If  is an uncertain continuous distribution with  ∈ Γ + (,  2 ), then the worst-case expected profit satisfies The optimal order quantity  *  maximizing () satisfies which is called "Scarf's ordering rule".

Robust Newsvendor Model with Discrete Demand
In this section, we are interested in solving the robust newsvendor model with discrete demand.Assume that  ∈ Γ + (,  2 ) is an uncertain discrete distribution with known points  1 ,  2 , . . .,   ( ≥ 4, {  } is a sequence sorted by size) and unknown probabilities  1 ,  2 , . . .,   .Motivated by the approaches adopted in Scarf [4], we also solve the model in two steps.Firstly, we use duality theory to analyze the minimization problem (M1), which is equivalent to the following linear programming problem: 1 , . . .,   ≥ 0. (P) where  0 ,  1 , and  2 are the dual variables corresponding to the probability-mass, mean, and variance constraints.The main idea is to construct a pair of primal-dual feasible solutions ( * 1 ,  * 2 , . . .,  *  ) for (P) and ( * 1 ,  * 2 ,  * 3 ) for (D), and make sure they satisfy the complementary slackness condition: for each  = 1, . . ., .
In the case of linear programming, this ensures optimality.That is to say, the point   has nonzero probability if and only if the dual optimal solution ( * 1 ,  * 2 ,  * 3 ) satisfies In other words, the same points of the smooth curve () fl  1 +  2  +  3  2 and the two-piece line (, ) should be the points where the primal optimal solution places all its masses.
A fundamental result of linear programming theory asserts that there exists an optimal solution such that the number of nonzero variables is no more than the number of constraints.Therefore, for (P), the number of nonzero  *  is no more than three.It indicates that the worstcase distribution is a three-point distribution.Therefore, we restrict ourselves to three-point distributions.

Mathematical Problems in Engineering
No other demand point Figure 1: The cases that satisfy the dual feasible condition.
As illustrated by Figure 1, the two functions () fl  1 +  2 +  2  3 and (, ) will have four intersection points at most.Based on the above analysis, we restrict ourselves to three corresponding demand points.Besides, in order to meet the dual feasibility, at least two of the three demand points are adjacent.
On the basis of the above analysis, we provide the closeform expression for the worst-case expected profit () fl min  1 ,...,  E[(, )], in the following theorem.Theorem 4. For three points   ,  +1 ,   in all known demand points, assume their probabilities as *  = 0,  = 1, . . ., ,  ̸ = , ,  + 1. ( Case 1.For arbitrary 2 ≤  ≤  − 1, 1 ≤  ≤  − 1 and a fixed , if the primal feasibility is satisfied, and the dual feasibility is satisfied, then the worst-case expected profit satisfies Case 2. For arbitrary 1 ≤  ≤  − 2,  + 2 ≤  ≤  and a fixed , if the primal feasibility is satisfied, and the dual feasibility is satisfied, then the worst-case expected profit satisfies Figure 2 provides a graphical representation of the primal feasibility in Theorem 4 in the mean-variance space.If (,  2 ) is in the red striped area, then the primal feasibility is satisfied.Table 1 provides the dual feasible intervals of  with respect to fixed  and different  in a list.
To sum up, Theorem 4 tells us that we can find out the worst-case distributions and the worst-case expected profit () by the following steps: (1) Judge whether arbitrary three demand points   ,  +1 , and   satisfy the primal feasibility.If they satisfy it, go to step 2; or else, consider other three points.
(2) Find out the dual feasible interval of .
(3) Obtain the close-form expression of () on the dual feasible interval.
Theorem 4 and Table 1 reveal that () must be a piecewise linear function.According to the expression for () in Theorem 4, we can draw the graph of the worstcase expected profit, to help decision-makers make order decisions.
Secondly, we consider the maximization problem (M2).We can not obtain the explicit expression of the optimal order quantity  *  maximizing (), since it is completely determined by the values of demand points.Fortunately, we can use MATLAB to draw the graph of the worst-case expected profit and find the optimal order quantity  *  that maximizes the worst-case expected profit.

Numerical Experiments
In contrast to the existing robust newsvendor models, the model presented by us takes discrete demand into consideration.Therefore, it is expected that our model may behave differently from the existing robust newsvendor models.In the following numerical experiments, we will first figure out the concrete value of  *  according to the solving steps in the previous section.Second, we will compare the value and performance of  *  with  *  , where  *  is obtained by "Scarf 's ordering rule" [4].Third, we will study the impact of various parameters on  *  and  *  .As indicated in Figure 3, the worst-case expected profit () is a piecewise linear function, and the worst-case demand distribution corresponding to each segment in the piecewise linear function is a three-point distribution.For example, the worst-case distribution corresponding to the first segment is the three-point distribution of ( 1 ,  3 ,  4 ).That is to say, the worst-case distribution of the demand is a piecewise combination of several three-point cases.The results of our numerical experiments are consistent with Theorem 4. The decision maker can find the highest worstcase expected profit point through Figure 3 and make a robust order quantity decision.
Secondly, we compare the performance of  *  with  *  in view of the expected profit in real situation.Both Scarf [4] and Gallego and Moon [5] analyzed the performance of ordering decisions for high-profit and low-profit products.Motivated by their numerical experiments, we further analyze the performance of ordering decisions for products with different marginal profits.
We set parameters as  = 10,  = {1, 3, 5, 7, 9},  = 0, and  = {6, 12, 24}.We denote profit margin by  = ( −)/( − ); then the corresponding profit margins are  = {90%, 70%, 50%, 30%, 10%}.For each profit margin, we randomly generate a set of 100 problems.In each instance,  demand points are randomly taken on the interval [0, 2000], and their corresponding probabilities are drawn from a uniform distribution.Under each random fixed discrete distribution, we use  *  and  *  instead of  *  to obtain the expected profits, respectively, and then compare them in Table 3 and Figure 4, in terms of   ( *  )/  ( *  ) and   ( *  )/  ( *  ).Table 3 shows that the number of total demand points has almost no effect on the performance of  *  and  *  .Furthermore, it is observed from both Table 3 and Figure 4 that, as the marginal profit decreases, the performance of  *  is getting worse and worse, but the performance of  *  gets worse first and then becomes better.We also found that for the super low-margin product ( = 10%)  *  performs significantly better than  *  ; and for the products with profit margin  = 30% and profit margin  = 90%, their performances are similar; and for the products with other profit margins, the performance of  *  is always better than  *  .It is worth mentioning that both  *  and  *  perform fairly well for the super high-margin product ( = 90%).
The results of the above numerical experiments conform with the reality.If we look at the discrete demand of the super low-margin product as continuous demand, the influence of the decision-making errors will be enlarged.That is to say, when the marginal profit of the product is very low, the decision maker should take the information of demand points into consideration; otherwise he should ignore this information.Therefore, we recommend the use of  *  as the Mathematical Problems in Engineering   optimal ordering decision for the decision maker in those circumstances when the product's marginal profit is super low and only partial information of discrete demand is known.

Effect of the Profit Margin.
To gain more insights, in our third numerical experiment we study the effect of the profit margin  = ( − )/( − ) on three different order decisions  *  ,  *  , and  *  .We set parameters as  = 10 and  = 0.In this example, 12 demand points are randomly taken on the interval [0, 2000], and their corresponding probabilities are drawn from a uniform distribution.
Figure 5 demonstrates that  *  ,  *  , and  *  are all increasing in profit margin .That is to say, the higher the marginal profit of the product is, the higher the optimal order quantity is, no matter what kind of optimal order quantity is.Specifically, in this random case, under different profit margins,  *  is always closer to  *  than  *  ; namely,  *  performs better than  *  .

Conclusion
Most previous researches about the robust newsvendor problem are based on continuous demand.However, in practice, the demand of many products is discrete.This paper integrates a distribution-free design with discrete demand, into a robust optimization approach.The close-form of expression for the worst-case expected profit is provided, and solving steps to obtain the optimal decision  *  are given.The important findings are summarized as follows: (1) For the same information of mean and variance,  *  is constant, but  *  adjusts according to the different information of demand points.
(2) The number of total demand points has almost no effect on the performance of  *  and  *  .
(3) The performance of  *  is significantly better than  *  only if the product's marginal profit is very low.The managerial implications of these findings are as follows: When the product's marginal profit is super low and only partial distributional information of the discrete demand is known,  *  is the most appropriate optimal ordering decision for the decision maker.Under this situation, besides the information of mean and variance, the information of demand points is also very important to the decision maker.In addition, the number of total demand points has no effect on the decision.
There are many questions that need to be further explored.For example, other extensions of our research include studying newsvendor problem with other partial information about the discrete demand distribution, e.g., range, median, symmetry, unimodality, and partial information of demand points.

Figure 2 :
Figure 2: Characterization of the primal feasibility in Theorem 4.

Figure 4 :
Figure 4: Comparison of the performance of  *  with  *  with respect to different profit margins.PS:  = 12.

Figure 5 :
Figure 5: Sensitivity of different optimal order quantities against the profit margin.
2): the class of all distribution functions with mean  and variance  2 Γ + (,  2 ): the subclass of distribution functions  of nonnegative random variables, i.e., ∫ Existing Research Result.Our second numerical study aims to compare the value and performance of  *  with  *  .Firstly, we compare the value of  *  with  *  in Table 2. Set the benchmark values of parameters as  = 50,  = 35,  = 25,  = 1000, and  = 500.From Table 2, it is clear that, for the same information of mean and variance,  *  is always unchanging, but  *  varies according to different information of demand points.

Table 1 :
The interval of fixed  to make sure of the dual feasibility.

Table 2 :
Comparison of the value of  *  with  *  .

Table 3 :
Comparison of the performance of  *  with  *  over 100 instances.