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A Constructive Methodology to Solving the Capacitated Newsvendor Problem: an Approximate Approach


The applications of the capacitated newsvendor problem are numerous. They range from inventory control, supply chain management, and airlines seat reservation to stock market applications. While several solution methodologies exist for this problem, most of them require advanced mathematical programming techniques or specifically developed iterative models. To complement these solution approaches and facilitate wider dissemination of this model particularly to upper-undergraduate and first-year graduate students as well as practitioners in related fields, in this paper, we develop an approximate solution methodology that is based on constructive approaches. These approaches already exist in standard textbooks of operations research, management science, and other similar disciplines. The two models utilized in the development are the classical newsvendor and the network flow model. Numerical comparisons are drawn between existing solution methodologies and the developed ones. Also, an example is presented to show the steps of the developed methodology.


The seminal work of Hadley and Whitin [1] in the capacitated multi-product newsvendor problem continues to receive considerable attention. This could be attributed to the wide range of its applicability in optimizing costs and profits for many systems. The spectrum of its real-world applications covers a wide range. In addition to its standard application in finding the optimum lot size of perishable commodities subject to a certain constraint, in the following, we cite some of its further utilization. Aloi et al. [2] treat radio transmission resources as short-term lifetime goods. In order to maximize both profit and quality of service offered to wireless users, an adaptive period inventory management policy is applied to a wireless cognitive telecommunication scenario, and the new benefits are realized through this model’s application. In order to select one or more entities from a large number of candidates subject to one or more constriants, Fan [3] develops an iterative process based on the newsvendor model to find an optimal profit margin value within the boundary (capacity). Then, the respective optimal quantity for each of the candidate entities is calculated by utilizing the optimal profit margin value. Khouja et al. [4] investigate the application of off-price retailer disposition channel for the excess in-season inventory of a retailer. The strategic consumer buys the product early or waits for a discount in order to maximize their excepted utility when this channel is provided. When the number of exclusive thrift off-price consumers is very large, the effect of the consumer’s strategic behavior will be reduced by the application of the off-price retailer. Hillier and Lieberman [5] mention an application of the reservations provided by an airline in their book. Overbooking can be treated as the overstocking of a perishable product, and empty seats can be treated as understocking. Many major airlines now are making extensive use of the newsvendor model to analyze how much overbooking to do.

In their book, Hadley and Whitin [1] introduce a Lagrangian iterative dynamic programming methodology to solve the problem. While their approach is adequate in tackling situations with a limited number of products (3 to 4 or so), it becomes intractable when the size of the problem becomes large. This limitation has propelled the publication of several articles to address larger size problems. Nevertheless, despite the interest of the community in the capacitated multi-product newsvendor problem, till today and to our knowledge, not many textbooks report on its solution methodologies. As will be seen in Section 2, this could stem from the fact that the existing solution methodologies utilize either uncommon iterative techniques or advanced mathematical programming methodologies. Applying these types of approaches may not be easy for upper-level undergraduate, first-year graduate students, and common practitioners in related fields. The absence of a straightforward solution methodology is what motivated our work. In this paper, a solution methodology for the capacitated multi-product newsvendor problem that is based on the classical newsvendor model and the network flow problem is developed. These two problems are included in standard textbooks of operations research, management science, and operations management as well as others.

Our taxonomy in this work is as follows. After this introduction, in Section 2, we briefly review the relevant literature. Then, in Sections 3 and 4, we present the methodology and numerical comparisons where the performance of the introduced model is compared with existing ones, respectively. Section 5 gives a step-by-step example for its application. Finally, Section 6 concludes with a summary.

Literature Review

There is a plethora of publications in the subject of the newsvendor problem. The interested reader is referred to the edited handbook by Choi [6] for further information on some of these applications. The handbook offers a compendium of articles covering a considerably wide range of the newsvendor models and their applications in various fields. Also, we would like to refer here to three earlier articles by Gallego and Moon [7], Khouja [8], and Janssen et al. [9] where they present reviews of the single-period newsvendor problem for different objectives and utility functions.

This review is divided into two subsections; the first one is devoted to some existing solution methods while the second references some recent extensions.

In the area of solution methodologies, Lau and Lau [10, 11] in one of their many published articles in this area introduce a numerical approach to a case of bakery. They are credited for being among the first to observe that relaxing the non-negativity constraints in the Lagrangian approach to the problem could lead to negative order quantities of some products particularly when the available budget is tight. They address this problem by setting the optimal order quantity of products with a negative service level to 0. Abdel-Malek and Montanari [12] introduce a generic iterative method (GIM) where the solution space is divided into three regions by developing two thresholds. The first region is where the budget is large enough to order the optimal batch size of each product, while the second region is where the budget allows the ordering of each of the products but some of the orders may be less than the optimal batch size, and the third region where the budget is tight so that some products have to be deleted from the list. Their iterative approach uses a specific numerical method that is particularly developed for this application. Niederhoff [13] develops a linear approximation method (LAM) for solving the newsstand problem. Then, she applied convex separable programming to find a close approximation of the optimal solution for any demand distribution. The convergence of the cost values is affected by the number of breakpoints used to interpolate the cost curve. The approach is iterative in nature and requires insights on how to break down and linearize the objective function. Abdel-Malek and Areeratchakul [14] develop a quadratic programming approach for solving the newsstand problem. The objective function is approximated as a quadratic form. Familiar linear programming software packages such as Excel Solver and Lingo can be utilized to obtain exact or approximate solutions depending on the type of demand distributions. Zhang et al. [15] propose a binary solution method (BSM) to solve the capacitated multi-product newsvendor problem. It can handle general demand distribution functions. They define a marginal budget benefit function and prove the properties of the optimal solution. A binary search algorithm is implemented to obtain the optimal value of the marginal budget benefit and order quantities.

In the application arena and extensions of the newsvendor problem, we refer to some of the recent ones. In the area of random yield, Abdel-Malek and Otegbeye [16] use separable programming and duality concept to find the optimum for the capacitated newsvendor problem. Then, the objective function can be expressed in a quadratic form, and the functional constraint is then arranged as separable problems for each product. Also, in the same vine, the random yield scenario is considered by Abdel-Malek et al. [17]. A modified generic iterative method (GIM2) based on Newton’s method, Leibniz rule, and Lagrangian approach is utilized to find the optimal order quantities. In the pricing and supplier quantity discount, Shi et al. [18] formulate the problem as a generalized disjunctive programming model. A Lagrangian iterative heuristic approach to achieve the near-optimal order quantities in an acceptable CPU time is developed. In many real-world applications, the demand patterns and the quality of products supplied are not known. Here, we cite four articles. Dash and Sahoo [19] consider fuzzy random demand and fuzzy cost for a newsvendor problem. They transform a fuzzy single-period inventory model into a crisp model to obtain the optimal order quantities and the expected profit. LINGO package can be implemented to solve the problems. Wang et al. [20] assume the market demands to be uncertain random variables in a constrained newsvendor environment. Situations in which the demand is originated from subjective evaluation and a small amount of data are considered. The uncertain random models are transferred to equivalent deterministic forms in some cases, and MATLAB is employed to find the optimal order quantities. They also perform sensitivity analysis by using the impact of products’ parameters and then the expected demands on the optimal order quantities. Moon et al. [21] present a distribution-free newsvendor problem with multiple discounts and upgrades. The binary solution method (BSM) is utilized to solve this kind of problem. Traditional cases and distribution-free cases are compared when the mean and the variance of the demand are the only available data to the decision-maker. Areerakulkan [22] introduces the product quality as a fuzzy variable into the newsvendor problem. Quadratic approximation and iterative solution finding steps using KKT’s conditions are utilized to achieve the optimum. He states that Excel Solver can be implemented to address the problem.

One can see that the recent development and extensions in this vine either modify and/or advance existing solution method or apply numerical and/or iterative methods that require specially developed techniques or apply sophisticated mathematical programming models. This is what motivated our work, developing a solution methodology that is based on known simple methods that are easy to apply. In essence, the highlights of this paper include presenting a constructive solution method to the capacitated newsvendor problem, developing an approximate but analytical solution based on the classical single period model of perishable commodity and the network flow algorithm, and an example exhibiting the ease of the solution methodology to this non-linear programming problem.

The Methodology

This section is divided into two subsections. The first subsection shows the formulation of the capacitated newsvendor problem. The second subsection presents the rational and necessary proof of our methodology.

The Model

One of the common classical formulations of the capacitated newsvendor problem is shown in Eqs. (1, 2, and 3).


$$ \sum \limits_{i=1}^nE\left({x}_i\right)=\sum \limits_{i=1}^n\left[{c}_i{x}_i+{h}_i{\int}_0^{x_i}\left({x}_i-{D}_i\right){f}_i\left({D}_i\right)d{D}_i+{v}_i\ {\int}_{x_i}^{\infty}\left({D}_i-{x}_i\right){f}_i\left({D}_i\right)d{D}_i\right] $$

Subject to:

$$ {\sum}_{i=1}^n{c}_i{x}_i\le B $$
$$ {x}_i\ge 0\ i=1,2\dots n $$

As can be seen, the objective function consists of three terms. The first term, cixi, expresses the unit purchase cost, ci, multiplied by the order quantity, xi, of product i. The second term evaluates the overstocking cost of the items where hi is the overstocking cost per unit and the integral \( {\int}_0^{x_i}\left({x}_i-{D}_i\right){f}_i\left({D}_i\right)d{D}_i \)is the expected number of overstocked items of product i. Similarly, the last term shows the expected understocking cost, where vi is the understocking cost per unit and the integral \( {\int}_{x_i}^{\infty}\left({D}_i-{x}_i\right){f}_i\left({D}_i\right)d{D}_i \) is the expected number of understocked items of product i. Equation (2) shows the budget constraint. It should be noted without loss of generality that the budget constraint can be replaced by space, weight, etc. Finally, Eq. (3) ensures the non-negativity of the order quantities. Table 1 lists further the notations that are used in the rest of our analysis and development.

Table 1 Notations

Rationale and Proof

For most common probability density functions, the total cost function of the newsvendor is known to be convex. Thus, as shown in Fig. 1, the more we order from product i, xi, the closer we get to the unconstrained optimum, \( {x}_i^{\ast } \). However, when there is a capacity constraint, the optimum lot size of all products can be ordered if not at all. Therefore, one has to decide the order of priority for each product. In order to simplify this procedure, we linearize the total cost function by Taylor’s expansion around the midpoint of the unconstrained optimal order quantity (\( {x}_i^{\ast }/2 \)) (as shown in Fig. 1). The slope of the approximate total cost function represents the cost reduction rate. In our analysis, we consider two probability density functions: the uniform and the exponential (the behavior of the normal distribution is known to lie in between, that is, its error lies in between these two, see Table 5). Equations (4–14) exhibit the calculation of the difference between the total cost of ordering policy between the proposed approximation and the current approaches.

Fig. 1

Total cost function for product i

Proof In the following, we analyze the difference due to the linearization for both the uniform and the exponential distributions and also the mapping of the problem into the minimum cost flow problem.

The total cost function for uniform and exponential distributions can be expressed respectively by Eq. (4) and Eq. (5):

$$ \sum \limits_{i=1}^n{E}_u\left({x}_i\right)=\sum \limits_{i=1}^n\left({c}_i-{v}_i\right)\ {x}_i+\frac{\left({v}_i+{h}_i\right)}{2\left({b}_i-{a}_i\right)}{x}_i^2+{v}_i\left(\frac{b_i-{a}_i}{2}\right) $$
$$ \sum \limits_{i=1}^n{E}_e\left({x}_i\right)=\sum \limits_{i=1}^n\left({c}_i-{v}_i\right)\ {x}_i+\left({v}_i+{h}_i\right)\left({x}_i+{\mu}_i{e}^{-\frac{x_i}{\mu_i}}\right)+{v}_i{\mu}_i $$

The slope of the total cost function for uniform and exponential distributions when \( {x}_i={x}_i^{\ast }/2 \) can be expressed respectively by Eq. (6) and Eq. (7):

$$ \sum \limits_{i=1}^n{E_u}^{\prime}\left({x}_i\right)=\sum \limits_{\mathrm{i}=1}^n\left({c}_i-{v}_i\right)+\frac{\left({v}_i+{h}_i\right)}{\left({b}_i-{a}_i\right)}\times \left({x}_i^{\ast }/2\right) $$
$$ \sum \limits_{i=1}^n{E_e}^{\prime}\left({x}_i\right)=\sum \limits_{i=1}^n\left({c}_i-{v}_i\right)+\left({v}_i+{h}_i\right)\left(1-{e}^{-\frac{x_i^{\ast }}{2{\mu}_i}}\right) $$

Applying Taylor’s expansion to Eq. (4) and Eq. (5) when \( {x}_i={x}_i^{\ast }/2 \) yields:

$$ {\displaystyle \begin{array}{c}\sum \limits_{i=1}^n{E}_u\left({x}_i\right)=\sum \limits_{i=1}^n\left({c}_i-{v}_i\right)\left({x}_i^{\ast }/2\right)+\frac{\left({v}_i+{h}_i\right)}{2\left({b}_i-{a}_i\right)}{\left({x}_i^{\ast }/2\right)}^2\\ {}+{v}_i\left(\frac{b_i-{a}_i}{2}\right)+\left[\left({c}_i-{v}_i\right)+\frac{\left({v}_i+{h}_i\right)}{\left({b}_i-{a}_i\right)}\left({x}_i^{\ast }/2\right)\right]\ast \left({x}_i-{x}_i^{\ast }/2\right)+\varepsilon \end{array}} $$
$$ {\displaystyle \begin{array}{c}\sum \limits_{i=1}^n{E}_e\left({x}_i\right)=\sum \limits_{i=1}^n\left({c}_i-{v}_i\right)\left({x}_i^{\ast }/2\right)+\left({v}_i+{h}_i\right)\left({x}_i^{\ast }/2+{\mu}_i{e}^{-{x}_i^{\ast }/2{\mu}_i}\right)+{v}_i{\mu}_i\\ {}+\left[\left({c}_i-{v}_i\right)+\left({v}_i+{h}_i\right)\left(1-{e}^{-\frac{x_i^{\ast }}{2{\mu}_{\mathrm{i}}}}\right)\right]\ast \left({x}_i-{x}_i^{\ast }/2\right)+\varepsilon \end{array}} $$

In Eqs. (8) and (9), it can be seen the linear approximation dominates the total cost function. This approximation is equivalent to the minimum cost flow algorithm shown in [23]. As exhibited in Fig. 2, a set of arcs on the left-hand side of the network represents the purchase cost assigned to each product with flow capacity between (0,\( {c}_i{x}_i^{\ast } \)); note that it will be not optimum to order more than \( {x}_i^{\ast } \). The unique arc on the right-hand side’s upper bound is the available budget (as proved by Zhang et al. [15], the available budget must be fully utilized to yield the optimum which is equivalent to maximizing the flow in the network, Li01 is the amount of flow in the ith arc from node 0 to node 1, and L12 is the amount of flow in the unique arc from node 1 to node 2).

$$ \operatorname{Max}\ \mathrm{Z}=\sum \limits_{i=1}^n{L}_{i01} $$
Fig. 2

Network presentation of the methodology

Subject to:

$$ \sum \limits_{i=1}^n{L}_{i01}={L}_{12} $$
$$ 0\le {L}_{i01}\le {c}_i{x}_i^{\ast } $$
$$ 0\le {L}_{12}\le B $$

This approximation is equivalent to the minimum cost flow algorithm shown in [23].

Since there are only three nodes in the network and according to the minimum cost flow algorithm, the arcs on the left-hand side will be filled from top to bottom until the arc on the right-hand side is full. In order to assign the arcs on the left-hand side to each product, the profitability ratio (vi/ci) is used to decide on the priority of assignment from top to bottom. This means the more profitable the product is, the higher priority it takes in ordering. To show the percentage of difference (∆E%) as expressed in Eq. (14), we consider a different profit ratio (vi/ci) and several budget tightness (B/Bopt). The results of these scenarios are summarized in Table 2. As the results in Table 2, one can see that the differences are small.

$$ \Delta E\%=\frac{\sum \limits_{i=1}^nE\left({x}_i^{\ast \ast}\right)-\sum \limits_{i=1}^n{E}^T\left({x}_{i,T}^{\ast \ast}\right)}{\sum \limits_{i=1}^nE\left({x}_i^{\ast \ast}\right)} $$
Table 2 The difference of the total cost for uniform and exponential distributions

The steps of the methodology are as follows:

  • Step 0: Find the optimum amount of each product independently by\( F\left({x}_i^{\ast}\right)=\frac{v_i-{c}_i}{v_i+{h}_i} \);

  • Step 1: Substitute the resulting amounts in the budget constraint, \( {\sum}_{i=1}^n{c}_i{x}_i\le B \); if satisfied, stop because a solution is found. Otherwise,

  • Step 2: Rearrange the products in the list in descending order of vi/ci and map the model as a network (see Fig. 2).

  • Step 3: Fill each arc on the left-hand side of the network in Fig. 2 to its maximum capacity \( {c}_i{x}_i^{\ast } \) , i = 1, 2, …n, till the arc on the right-hand side reaches its upper bound (note that the last arc to be filled on the left-hand side may not reach its maximum capacity).

  • Step 4: Use \( {x}_i^{\ast \ast }=\frac{L_{i01}}{c_i}\kern0.5em \)to calculate \( {x}_i^{\ast \ast } \), i = 1, 2, …nLi01 is the amount of flow in the ith arc on the left-hand side of the network).

  • Step 5: Use \( {x}_i^{\ast \ast }, \) i = 1, 2, . . n and Eq. (1) to calculate the optimal total cost.

For more illustrative details, see the flowchart of the methodology in Fig. 4.

Numerical Comparison to Existing Approaches

In this section, we compare the performance of the developed methodology to the ones developed earlier, such as BSM, GIM and LAM. These three methodologies are known to yield exact optima or close to it depending on the probability distribution function of the demand. While the exhaustive number of numerical experiments conducted to ascertain the accuracy of the developed methodology, the comparison reported here is based on the example which is extracted from Lau and Lau [10] where ten products are considered (their parameters are shown in Table 3). To show the adaptability of the proposed methodology to different types of demand distribution functions, we also apply it to an example of nine products where their parameters are listed in Table 4. Table 5 shows the narrow differences between the optimal total cost obtained by the proposed methodology, BSM, GIM, and LAM when demand distributions are uniform, exponential, normal, and combinative. The comparisons with BSM, GIM, and LAM cover a reasonable range of budgets. We calculate the differences of optimal total costs when the available budget is 50%, 70%, and 90% of the minimum one (Bopt) required to order the optimal quantity of all products.

Table 3 Parameters for numerical comparisons
Table 4 Parameters of products when demand distributions are considered from several distributions
Table 5 Numerical comparisons between developed methodology and existing approaches

The numerical results support the analytical ones which are presented in Section 3 regarding the dominance of vi/ci on the optimal solution. One can see from Table 5 that the differences resulting from the procedure are most of the time negligible (the range is 0.49–2.18%).

In Table 5, the following insights can be summarized:

  1. a)

    As the available budget becomes closer to that needed to order the optimal number of items of each product in the list, the differences between optimal total cost obtained by the developed methodology, BSM, GIM, and LAM narrows.

  2. b)

    The ratio vi/ci dominates on deciding the priority of product purchases when the tightness of the budget constraint is in a reasonable range.

  3. c)

    The optimal total cost is not very sensitive to the order quantities of each product


To clarify the application of the developed methodology, in this section, we give an example of ten products from Table 3 with exponentially distributed demands. The optimal order quantities and optimal total cost obtained by the developed methodology, BSM, GIM, and LAM are presented in Table 6. The available budget is set to $4500. The following are the steps of the developed methodology:

  • Step 0: Find the optimal amount for each product independently by\( F\left({x}_i^{\ast}\right)=\frac{v_i-{c}_i}{v_i+{h}_i} \).

  • Step 1: Then, substitute the resulting amounts in the budget constraint and obtain Bopt = $8008. Because B = $4500 and B < Bopt, so the budget constraint is binding.

  • Step 2: Rearrange the products in Table 3 in descending order of the corresponding vi/ci as shown in Table 6 and map it as a network as in Fig. 3.

  • Step 3: Fill each arc on the left-hand side of the network in Fig. 3 to its maximum capacity \( {c}_i{x}_i^{\ast } \), till the unique arc on the right side reaches its maximum B.

  • Step 4: Then, use \( {x}_i^{\ast \ast }=\frac{L_{i01}}{c_i} \)to calculate\( {x}_i^{\ast \ast } \), i = 1, 2, …n. The results are presented in Table 6, \( {x}_6^{\ast \ast } \) = 27,\( {x}_8^{\ast \ast } \) = 59, \( {x}_4^{\ast \ast } \) = 48..., \( {x}_2^{\ast \ast } \) = 0, \( {x}_9^{\ast \ast } \) = 0.

  • Step 5: Use \( {x}_i^{\ast \ast } \), i = 1, 2, …n and Eq. (1) to calculate the optimal total cost and it is $28,890 as shown in Table 6.

Table 6 The parameters and results of the application
Fig. 3

Network flow representation of the application

It should be mentioned here that the optimal total cost obtained by the introduced methodology is very close to those yielded by BSM, GIM, and LAM (the difference is around 1.26% and 1.16%).


In this paper, a constructive methodology to solve the capacitated newsvendor problem is developed. Among its salient features is the fact that it utilizes solution techniques that exist in the introductory operations research, operation management, management science, and other related fields’ textbooks. These techniques are the unconstrained newsvendor problem and the network flow algorithm. The introduced methodology should enable the dissemination of the capacitated newsvendor solution approach to a wider range of an0 interested reader. Extensive numerical experiments are conducted to compare the results between existing methods (BSM, GIM, and LAM) that are known to yield exact optima or near optimum, depending on the type of demand distributions, and the one developed here. Over a wide range of the governing parameters as well as the most common types of the demand probability distribution functions, the developed methodology, in addition to its ease of applying, yields narrow differences (0.49–2.18%) between its optimal total cost and those rendered by the foresaid methods. Additionally, the example presented where ten products are considered exhibits detailed solution steps showing the simplicity of the computational effort where it does not require advanced expertise or specialized codes and/or knowledge to attain the solution.

It should be noted further that one can easily alternate among different capacity constraints and decide on which of them is more restricting. More clearly, if there are constraints other than budget, such as space or weight, each can be considered alternately and the results can be assessed and compared.

Finally, one should note that in addition to its ease of implementation, the developed methodology can serve as the first cut approach for decision-makers to ascertain whether they should choose to pursue one of the more involved iterative techniques that are available in the open literature. Additionally, as reported in Section 4, the approach and its ensuing analysis revealed three important managerial insights. They are (1) the dominance of vi/ci for each product on the acquisition decision (the higher the profit of the product, the higher the priority of ordering), (2) the optimal total cost is diminutively sensitive to the optimum order quantity of each product, and (3) the difference between the optimal total cost obtained by the developed approach and existing ones becomes negligible (less than 2.18%) when the budget is 50% or more than that required to order the optimum of items for each product, which is reasonable from a practical point of view. Moreover, we would like to mention that our methodology is constructive in nature and the existing solution methods which are either non-polynomial (NP) or pseudopolynomial. This developed methodology is polynomial (P) in time.


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It would be remiss if we do not express our sincere appreciation for the invaluable feedback of two anonymous reviews.

We would also like to acknowledge the participation of the many students who contributed to make this paper possible.


This study has not received any external funding.

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Correspondence to Layek Abdel-Malek.

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The authors declare that they have no conflict of interest.



Fig. 4

Flow chart exhibiting the steps of the proposed methodology

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Abdel-Malek, L., Shan, P. & Montanari, R. A Constructive Methodology to Solving the Capacitated Newsvendor Problem: an Approximate Approach. SN Oper. Res. Forum 1, 8 (2020).

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  • Inventory
  • Newsvendor problem
  • Network flow
  • Supply chain management