Multi-Sourcing Procurement Management: A Closed-Form Mean–Variance Approach

Abstract

In the post-pandemic era, multi-sourcing is rapidly becoming the preferred approach for companies to drive optimized cost, quality and turnaround times. However, multi-sourcing systems present a myriad of challenges in designing an effective planning model. In this article, we propose a robust capacity planning model with multiple supply sources, and demonstrate how the capacity plan can be efficiently solved via a parsimonious mean–variance approach. We apply the max-min criterion to design a distributionally robust multi-source capacity plan. We use a new approach to deriving a set of optimality conditions on the robust capacity plan. These conditions enable us to derive the worst distribution, the optimal robust capacity vector, and the worst-case expected profit in closed form. The new approach bypasses numerous tedious intermediate procedures in the traditional distributionally robust optimization literature and allows us to derive the optimal solution based on exogenous cost parameters. This closed-form solution appears to be hitherto unknown and has important ramifications for the multi-sourcing capacity planning problem. Our findings reveal that, despite the complexity of multi-sourcing, the worst-case demand scenario can be reduced to just $n+1$ distinct outcomes, structured by two key sequences derived from supplier characteristics. The optimal capacity plan has a clear structure–allocations align with the midpoints between these demand scenarios. Surprisingly, we also find that most of the robustness benefits of a full supplier portfolio can be achieved by engaging just the two best-matched sources. This provides a practical and cost-effective road map for robust capacity planning in multi-sourcing environments.

1. Introduction

Sourcing strategy is crucial for firms to thrive in today’s volatile business environment. While single sourcing may have low administrative costs, it exposes firms to significant risks caused by demand or supply uncertainties. To gain a competitive advantage, many firms diversify their supply sources to increase their flexibility to cope with uncertainties. For example, Hewlett-Packard is a pioneer in using option contracts to manage the supply of memory devices (Fu et al., 2010). Each option contract specifies a reservation fee for locking in the capacity and an exercising fee for executing the procurement option after observing the realized demand. Any unexercised option expires without incurring the exercising fee. By engaging a portfolio of option contracts with diversified costs and flexibility, Hewlett-Packard has achieved enormous savings in the procurement process.

The multi-sourcing problem also arises when designing a supply network, where a company sets up multiple warehouses to replenish a set of retailers serving different market regions. Each retailer, facing a random demand replenishes inventory from multiple warehouses, capitalizing on the varying holding costs at each warehouse (equivalent to the reservation cost in an option contract) and the different transportation costs between the warehouses and the retail store (equivalent to the execution cost in an option contract). The company faces the challenge of selecting warehouses from a list of potential locations and fulfilling retailer demands from those selected warehouses. This decision-making process considers various factors such as the fixed costs associated with warehouse establishment, holding costs, and transportation expenses.

The third example is Xerox Australia, which recycles and remanufactures photocopiers (Kerr and Ryan, 2001). When the photocopiers are returned to Xerox after the end of lease contracts, Xerox inspects their condition and sorts them into four grades: grade $1$ (suitable for refurbishment), grade $2$ (suitable for reprocessing), grade $3$ (suitable for remanufacturing), and grade $4$ (suitable for asset recovery or disposal). After an order for used photocopiers arrives, Xerox implements a priority rule that exhausts an alphabetically lower grade (which has a better condition) before using another grade. Photocopiers with grade $1$ and $2$ are cleaned and repaired. High-frequency-service parts are replaced regardless of their condition or use. Other parts are replaced depending on their condition and expected remaining life. Photocopiers with grade $3$ and $4$ must undergo a labor-intensive disassembly process. The good-quality parts are cleaned, tested, and reconditioned. Some photocopiers are then reassembled, whereas others are sent without reassembly to a disposal facility. Due to varying grades of conditions, the used photocopiers have different acquisition and remanufacturing costs. Photocopiers with grade $1$ have better condition, and thus require a higher acquisition cost but a lower remanufacturing cost. Conversely, photocopiers with grades $3$ or $4$ are less expensive to acquire but have a higher remanufacturing cost. Xerox remanufactures different grades of used photocopies to the same specification to satisfy demands. If the quantity of used photocopiers is insufficient, the demand will be lost.

The operations of Hewlett-Packard, a multi-sourcing supply network, and Xerox Australia share two common characteristics: i) the firm’s sourcing decision involves two distinct stages: reservation and fulfillment, and i) the firm employs multiple supply sources with different two-stage cost structures. When choosing among multiple supply sources, the firm balances reservation and execution costs. If the first source charges a low execution cost but a high reservation cost, the firm may benefit from engaging with a second source that charges a low reservation cost but a high execution cost. The multi-sourcing procurement problem has been well studied in the literature assuming that the demand distribution is precisely known (e.g., Fu et al., 2010; Mutha et al., 2019).

However, in many practical circumstances, the exact demand distribution is usually difficult to specify (e.g., Natarajan et al., 2018). Managers often need to make decisions without precisely knowing the underlying distribution that governs the (random) operating environment. For example, forecasting demand for new products is notoriously difficult (see Chapter 5 in Lilien et al., 2017: page 135 to 168). Researchers have dedicated efforts to finding “robust” solutions to address ambiguity in planning problems, using only a parsimonious set of information such as the means and variances (or higher moments) to characterize the distributional ambiguity in the operating environment (e.g., Li and Kirshner, 2021; Natarajan et al., 2018). The mean and variance parameters of demands are often readily available or commonly employed by practitioners in their planning process. In this body of literature, the robust max-min decision rule has gained popularity (see page 1931 in Lu and Shen, 2021).

In a typical max-min optimization problem with known moments, a firm chooses actions to maximize the worst-case expected profit based on the available moments information. This max-min problem can be formulated as a two-stage model, where the second stage solves the firm’s worst-case expected profit for any given action, and the first stage optimizes the actions. While the practical reason is that the worst-case performance is of special interest to risk managers (van Eekelen et al., 2025) who may favor a conservative approach, the technical advantage is that with any given action, the second-stage problem can be reformulated as a semi-infinite programming (SIP) model using duality. We label this model as the firm’s SIP model, which is linear and hence, the Karush–Kuhn–Tucker (KKT) conditions are sufficient and necessary. The practical relevance and computational tractability of this class of max-min models under moment conditions make them prevalent in the literature (see Section 2.1.2 of Lu and Shen, 2021: and the references therein).

Obtaining analytical solutions for the max-min problem via the two-stage optimization approach becomes increasingly challenging and often intractable as the size of the multi-sourcing problem grows. First, as the number of first-stage decision variables increases, the number of cases that need to be considered for solving the second-stage problem becomes overwhelming. Second, to determine the optimal actions for the first-stage problem, it is necessary to derive an explicit functional form for the worst-case profit of the second stage, given any specific first-stage decisions. However, the second-stage action-dependent SIP model frequently results in a piecewise and even implicit objective function, as the problem size increases due to the increasing number of binding SIP constraints. This makes the first-stage problem intricate and often impossible to solve analytically.

Scarf (1958) solved the following robust newsvendor problem with a single source:

$$\underset{Q\ge 0}{min}\underset{\theta \in \mathcal{D}(\mu ,\sigma )}{max}\mathbb{E}[(p-c)(\theta -Q{)}^{+}+c(Q-\theta {)}^{+}],$$

where $\mathcal{D}(\mu ,\sigma )$ denote the set of distributions with mean $\mu$ and standard deviation $\sigma$ . The optimal decision rule for the order quantity ${\hat{Q}}_{SF}$ is:

$$\begin{array}{r}{\hat{Q}}_{SF}=\{\begin{array}{ll}{\displaystyle \mu +\frac{\sigma }{2}(\sqrt{\frac{p-c}{c}}-\sqrt{\frac{c}{p-c}}),}& \text{if }0\le \frac{p-c}{p}\le \frac{\mu }{\mu +\sigma },\\ 0,& \text{otherwise}.\end{array}\end{array}$$

Because $pmin(\theta ,Q)-cQ=(p-c)\theta -(p-c)(\theta -Q{)}^{+}-c(Q-\theta {)}^{+}$ , ${\hat{Q}}_{SF}$ is also the optimal solution to

$$\underset{Q\ge 0}{max}\underset{\theta \in \mathcal{D}(\mu ,\sigma )}{min}\mathbb{E}[pmin(\theta ,Q)-cQ].$$

It is easy to verify that if this ordering policy is used, the minimum expected profit is either

$$(p-c)\mu -\sigma \sqrt{c(p-c)}$$

or $0$ , whichever is greater. Scarf’s min-max model is a cornerstone of robust decision-making, showing how to hedge against demand uncertainty using only mean and variance. Its simplicity and closed-form solution make it both practical and powerful. We propose a new method to generalize Scarf’s closed-form solution to the multi-sourcing procurement problem. This provides robust, data-light planning tools for today’s fragmented and multi-sourcing environments.

In our problem, we can source from $n$ suppliers (where the $(n+1)$ -th supplier represents lost sales), with holding cost $c_1=c>c_2>;\dots >c_{n+1}=0$ , and execution cost $r_1=0<r_2<\dots <r_{n+1}=p$ . The ability to derive a closed-form solution for such a complex two-stage optimization model is somewhat surprising. Typically, such models, given their layered and multifaceted nature, require numerical approaches or simulation techniques due to the intricate interdependencies between variables and stages. However, our approach simplifies the decision-making process by simultaneously solving for the firm’s robust optimal solution, effectively bypassing the usual iterative and computationally intensive second-stage analysis. This novel result significantly enhances the practical applicability and theoretical understanding of multi-sourcing strategies. The fact that the worst-case distribution, the optimal robust multi-source procurement vector, and even the worst-case expected profit can all be expressed in elegant, closed forms is both intriguing and highly beneficial, providing clear and actionable insights directly from the model’s analytical expressions. We demonstrate that, under a max-min criterion, the effects of demand-side variability and supply-side multi-sourcing contract parameters on robust capacity planning and profit performance can be effectively decoupled. On the demand side, increased variability leads to higher robust capacity levels but reduces the worst-case profit. On the supply side, we identify a critical constant, $\mathrm{\Delta }$ , defined entirely by the exogenous cost parameters of the supplier portfolio. This quantity, which replaces the classical term $\sqrt{c(p-c)}$ in Scarf’s single-source model, serves as a measure of the dispersion in net adjusted execution costs across the supply base. Our analysis reveals that a portfolio with a smaller $\mathrm{\Delta }$ yields higher robust capacity and improved profit performance. As a result, comparing supplier portfolios reduces to comparing their $\mathrm{\Delta }$ values—eliminating the need for distribution-specific integral calculations and offering clear, actionable guidance for multisourcing strategy evaluation in complex networks such as those of HP, Xerox, and modern multi-sourcing supply chains.

Moreover, the closed-form solution enables us to extend our analysis to other variants of the problems, with price dependent demand and fixed cost in inventory replenishment. It also facilitates the analysis of the portfolio effect, namely, the benefit of multisourcing relative to single sourcing. Interestingly, the firm can attain the full potential of the portfolio effect by engaging with only two sources with the right cost parameters. As a Chinese proverb indicates, “The excellence of soldiers outweighs the number.” We demonstrate that, from the worst-case perspective, contracting with the two most suitable suppliers is more important than naively increasing the size of the portfolio.

The remaining sections are organized as follows. Section 2 reviews the relevant literature. Section 3 introduces the model. Section 4 describes our methodology and presents the analytical results. Section5 discusses several extensions. Section 6 conducts a numerical study to demonstrate various results, investigate the portfolio effect, and quantify the value of information. In Section 7, we conclude the article. We present all the technical proofs in the Appendix.

2. Literature Review

Our article is related to two streams of literature. The first stream is inventory models with multiple supply sources. There has been a large body of literature in this stream. Studies have indicated the advantages of multi-sourcing, including hedging supply risks, reducing cost, improving flexibility, etc. In this stream of literature, supply sources differ not only in product prices but also in other critical dimensions such as lead time (e.g., Allon and Van Mieghem, 2010), flexibility (e.g., Saghafian and Van Oyen, 2012), quality (e.g., Mutha et al., 2019), yield (e.g., Federgruen and Yang, 2009), service level (e.g., Jin and Ryan, 2012), etc. As a result, the optimal procurement decision involves trade-offs between price and the other characteristics. Papers that are closely related to our study are Martínez-de Albéniz and Simchi-Levi (2009) and Fu et al. (2010). These papers consider supply sources with different reservation and execution costs (and thus different flexibility) and examine the optimal portfolio of sources as well as the optimal sourcing decisions that can increase the flexibility of the procurement process and thereby reducing the procurement cost. However, in this stream of research, the optimal inventory decisions are characterized on the basis of specific demand distributions. In contrast, in the this work we assume that the decision maker does not know the exact demand distribution and therefore, adopts a distributionally robust approach to planning sourcing decisions.

Our article also relates to the literature on robust operations management. We refer readers to Zhang et al. (2025) for an updated literature review. The research on optimal ordering decisions with limited information about the demand distribution is inspired by Scarf (1958) and has attracted growing interest. In this literature stream, the firm knows only the mean and variance of demand and aims to find an order quantity to maximize her expected profit against the worst possible distribution. With a single supply source, the worst distribution is a two-point distribution. The robust optimal order quantity and the resulting profit can be derived in closed forms. Gallego and Moon (1993) provide a more concise proof of Scarf’s result by using the Cauchy–Schwartz inequality. Natarajan et al. (2018) examine the impact of an asymmetric demand distribution by using second-order partitioned statistics to measure distributional asymmetry. Das et al. (2021) extend the moment information to include the $t$ -th moment of the demand distribution. It is possible that multiple decision makers could simultaneously face ambiguity. For instance, Fu et al. (2018) consider an agricultural supply chain where both the upstream and downstream entities face mean–variance ambiguity. They extend Scarf’s result to the case when there is distribution ambiguity in both the demand and selling price. Li and Kirshner (2021) label this ambiguous environment “two-sided ambiguity” and consider the contracting issue between the firm and her sales agent. Minimax regret is another robust decision rule commonly used in the literature. Yue et al. (2006) define the value of information (VOI) as the difference between knowing and not knowing the underlying demand distribution, while Perakis and Roels (2008) refer to VOI as regret (which measures forgone profit in the absence of full information on the underlying demand distribution). Yue et al. (2006) focus on mean–variance ambiguity, whereas Perakis and Roels (2008) consider a variety of partial information on the distribution, such as its mode, range, mean, and variance.

In the robust optimization literature, the difference between risk aversion and ambiguity aversion is highlighted as follows. A risk-averse decision maker prefers an order quantity that avoids profit volatility in addition to the expected profit, whereas an ambiguity-averse decision maker, lacking complete knowledge of the demand distribution, prefers an order quantity that is distributionally robust (Han et al., 2014). Incorporating the variance of the profit, Han et al. (2014) study a distributionally robust newsvendor model by combining both risk aversion and ambiguity aversion. Several recent articles (Kouvelis et al., 2021; Yang et al., 2021, 2018) employ the conditional value at risk as a measure of risk tolerance. With advances in machine learning, several recent articles have proposed data-driven methods to determine the robust order quantity. For instance, Chen and Xie (2021) consider an unknown joint distribution for demand and yield, whereas He and Lu (2021) and Li and Lu (2023) consider jointly optimizing the price and inventory policies based on limited information. Xing et al. (2025) consider a responsive pricing strategy to mitigate yield ambiguity from a single source. We combine the aforementioned two streams of literature by examining the distributionally robust capacity planning problem with multiple supply sources and propose a joint optimization approach.

3. Model

We consider a firm that faces random demand and procures from $n$ different sources, where each source can represent a different supplier, grade, or option contract. For ease of exposition, we use feminine pronouns for the firm. Let $\theta$ be a nonnegative random variable denoting the external demand for the firm’s end product. The cumulative distribution function of demand $\theta$ is $F(\theta )$ , which remains unknown to the firm. However, the firm knows the values of the mean ( $\mu >0$ ) and variance ( ${\sigma }^2>0$ ) associated with the demand distribution. Without complete knowledge of the demand distribution, the firm applies the robust max-min criterion to plan her capacity. Let $\rho =\sigma /\mu$ be the coefficient of variation of the demand.

The sourcing decisions are made in two stages. In Stage $1$ (which we refer to as the procurement stage before the selling season), the firm reserves $q_i$ units of capacities from each source $i$ ( $i=1,2,\dots ,n$ ) by paying a reservation cost of $ $c_i$ per unit. In Stage $2$ (which we refer to as the fulfillment stage), the external demand $\theta$ is realized, and the firm uses available capacities to produce the end products. When using the capacity of source $i$ to produce one unit of the end product, the firm incurs an execution cost of $ $r_i\ge 0$ per unit. The end products delivered from different sources have the same quality and functionality and thus are indistinguishable by customers. The firm sells the end products at a price $ $p$ per unit. If the reserved capacities are insufficient to satisfy all the external demands, the excess demand is lost. Any unused capacity expires with zero salvage value but the execution cost is avoided. We index the supply sources in decreasing order of the reservation costs $c_1>c_2>\cdots >c_n$ , which implies an increasing order in the execution cost $r_1<r_2<\cdots <r_n$ , because a supply source with both reservation and execution costs higher than another source must be dominated. For simplicity, we define source- $(n+1)$ as an artificial source representing lost sales, such that $c_{n+1}=0$ and $r_{n+1}=p$ . We use a bold capital letter to represent a vector. Let $\mathbf{q}=(q_1,q_2,\dots ,q_n)$ represent the capacity vector chosen by the firm in the procurement stage.

Note that the artificial source- $(n+1)$ can represent either lost sales or the spot market, depending on the interpretation of the mutli-sourcing newsvendor problem. From a profit maximization perspective, if the realized demand $\theta$ exceeds the total reservation quantity, the excess demand leads to lost sales, incurring a unit revenue loss of $p$ . In this context, source- $(n+1)$ represents lost sales. Conversely, from a cost minimization standpoint, the firm seeks to fulfill all realized demand $\theta$ . If the total reserved quantity is insufficient, the firm must procure additional units from the spot market at a unit price $p$ to meet the excess demand. Then the spot market serves as the (n+1)- $th$ resource. Under condition of a fixed selling price or spot price $p$ , the profit maximization and cost minimization problems are equivalent yielding the same optimal solution.

In the fulfillment stage, the capacity reservation decisions are sunk and the firm observes the realized demand $\theta$ . The firm solves the following linear programming model to maximize her ex post profit:

$$Z\left(\theta |\mathbf{q}\right)=-\sum _{i=1}^n{c}_i{q}_i+\underset{x_i\ge 0}{max}\left\{\sum _{i=1}^n(p-r_i)x_i\right\},$$

(1)

subject to the capacity availability constraints:

$$x_i\le q_i,\mathrm{\forall }i\in \{1,2,\dots ,n\},$$

(2)

and the total demand constraint:

$$x_1+x_2+\cdots +x_n\le \theta .$$

(3)

The nonnegative decision variable $x_i$ represents the quantity of the end products executed from source- $i$ . The capacity availability constraints (2) ensure that the quantity of the end products delivered by source- $i$ does not exceed the available capacity $q_i$ , and the total demand constraint (3) ensures that the total delivered quantities do not exceed the realized demand $\theta$ .

In the procurement stage, as the complete demand distribution is unknown, the firm employs the max-min robust decision rule to maximize the worst-case expected profit based on the mean and variance of demand. Let $\mathrm{\Omega }$ be the collection of all probability distributions (which can be continuous, discrete or mixed) satisfying the mean–variance constraints. Formally, we define the ambiguity set $\mathrm{\Omega }$ as follows:

$$\begin{array}{rl}\mathrm{\Omega }& =\{F\left(\theta \right)|{\int }_0^{\mathrm{\infty }}dF\left(\theta \right)=1,{\int }_0^{\mathrm{\infty }}\theta dF\left(\theta \right)\\ & =\mu ,{\int }_0^{\mathrm{\infty }}{\theta }^{2}dF\left(\theta \right)={\mu }^2+{\sigma }^2\}.\end{array}$$

(4)

The firm’s robust multisource capacity planning model is as follows:

$$Z=\underset{\mathbf{q}\ge \mathbf{0}}{max}\left\{\underset{F\in \mathrm{\Omega }}{inf}{\int }_0^{\mathrm{\infty }}Z\left(\theta |\mathbf{q}\right)dF\left(\theta \right)\right\},$$

(5)

in which the firm wishes to maximize her expected profit $Z$ by choosing a capacity vector $\mathbf{q}$ but nature wishes to minimize the firm’s expected profit by choosing a demand distribution from the ambiguity set $\mathrm{\Omega }$ .

4. Analysis

We first characterize the ex post profit function $Z(\theta |\mathbf{q})$ in the second stage of fulfillment and then analyze the robust capacity decisions in the first stage of procurement.

4.1. Fulfillment Stage

In the fulfillment stage, the total cost $\sum _{i=1}^n{c}_i{q}_i$ in equation (1) is sunk, and the coefficient $(p-r_i)$ decreases in $i$ as the execution cost $r_i$ increases in $i$ . Hence, the firm must prefer source $i$ to source $(i+1)$ , implying that the firm’s optimal fulfillment plan follows a priority rule such that source $(i+1)$ is not used unless the reserved capacity of source $i$ is exhausted.

For ease of exposition, we denote the total capacity for the first $i$ sources as $Q_i=\sum _{j=1}^i{q}_i$ . By default, we let $Q_0=0$ such that $Q_i\ge Q_{i-1}$ , which we refer to as the monotonic constraints on $Q_i$ . The definition of $Q_i$ resembles the echelon inventory level that Veinott (1965) proposed. Because the mapping between vectors $\mathbf{Q}=(Q_1,Q_2,\dots ,Q_n)$ and $\mathbf{q}=(q_1,q_2,\dots ,q_n)$ is unique, hereafter, we regard $Q_i$ as the decision variables to facilitate analysis. Let $(\cdot {)}^{+}=max(0,\cdot )$ .

Lemma 1

a) For any given capacity vector $\mathbf{Q}$ , the firm’s ex post profit equals

$$\begin{array}{rl}Z\left(\theta |\mathbf{Q}\right)& =\sum _{i=1}^n[(p-r_i)(min(Q_i,\theta )-min(Q_{i-1},\theta ))\\ & -c_i(Q_i-Q_{i-1})],\end{array}$$

(6)

which is continuous and concave with respect to the realized demand $\theta$ . b) In addition, for $\theta \in (Q_{k-1},Q_k]$ , the first derivative satisfies

$$\frac{Z\left(\theta |\mathbf{Q}\right)}{\mathrm{\partial }{Q}_i}=\frac{Z\left(\theta |\mathbf{Q}\right)}{\mathrm{\partial }{q}_i}=\{\begin{array}{ll}{r}_k-r_i-c_i& \text{if }k\ge i+1\\ -c_i& \text{if }k\le i.\end{array}$$

(7)

The ex post $Z(\theta |\mathbf{Q})$ function in equation (6) is piecewise linear and has $(n+1)$ different cases, depending on the value of realized demand $\theta$ . Specifically, the first derivative $(\mathrm{\partial }Z(\theta |\mathbf{Q}))/\mathrm{\partial }\theta$ is a multistep function that is nonnegative and decreasing in $\theta$ , making $Z(\theta |\mathbf{Q})$ concave and increasing in $\theta$ . To facilitate further analysis, we also characterize the first derivatives $(\mathrm{\partial }Z(\theta |\mathbf{Q}))/\mathrm{\partial }{Q}_i$ and summarize their expressions in the following Table 1. Additionally, the value of $Z(\theta |\mathbf{Q})$ is finite for any $\theta$ due to lost sales. Thus, for any finite $\mathbf{Q}$ vector, the expected profit ${\mathbb{E}}_{\theta }[Z(\theta |\mathbf{Q})]$ is finite, where $\mathbb{E}$ is the expectation operator.

Table 1.

The patterns in the first derivative $\mathrm{\partial }Z(\theta |\mathbf{Q})/\mathrm{\partial }{Q}_i$ .

Interval	$\frac{\mathrm{\partial }Z(\theta |Q)}{\mathrm{\partial }{Q}_1}$	$\frac{\mathrm{\partial }Z(\theta |Q)}{\mathrm{\partial }{Q}_2}$	$\frac{\mathrm{\partial }Z(\theta |Q)}{\mathrm{\partial }{Q}_3}$	$\cdots$	$\frac{\mathrm{\partial }Z(\theta |Q)}{\mathrm{\partial }{Q}_n}$
$[0,Q_1]$	$-c_1$	$-c_2$	$-c_3$		$-c_n$
$(Q_1,Q_2]$	$r_2-r_1-c_1$	$-c_2$	$-c_3$		$-c_n$
$(Q_2,Q_3]$	$r_3-r_1-c_1$	$r_3-r_2-c_2$	$-c_3$		$-c_n$
$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
$(Q_{n-1},Q_n]$	$r_n-r_1-c_1$	$r_n-r_2-c_2$	$r_n-r_3-c_3$		$-c_n$
$(Q_n,\mathrm{\infty })$	$p-r_1-c_1$	$p-r_2-c_2$	$p-r_3-c_3$		$p-r_n-c_n$

4.2. Unambiguous Benchmark

Before formulating the robust model, we examine the unambiguous benchmark, which assumes that the demand distribution $F(\theta )$ is known. The firm’s expected profit equals $Z(\mathbf{Q})={\int }_0^{\mathrm{\infty }}Z(\theta |\mathbf{Q})dF(\theta )$ . Let $\tilde{\mathbf{Q}}$ be the firm’s distribution-dependent optimal capacity plan. For ease of expression, we define two sequences as follows.

Definition 1

Let ${\alpha }_0=0$ and ${\alpha }_{n+1}=1$ . Define

$$\begin{array}{r}\{\begin{array}{ll}{\alpha }_i=1-\frac{c_i-c_{i+1}}{r_{i+1}-r_i},& \text{for }i=1,2,\dots ,n,\\ {\beta }_i={\alpha }_i-{\alpha }_{i-1},& \text{for }i=1,2,\dots ,n+1.\end{array}\end{array}$$

(8)

We note that ${\alpha }_i$ and ${\beta }_i$ are invariant with the demand distribution and depend only on the exogenous cost parameters of the supply sources.

Lemma 2

In the benchmark without ambiguity, the firm’s expected profit $Z(\mathbf{Q})$ is concave in $\mathbf{Q}$ , and the optimal capacity vector $\tilde{\mathbf{Q}}$ satisfies $F({\tilde{Q}}_i)={\alpha }_i$ , implying that the firm’s optimal capacity level for source- $i$ equals

$${\tilde{q}}_i=F^{-1}\left({\alpha }_i\right)-F^{-1}\left({\alpha }_{i-1}\right)\text{ for any }i=1,2,\dots ,n,$$

(9)

where $F^{-1}$ is the inverse function of $F$ .

Lemma 2 indicates that the optimal capacity plan $(Q_1,Q_2,\dots ,Q_n)$ can be characterized by a sequence of percentiles $({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$ on the demand distribution, so ${\alpha }_i$ must be increasing in $i$ to ensure that ${\tilde{q}}_i>0$ . In the proof of Lemma 2 (we refer readers to equation (EC.4) in Appendix A of the E-Companion), we expand the firm’s expected profit as the sum of $n$ separate identities. Each of these $n$ identities involves only one $Q_i$ (which is the total quantity of the first $i$ sources) and is concave in $Q_i$ . The definition of $Q_i$ must imply that $Q_i\ge Q_{i-1}$ . After relaxing the monotonic constraint on $Q_i$ and solving the FOC, we obtain that $F({\tilde{Q}}_i)={\alpha }_i$ . Thus, if ${\alpha }_i$ is monotonic in $i$ , then the candidate solution ${\tilde{Q}}_i$ is increasing in $i$ , making ${\tilde{q}}_i={\tilde{Q}}_i-{\tilde{Q}}_{i-1}$ optimal and positive. If ${\alpha }_i\le {\alpha }_{i-1}$ , then the monotonic constraint $Q_i\ge Q_{i-1}$ must be binding, making $q_i=Q_i-Q_{i-1}=0$ (implying that source $i$ is not used in the optimal solution). We emphasize that the binding status of the monotonic constraint $Q_i\ge Q_{i-1}$ depends on the monotonicity of ${\alpha }_i$ rather than the demand distribution $F$ . As such, ${\beta }_i$ is a mass probability sequence. We can verify that i) $\sum _{i=1}^{n+1}{\beta }_i=1$ and that ii) ${\beta }_i\ge 0$ if and only if ${\alpha }_{i-1}\le {\alpha }_i$ . To ensure that all supply sources are used, we require that ${\beta }_i>0$ for all $i=1,2,\dots ,n+1$ . If the mass probability ${\beta }_i$ is negative, source $i$ is never used in the optimal solution (i.e., $q_i=0$ is optimal). Thus, we can eliminate source $i$ from the analysis and reduce the number of available sources by one.

We refer to all supply sources as a portfolio. We can visualize a portfolio by depicting each source as a point in a two-dimensional plane, where the horizontal axis is the execution cost $r_i$ and the vertical axis is the reservation cost $c_i$ . By connecting the points from source $1$ to source $(n+1)$ (the artificial source representing lost sales), we obtain a path with $n$ linear segments visualizing the portfolio. From Definition 1 and Lemma 2, the contracts with a positive order quantity must lie on the lower convex hull, characterized by the property in Remark 1. This property has been shown in the literature (e.g., Fu et al., 2010; Martínez-de Albéniz and Simchi-Levi, 2009).

Remark 1

The path of the optimal portfolio is decreasing and convex, with the sequence of slopes $0<(c_i-c_{i+1})/(r_{i+1}-r_i)<1$ , which is strictly decreasing in $i$ , implying that the parameters satisfy (i) $c_1+r_1<c_2+r_2<\cdots <c_n+r_n<p$ and (ii) $r_1<r_2<\cdots <r_n$ and $c_1>c_2>\cdots >c_n$ .

Remark 1 explains the underlying logic for the multiple sourcing strategy. Source $1$ has the lowest overall cost (reservation plus execution costs) despite having the highest reservation cost. A supply source is less flexible if it requires a higher reservation cost, because unused capacity is wasted. As the index $i$ increases, the reservation cost decreases, but the total cost increases, thus reflecting the trade-off between reservation (flexibility) and execution costs among the $n$ sources.

The property in Remark 1 remains valid in the distributionally robust setting. Thus, for any given portfolio, we can ex ante (rather than ex post) identify which source to exclude. Specifically, the source appearing above the lower convex hull has zero quantity and hence, can be eliminated. Without loss of generality, we focus on the supply sources with positive order quantities. Equivalently, ${\beta }_i$ is positive for all $i$ (or equivalently, ${\alpha }_i$ ’s are increasing in $i$ ).

With the optimal capacity vector $\tilde{\mathbf{Q}}$ , the optimal expected profit for the unambiguous benchmark equals:

$$\begin{array}{rl}{Z}_F^{*}& =\sum _{i=1}^n[(p-r_i)(Emin(\theta ,{\tilde{Q}}_i)-Emin(\theta ,{\tilde{Q}}_{i-1}))\\ & -c_i({\tilde{Q}}_i-{\tilde{Q}}_{i-1})],\end{array}$$

(10)

where the first and second terms correspond to the expected second-stage profit and first-stage reservation costs, respectively. Utilizing the optimal capacity levels $F({\tilde{Q}}_i)={\alpha }_i$ in Lemma 2, we can simplify $Z_F^{*}$ as follows.

Lemma 3

In the benchmark without ambiguity, the firm’s optimal expected profit is:

$$\begin{array}{rl}{Z}_F^{*}& =p{\int }_0^{{\tilde{Q}}_n}\theta dF(\theta )-r_1{\int }_0^{{\tilde{Q}}_1}\theta dF(\theta )-\sum _{i=2}^n{r}_i{\int }_{{\tilde{Q}}_{i-1}}^{{\tilde{Q}}_i}\theta dF(\theta )\\ & =p\mu -r_1{\int }_0^{{\tilde{Q}}_1}\theta dF(\theta )-\sum _{i=2}^n{r}_i{\int }_{{\tilde{Q}}_{i-1}}^{{\tilde{Q}}_i}\theta dF(\theta )\\ & -p{\int }_{{\tilde{Q}}_n}^{\mathrm{\infty }}\theta dF(\theta ).\end{array}$$

(11)

The optimal expected profit in the unambiguous case equals the mean revenue minus the weighted sum of the conditional means of demand, which involves calculating the integrals based on the demand distribution and the optimal capacity plan.

4.3. Procurement Stage

In the first stage of procurement, the firm makes the capacity reservation decisions $\mathbf{Q}$ to maximize her worst-case profit in equation (5). The ambiguity set $\mathrm{\Omega }$ is defined by the total probability, mean, and variance constraints. Because the ex post profit function $Z(\theta |\mathbf{Q})$ is continuous with respect to $\theta$ , we can establish that the firm’s worst distribution reduces to a class of discrete distributions. Thus, we reformulate equation (5) as the following optimization model:

$$\begin{array}{rl}P& =\underset{\mathbf{Q}\ge \mathbf{0}}{max}\underset{y_0,y_1,y_2}{max}\{y_0+y_1\mu +y_2({\mu }^2+{\sigma }^2)\}\\ & \text{s.t. }{y}_0+y_1\theta +y_2{\theta }^2\le Z\left(\theta |\mathbf{Q}\right),\mathrm{\forall }\theta \ge 0,\end{array}$$

(12)

where $y_0$ , $y_1$ , and $y_2$ are the shadow prices associated with the total probability, mean, and variance constraints in the general finite sequence space.

Lemma 4

For any finite $\mathbf{Q}$ vector, $P=Z$ holds, and hence, the maximization problems in equations (5) and (12) are equivalent.

The optimization model in equation (12) is an SIP model with a finite number of decision variables (i.e., $y_i,Q_i$ ) but an infinite number of constraints (because $\theta$ is an exogenous nonnegative state variable and each $\theta$ specifies one constraint). The SIP model in equation (12) has a neat interpretation. While nature sells, the firm purchases three types of probability resources in a hypothetical trade. The prices for the total probability resource, the mean resource, and the variance resource are $y_0$ , $y_1$ , and $y_2$ , respectively. When the realized demand is $\theta$ , the firm must purchase one unit of the total probability resource, $\theta$ units of the mean resource, and ${\theta }^2$ units of the variance resource from nature. Thus, the expected sales revenue of nature equals $y_0+y_1\mu +y_2({\mu }^2+{\sigma }^2)$ . However, nature cannot set the prices of the probability resources prohibitively high. Specifically, the ex post revenue cannot exceed the firm’s ex post profit such that for any realized state $\theta$ , there must be a trade between the two parties, giving rise to the SIP constraints $y_0+y_1\theta +y_2{\theta }^2\le Z(\theta |\mathbf{Q})$ .

The conventional approach to solving equation (12) starts with the inner maximization and then the outer maximization model. For any given $\mathbf{Q}$ , the inner maximization problem in equation (12) is a linear SIP model such that the KKT conditions are sufficient and necessary. After solving the inner maximization problem, we can explicitly derive the objective function $Z_{\mathrm{wst}}(\mathbf{Q})$ , where the subscript $\mathrm{wst}$ denotes the worst case, and the next step is to solve the first-order conditions (FOCs) to obtain ${\mathbf{Q}}^{*}$ .

In contrast, separating equation (12) into two stages is unnecessary and may hinder the analysis. For example, Das et al. (2021: pages 1097-1098) explain the “implausibility” in deriving $Z_{\mathrm{wst}}(q)$ when the mean and the $t$ -th (where $t>1$ is a real number) moment are known. This problem is often referred to as the $1+t$ problem. When the explicit expression of the objective function $Z_{\mathrm{wst}}(q)$ remains analytically unavailable, the extant literature fails to produce the robust inventory level $q^{*}$ in closed form. We propose a joint optimization method to overcome the relevant technical challenge. ¹ In the multisource capacity planning problem, the linear SIP model in the second stage remains difficult to solve or produces an objective function $Z_{\mathrm{wst}}(\mathbf{Q})$ that is too complex to advance the analysis. The specific reason is that the ex post $Z(\theta |\mathbf{Q})$ function, which appears on the right-hand-side (RHS) of the SIP constraints, has $(n+1)$ pieces (or linear segments). The binding SIP constraints involve at least two of these linear segments, resulting in $(2^{n+1}-n-2)$ possible cases. As $n$ increases, the complexity of the inner maximization model grows, making the outer maximization model difficult to solve. The greatest convenience of our joint optimization approach is that we can directly attack the robust optimal solution ${\mathbf{Q}}^{*}$ without being stuck in the intermediate process of solving the inner model of equation (12).

After merging the two maximization operators in equation (12), we obtain the following joint optimization model:

$$\begin{array}{rl}P& =\underset{\mathbf{Q}\ge \mathbf{0}\mathbf{,}{y}_0,y_1,y_2}{max}\{y_0+y_1\mu +y_2({\mu }^2+{\sigma }^2)\}\\ & \text{s.t. }{y}_0+y_1\theta +y_2{\theta }^2\le Z\left(\theta |\mathbf{Q}\right),\mathrm{\forall }\theta \ge 0,\end{array}$$

(13)

in which the objective function is linear with respect to the decision variables $(Q_i,y_i)$ . As $\theta$ is a state variable (rather than a decision variable), the constraints in equation (12) are convex with respect to decision variables. We conclude that KKT conditions are sufficient and necessary for solving this joint optimization model, giving rise to the following three steps:

• First, we conjecture where the binding constraints are located. Via visualization, Figure 1(a) shows that the left-hand-side (LHS) of the constraint (12) is a quadratic function of the state variable $\theta$ . The RHS is the ex post function $Z(\theta |\mathbf{Q})$ , which is continuous, concave, and piecewise linear in $\theta$ with $(n+1)$ pieces. The constraint results in $(n+1)$ binding points, with one binding point in each piece of the ex post function $Z(\theta |\mathbf{Q})$ , which implies that for any distribution $F$ , this $(n+1)$ -point discrete distribution yields the same expected profit. Thus, the SIP model has $(n+1)$ binding points. At these $(n+1)$ binding points, the LHS and RHS curves are also tangent to each other. We visualize the tangent conditions in Figure 1(b). When the shadow prices $y_1$ and $y_2$ are fixed, the LHS of the tangent condition is $y_1+2y_2\theta$ , forming a straight line with respect to $\theta$ . The RHS of the tangent condition is the first derivative of the ex post function with respect to $\theta$ , which is a multistep function according to part a) of Lemma 1. The height of each step depends on the cost parameters but the length of each step depends on $Q_i$ . A sub-optimal $\mathbf{Q}$ vector could result in less than $(n+1)$ intercepts between the multistep function and the straight line, whereas the optimal vector always creates $(n+1)$ intercepts. Thus, the marginal view of the constraints is also useful.

• Second, based on the conjectured binding constraints, we apply a relaxation method to establish the following Lagrangian function:

  $$\begin{array}{rl}\mathcal{L}& =y_0+y_1\mu +y_2({\mu }^2+{\sigma }^2)\\ & -{\lambda }_1[y_0+y_1{\theta }_1+y_2{\theta }_1^2-Z\left({\theta }_1|\mathbf{Q}\right)]\\ & -{\lambda }_2[y_0+y_1{\theta }_2+y_2{\theta }_2^2-Z\left({\theta }_2|\mathbf{Q}\right)]\\ & -\cdots -{\lambda }_{n+1}[y_0+y_1{\theta }_{n+1}+y_2{\theta }_{n+1}^2-Z\left({\theta }_{n+1}|\mathbf{Q}\right)],\end{array}$$

  (14)
  where points ${\theta }_i$ ( $i=1,2,\dots ,n+1$ ) make the SIP constraints (12) binding. When optimizing the Lagrangian, we apply the KKT conditions to the decision variables $(Q_i,y_i)$ , the Lagrangian multipliers, and the tangent conditions on ${\theta }_i$ , which is equivalent to the FOCs with respect to ${\theta }_i$ . Notably, the tangent condition is the necessary condition for the omitted SIP constraints to hold. For instance, if the SIP constraint is binding at point ${\theta }_i$ , then the omitted constraints include a continuum of $\theta$ in the neighborhood of ${\theta }_i$ (i.e., $\theta \in [{\theta }_i-ϵ,{\theta }_i+ϵ]$ where $ϵ>0$ is an arbitrarily small real number). When the FOC $(\mathrm{\partial }\mathcal{L}({\theta }_i|\mathbf{Q}))/\mathrm{\partial }{\theta }_i=0$ and the second order condition (e.g., $y_2<0$ ) hold, the omitted SIP constraints neighboring the point ${\theta }_i$ still hold.

• Third, we thoroughly verify whether the relaxed solution based on the Lagrangian in equation (14) is feasible for the omitted SIP constraints. If the relaxed solution is indeed feasible for all the constraints, the relaxed solution is globally optimal.

Figure 1.

SIP constraints with $n=3$ , $(c_1,r_1)=(10,2)$ , $(c_2,r_2)=(5,8)$ , $(c_3,r_3)=(3,14)$ , $p=25$ , $\mu =5$ , $\sigma =3$ . (a) Original View; (b) Marginal View.

The next Proposition 1 summarizes all the optimality conditions for the joint optimization model.

Proposition 1

The optimal solution to the distributionally robust model in equation (12) satisfies the following four sets of optimality conditions:

The firm’s FOCs: $\mathrm{\partial }\mathcal{L}/\mathrm{\partial }{Q}_i=0$ , for each $Q_i$ , where $i=1,2,\dots ,n$ ;

Binding conditions: $y_0+y_1{\theta }_i+y_2{\theta }_i^2=Z({\theta }_i|\mathbf{Q})$ at each binding point ${\theta }_i$ , where $i=1,\dots ,n+1$ ;

Tangent conditions: $y_1+2y_2{\theta }_i=(\mathrm{\partial }Z({\theta }_i|\mathbf{Q}))/\mathrm{\partial }{\theta }_i$ at each tangent point ${\theta }_i$ , where $i=1,\dots ,n+1$ ;

Moment constraints.

Overall, we encounter the following four sets of FOCs:

$$\begin{array}{rl}\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{Q}_i}& =0,\text{for}i=1,2,\dots ,n,(n\text{ FOCs on decision variables)},\\ \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\lambda }_i}& =0,\text{for}i=1,2,\dots ,n+1,(n+1\text{ binding conditions)},\\ \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\theta }_i}& =0,\text{for}i=1,2,\dots ,n+1,(n+1\text{ tangent conditions)},\\ \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{y}_i}& =0,\text{for}i=0,1,2,\text{(3 moment conditions)}.\end{array}$$

The entire system has $(3n+5)$ unknown variables, including the original $n$ decision variables $(Q_1,Q_2,\dots ,Q_n)$ , $(n+1)$ binding points $({\theta }_1,{\theta }_2,\dots ,{\theta }_{n+1})$ , $(n+1)$ Lagrangian multipliers, and $3$ dual variables $(y_0,y_1,y_2)$ associated with moment constraints. The system also establishes $(3n+5)$ equations, including $n$ FOCs with respect to $Q_i$ ; $(n+1)$ FOCs with respect to ${\theta }_i$ ; $(n+1)$ FOCs with respect to Lagrangian multipliers; and $3$ FOCs with respect to $(y_0,y_1,y_2)$ . The complexity of solving the above system of $(3n+5)$ equations depends on the functional form of $Z(\theta |\mathbf{Q})$ and the moment conditions. Notably, the traditional two-stage method isolates the $n$ FOCs with respect to $Q_i$ , which prompts us to consider many different cases (as the changes in vector $\mathbf{Q}$ inevitably affect the binding constraints). Instead, we bring these $n$ FOCs forward, enabling us to obtain closed-form solutions more conveniently. We recognize that a joint optimization method may have been numerically used; however, we are unaware of any published articles that used a joint optimization method to derive closed-form solutions as neat as those we present in the next few sections. ²

4.4. The Worst Case Distribution

Before characterizing the worst case distribution associated with equation (12), we define several important constants as follows.

Definition 2

Let ${\delta }_i=r_i-r_1-c_1,\text{ for }i=1,2,\dots ,n+1$ . We define $\mathrm{\Delta }$ as a positive constant satisfying $\mathrm{\Delta }=\sqrt{\sum _{i=1}^{n+1}{\beta }_i{\delta }_i^2}$ , where $\{{\beta }_i\}$ is the mass probability sequence given by Definition 1.

In Definition 2, ${\delta }_i=(Z(\theta |\mathbf{Q}))/\mathrm{\partial }{Q}_1$ is the first derivative with respect to $Q_1$ for the $i$ th case in equation (7) or the second column in Table 1. We can interpret the value of ${\delta }_i$ as a marginal impact (or net adjusted execution cost with respect to the baseline supplier $1$ ). $\mathrm{\Delta }$ is therefore a measure of the dispersion of these net adjusted execution cost parameters. In our model, the firm can access $n\ge 1$ sources with source $1$ being her most preferred source. If the demand is deterministic, the firm can use only source $1$ in her capacity plan. However, due to random demand, the firm chooses multiple sources with lower reservation costs to enhance flexibility. When the realized demand satisfies that $Q_{i-1}\le \theta <Q_i$ (where by default $Q_0=0$ ), source- $i$ still has some unused capacity. If the firm increases the capacity of source- $1$ from $q_1$ to $q_1+\epsilon$ and keeps all the other $q_i$ ’s ( $i\ge 2$ ) unchanged, then the sales quantity of source- $1$ increases by $\epsilon$ while that of source- $i$ deceases by $\epsilon$ units (where $\epsilon \in (0,Q_i-\theta )$ is a small positive number). The marginal impact includes two parts: (i) the change in the sales revenue due to the additional $\epsilon$ units of source- $1$ capacity and(ii ) the reservation cost of the additional $\epsilon$ units of source- $1$ capacity. The net impact equals

$$(p-r_1)-(p-r_i)-c_1=r_i-r_1-c_1\stackrel{def}{=}{\delta }_i.$$

In a notable special case where $i=(n+1)$ and the realized demand satisfies $\theta >Q_n$ , the firm suffers lost sales. With the artificial source $(n+1)$ , where $r_{n+1}=p$ and $c_{n+1}=0$ , we observe that ${\delta }_{n+1}=r_{n+1}-r_1-c_1=p-r_1-c_1>0$ .

The next proposition identifies the firm’s worst demand distribution.

Proposition 2

The firm’s worst demand distribution $F^{*}$ is a discrete $(n+1)$ -point distribution characterized as follows. The $i$ -th possible realization of $\tilde{\theta }$ is given by:

$${\theta }_i^{*}\stackrel{def}{=}\mu +{\delta }_i\frac{\sigma }{\mathrm{\Delta }}\text{, for }i=1,2,\dots .,n+1,$$

(15)

with the following probability:

$$Pr(\theta ={\theta }_i^{*})={\beta }_i,\text{ for }i=1,2,\dots ,n+1.$$

(16)

The worst distribution in Proposition 2 has a nice and neat form. The marginal impact ${\delta }_i$ is an increasing sequence by definition. The $(n+1)$ points are characterized by a linear transformation of ${\delta }_i$ , that is, adding a constant $\sigma /\mathrm{\Delta }$ scaled by ${\delta }_i$ to the mean of demand $\mu$ , which increases in $i$ . The probability associated with the $i$ -th demand realization ${\theta }_i^{*}$ is ${\beta }_i$ , which is the same probability as that under unambiguous demand for the demand realization to fall into the $i$ -th interval of $[Q_{i-1},Q_i]$ given the same cost parameters. We recall that ${\theta }_i\in [Q_{i-1},Q_i]$ ; thus, the $(n+1)$ -point distribution has the same partition in probability space but aggregates the cumulative probability within the interval to a probability mass at a discrete point in the interval. Because the demand is nonnegative, the lowest demand realization must satisfy ${\theta }_1^{*}=\mu -c_1(\sigma /\mathrm{\Delta })\ge 0$ . We assume that $(p-r_1-c_1)\mu \ge \mathrm{\Delta }\sigma$ to avoid the uninteresting case in which the firm ceases operations by employing a zero capacity vector. We show a critical relationship between the two coefficient sequences ${\beta }_i$ and ${\delta }_i$ for $i=1,\dots ,n+1$ in the next proposition.

Proposition 3

It holds that $\sum _{i=1}^{n+1}{\beta }_i{\delta }_i=0$ .

Proposition 3 uncovers an intriguing finding that the mean of the sequence of marginal impacts, encompassing all possible demand scenarios of $\theta$ , is zero. This means that although the marginal impact transitions from negative to positive as $\theta$ increases, the expected effect on the profit balances out to zero. With this property, we can verify that the discrete distribution shown in Proposition 2 satisfies the mean and variance constraints. Equation (15) indicates that ${\theta }_i^{*}$ is a linear transformation of ${\delta }_i$ . Thus, we immediately obtain that

$$\sum _{i=1}^{n+1}{\beta }_i{\theta }_i^{*}=\sum _{i=1}^{n+1}{\beta }_i(\mu +\frac{\sigma }{\mathrm{\Delta }}{\delta }_i)=\mu +\frac{\sigma }{\mathrm{\Delta }}\sum _{i=1}^{n+1}{\beta }_i{\delta }_i=\mu$$

and

$$\begin{array}{rl}\sum _{i=1}^{n+1}{\beta }_i{({\theta }_i^{*}-\mu )}^2& =\sum _{i=1}^{n+1}{\beta }_i{\left(\frac{\sigma }{\mathrm{\Delta }}{\delta }_i\right)}^2\\ & =\frac{{\sigma }^2}{{\mathrm{\Delta }}^2}\sum _{i=1}^{n+1}{\beta }_i{\delta }_i^2=\frac{{\sigma }^2}{{\mathrm{\Delta }}^2}{\mathrm{\Delta }}^2={\sigma }^2,\end{array}$$

where the third equality is based on the definition of $\mathrm{\Delta }$ . Hence, the discrete distribution shown in Proposition 2 is an element of the ambiguity set $\mathrm{\Omega }$ satisfying the mean and variance constraints. Propositions 2 and 3 significantly advance our analysis by paving the way toward determining a robust multisource capacity plan and the firm’s worst-case expected profit (or the firm’s optimal utility).

4.5. Robust Optimal Capacity Vector

After obtaining the most unfavorable distribution, the binding and tangent conditions are linear with respect to the remaining decision variables. We can then completely characterize the firm’s robust capacity plan.

Proposition 4

The firm’s robust optimal capacity vector satisfies ${\mathbf{Q}}^{*}=(Q_i^{*})=(({\theta }_i^{*}+{\theta }_{i+1}^{*})/2)$ for $i=1,2,\dots ,n$ , implying that the robust optimal capacity level for source- $i$ (where $i\ge 2$ ) equals

$$q_i^{*}=\frac{{\theta }_i^{*}+{\theta }_{i+1}^{*}}{2}-\frac{{\theta }_{i-1}^{*}+{\theta }_i^{*}}{2}=\frac{{\theta }_{i+1}^{*}-{\theta }_{i-1}^{*}}{2}=\frac{{\delta }_{i+1}-{\delta }_{i-1}}{2}\frac{\sigma }{\mathrm{\Delta }}.$$

(17)

Proposition 4 solves the robust optimal capacities in closed form by demonstrating that in the robust optimal solution, the total capacities of the first $i$ sources equal to the midpoint of the closed interval $[{\theta }_i^{*},{\theta }_{i+1}^{*}]$ . The discrete distribution in Proposition 2 is critical for unlocking all the results. Because we can ex ante construct the sequences $\{{\beta }_i\}$ and $\{{\delta }_i\}$ by using exogenous cost parameters, the robust optimal capacity vector ${\mathbf{Q}}^{*}$ is easy to compute.

Proposition 5

The optimal solution for equation (12) satisfies $y_0^{*}=-(\mathrm{\Delta }/2\sigma )({\mu }^2+{\sigma }^2)$ , $y_1^{*}=(p-r_1-c_1)+\mathrm{\Delta }(\mu /\sigma )$ , $y_2^{*}=-(\mathrm{\Delta }/2\sigma )$ , and $Q_i^{*}=(({\theta }_i^{*}+{\theta }_{i+1}^{*})/2)$ . The firm’s optimized utility equals

$$Z^{*}=Z\left({\mathbf{Q}}^{*}|F^{*}\right)=(p-r_1-c_1)\mu -\mathrm{\Delta }\sigma .$$

(18)

Proposition 5 characterizes the firm’s optimal utility $Z^{*}$ in a clean and neat form. We observe that $(p-r_1-c_1)$ is the understock cost of the firm’s most profitable source- $1$ capacity. With deterministic demand, the firm can solely use source- $1$ and gain a profit of the first term $(p-r_1-c_1)\mu$ .

With random demand, the term $\mathrm{\Delta }\sigma$ in the robust objective neatly decouples the effects of demand variability (via $\sigma$ ) and supply-side cost heterogeneity (via $\mathrm{\Delta }$ ). In contrast, in the distribution-dependent case, using the definition of ${\delta }_i$ , the expected profit in Equation (11) becomes:

$$\begin{array}{rl}{Z}_F^{*}& =(p-r_1-c_1)\mu -{\delta }_1{\int }_0^{{\tilde{Q}}_1}\theta dF(\theta )\\ & -\sum _{i=2}^n{\delta }_i{\int }_{{\tilde{Q}}_{i-1}}^{{\tilde{Q}}_i}\theta dF(\theta )-{\delta }_{n+1}{\int }_{{\tilde{Q}}_n}^{\mathrm{\infty }}\theta dF(\theta ),\end{array}$$

where the demand variability and multi-sourcing effects are intricately entangled. These integrals are generally intractable, and the integration limits ${\tilde{Q}}_i$ , which depend on the optimal capacity plan, lack closed-form expressions.

By contrast, Equation (18) offers a transparent characterization of multi-sourcing performance and the fundamental mean–variance tradeoff. The worst-case profit increases with average demand $\mu$ but is penalized by demand variability through the term $\sigma \mathrm{\Delta }$ . The constant $\mathrm{\Delta }$ , defined in Definition 2, is determined entirely by exogenous supply-side cost parameters. It not only specifies the worst-case distribution and robust capacity allocation but also serves as a measure of portfolio quality: a smaller $\mathrm{\Delta }$ implies better worst-case profit performance.

This insight enables efficient portfolio evaluation. For instance, if the optimal supplier pool is large, managing it may incur high administrative costs. A firm may prefer to select a smaller subset of $k$ key suppliers. While directly solving Equation (11) for all $\left(\begin{array}{c}n\\ k\end{array}\right)$ subsets is computationally prohibitive, the robust formulation in Equation (18) reduces the task to computing $\mathrm{\Delta }$ for each candidate subset. The subset with the smallest $\mathrm{\Delta }$ yields the best robust performance, simplifying supplier selection without sacrificing quality.

If we regard the marginal impact $\delta$ as a random variable (caused by the randomness in demand), $\mathrm{\Delta }$ can be interpreted as the standard deviation of the marginal impact. With the term $-r_1-c_1$ in ${\delta }_i$ being unchanged, $\mathrm{\Delta }$ can also be interpreted as the standard deviation of the processing cost $r_i$ . The probability sequence ${\beta }_i$ determines the probability that the firm incurs the execution cost $r_i$ . Overall, equation (18) reveals that the understock cost of source- $1$ and the standard deviation of the marginal impact ${\delta }_i$ determines the balance of the mean–variance tradeoff.

5. Extensions

This section discusses a few extensions of the baseline model.

Figure 2.

Price-dependent demand with $n=4$ sources. (a) Additive Demand; (b) Multiplicative Demand.

5.1. Price-Dependent Demand

Let $\mu (p)=E(\theta )$ and $\sigma (p)=\sqrt{Var(\theta )}$ be the price-dependent mean and standard deviation of the random demand $\theta$ . To jointly determine the price and capacity plan, we solve the following model:

$$Z(p)=\underset{p,\mathbf{Q}}{max}\{(p-r_1-c_1)\mu \left(p\right)-\sigma \left(p\right)\mathrm{\Delta }\left(p\right)\}.$$

(19)

We consider two commonly used price-dependent demand models: the additive and multiplicative demand models. First, in an additive demand model, $\mu =A-bp$ and $\sigma ={\sigma }_0$ for any $p$ , where $(A,b,{\sigma }_0)$ are positive constants. We obtain the following objective function:

$$Z=\underset{p,\mathbf{Q}}{max}\{(p-r_1-c_1)(A-bp)-{\sigma }_0\mathrm{\Delta }(p)\}.$$

In a multiplicative demand model, $\mu =A-bp$ and $\sigma ={\rho }_0(A-bp)$ , where ${\rho }_0$ is a constant unaffected by $p$ . We obtain the following objective function:

$$Z=\underset{p,\mathbf{Q}}{max}\{((p-r_1-c_1)-{\rho }_0\mathrm{\Delta }(p))(A-bp)\}.$$

However, the FOC with respect to $p$ does not have a neat expression due to the following reasons. First, the price $p$ affects the cumulative probability ${\alpha }_n=1-(c_n/p-r_n)$ and the marginal impact ${\delta }_{n+1}=p-r_1-c_1$ . When $p\ge r_n+(c_n/(1-{\alpha }_{n-1}))$ , the price affects the triplet $({\beta }_n,{\beta }_{n+1},{\delta }_{n+1})$ and none of the $n$ sources is excluded. In contrast, when $p<r_n+(c_n/(1-{\alpha }_{n-1}))$ , source $n$ and possibly other sources are excluded and price $p$ affects the new triplet $({\beta }_k,{\beta }_{k+1},{\delta }_{k+1})$ , where $k$ represents the highest index of the sources included in the optimal portfolio. This observation implies that the objective function is piece-wise. Nonetheless, we can numerically solve these price-capacity joint control models by capitalizing on the closed form expression in equation (18).

Figure 2 illustrates a numerical example with four supply sources under additive or multiplicative demand cases. The parameters include: $A=50$ , $b=1$ , ${\sigma }_0={\rho }_0=0.6351$ , $(c_1,r_1)=(8,0)$ , $(c_2,r_2)=(6,3)$ , $(c_3,r_3)=(3,10)$ , and $(c_4,r_4)=(1,20)$ , are the same for both additive and multiplicative demand cases. In the optimal portfolio, the sum of the unit reservation and execution costs increases with the contract index $n$ ; thus, a lower price $p$ will exclude the higher-cost sources, leading to a reduction in the optimal number of contracts. We use different colors and markers to represent different segments of the objective function arising from changes in the optimal portfolio when the price $p$ varies. The segment $z_1$ corresponds to the lowest price $p$ , where only the first contract is included in the optimal portfolio. As $p$ increases, segments $z_2$ , $z_3$ , and $z_4$ represent the scenarios with $2$ , $3$ , and $4$ contracts in the optimal portfolio, respectively. Despite the piece-wise nature, we observe a smooth and concave pattern in both Figures 2 (a) and (b).

There exist significant differences in the optimal prices $p^{*}$ and the resulting worst-case profits $z^{*}$ between the additive and multiplicative demand models. In the latter, not only the mean, but also the standard deviation of demand changes with the price, negatively impacting the term $\mathrm{\Delta }(p)$ in the objective function. Consequently, the multiplicative demand model leads to a higher selling price but results in a much lower worst-case profit than the additive demand model.

5.2. Fixed Cost to Open Any Source

In addition to the linear reservation cost $c_i$ and $r_i$ , we assume that the firm also incurs a fixed cost $K_i\ge 0$ when opening source $i$ . Let $S\subseteq \{1,2,\dots ,n\}$ be a feasible supply portfolio representing the sources to be opened. The procedures to compute the worst-case total profit under a given portfolio $S$ involve the following steps.

First, we update the sequences of $\{{\alpha }_i^S\}$ and $\{{\delta }_i^S\}$ based on portfolio $S$ . Let $(i)$ be the $i$ -th smallest index in $S$ and $|S|$ be the number of sources that are open. We also call $|S|$ the size of portfolio $S$ . For instance, when $n=6$ and $S=\{2,4,5\}$ , $(1)=2$ , $(2)=4$ , and $(3)=5$ . As $|S|=3$ in this example, the artificial source representing lost sales is source- $(4)$ . Using the updated (and shorter) sequences of $\{c_{(i)},r_{(i)}\}$ , we can compute the cumulative probabilities ${\alpha }_{(i)}^S=1-((c_{(i)}-c_{(i+1)})/(r_{(i+1)}-c_{(i)}))$ and the marginal impacts ${\delta }_{(i)}^S=r_{(i)}-r_{(1)}-c_{(1)}$ . The marginal probabilities satisfy ${\beta }_{(i)}^S={\alpha }_{(i)}^S-{\alpha }_{(i-1)}^S$ , and by default, ${\alpha }_{(0)}^S=0$ and ${\alpha }_{(n+1)}^S=1$ .

Second, we apply Proposition 4 to compute the robust capacity plan ${\mathbf{Q}}_S$ using the updated sequences of $\{{\beta }_i^S\}$ and $\{{\delta }_i^S\}$ . We also compute the firm’s worst-case expected profit via equation (18) by subtracting the fixed cost $\sum _{i\in S}{K}_i$ .

Finally, we identify the optimal supply portfolio $S^{*}$ by solving the following equation:

$$\begin{array}{rl}& \underset{S\subseteq \{1,2,\dots ,n\},\mathbf{Q}\ge \mathbf{0}}{max}\{\phantom{\sum _{i\in S}}Z\left(S\right)=(p-r_{(1)}-c_{(1)})\mu \\ & -\sigma \mathrm{\Delta }\left(S\right)-\sum _{i\in S}{K}_i\}.\end{array}$$

(20)

We find numerous examples showing that $Z(S)$ in equation (20 ) is neither subadditive nor superadditive with respect to set $S$ . Without these properties, various well-known greedy algorithms (e.g., Jain et al., 2003) could lose their effectiveness. As equation (20) overlaps with a generic facility-location model, which is NP-hard (see Theorem 3.1 in Guha and Khuller, 1999), we conclude that the optimal supply portfolio $S^{*}$ must be solved by conducting a full search. To illustrate the scenario with fixed costs, we discuss a case study on Supply Network Design in Section 6.3.

5.3. Mean-Absolute Deviation

Similar to van Eekelen et al. (2022), we define the ambiguity set as

$$\begin{array}{rl}{\mathrm{\Omega }}_d& =\{F\left(\theta \right)|{\int }_L^{H}dF\left(\theta \right)=1,{\int }_L^H\theta dF\left(\theta \right)\\ & =\mu ,{\int }_L^H|\theta -\mu |dF\left(\theta \right)=d\},\end{array}$$

(21)

where $L$ and $H$ represent the limits of the bounded support and $d$ represents the mean-absolute deviation (MAD). We assume that $\mu \in (L,H)$ and $0<d<(2(H-\mu )(\mu -L))/(H-L)$ so that the ambiguity set ${\mathrm{\Omega }}_d$ contains an infinite number of distributions (otherwise, ${\mathrm{\Omega }}_d$ is either empty or a singleton). The joint optimization model is then formulated as follows:

$$\begin{array}{rl}P& =\underset{\mathbf{Q}\ge \mathbf{0},y_0,y_1,y_2}{max}\{y_0+y_1\mu +y_{2}d\}\\ & \text{s.t. }{y}_0+y_1\theta +y_2|\theta -\mu |\le Z\left(\theta |\mathbf{Q}\right),\mathrm{\forall }\theta \ge 0.\end{array}$$

(22)

The LHS of (22) is continuous and piece-wise linear in $\theta$ with a turning point at $\theta =\mu$ . It is well-known that for any $\mathbf{Q}$ , the worst distribution is a three-point distribution (see Theorem 1 of van Eekelen et al., 2022: on page 1683) with realized values of ${\theta }_1=L$ , ${\theta }_2=\mu$ , and ${\theta }_3=H$ . Using these three binding constraints, we write down the Lagrangian as follows:

$$\begin{array}{rl}\mathcal{L}& =y_0+y_1\mu +y_{2}d-{\lambda }_1(y_0+y_{1}L+y_2|L-\mu |-Z\left(L|\mathbf{Q}\right))\\ & -{\lambda }_2(y_0+y_1\mu +y_2|\mu -\mu |-Z\left(\mu |\mathbf{Q}\right))\\ & -{\lambda }_3(y_0+y_{1}H+y_2|H-\mu |-Z\left(H|\mathbf{Q}\right)).\end{array}$$

(23)

Lemma 5 solves the special case of equation (23) with only one source.

Lemma 5

With MAD and only source $1$ available, the optimal solution displays the following properties.

$$Z_d^{*}=\{\begin{array}{ll}(p-r_1-c_1)L,& \text{if }\frac{p-c_1-r_1}{p-r_1}\le \frac{d}{2(\mu -L)}\\ (p-c_1-r_1)\mu -\frac{1}{2}(p-r_1)d,& \text{if }\frac{d}{2(\mu -L)}<\frac{p-c_1-r_1}{p-r_1}\le 1-\frac{d}{2(H-\mu )}\\ (p-r_1)\mu -c_{1}H,& \text{if }\frac{p-c_1-r_1}{p-r_1}>1-\frac{d}{2(H-\mu )}\end{array}$$

(24)

The optimal robust capacity level $q_1^{*}$ satisfies

$$q_1^{*}=\{\begin{array}{ll}L,& \text{if }\frac{p-c_1-r_1}{p-r_1}\le \frac{d}{2(\mu -L)}\\ \mu ,& \text{if }\frac{d}{2(\mu -L)}<\frac{p-c_1-r_1}{p-r_1}\le 1-\frac{d}{2(H-\mu )}\\ H,& \text{if }\frac{p-c_1-r_1}{p-r_1}>1-\frac{d}{2(H-\mu )}\end{array}$$

(25)

When extending to the case with multiple sources, we make use of nature’s equilibrium strategy, which is a three-point distribution with realized values of ${\theta }_1=L$ , ${\theta }_2=\mu$ , and ${\theta }_3=H$ , irrespective of $\mathbf{Q}$ . Graphically, the LHS of (22) is continuous with two linear pieces. When changing $(Q_1,Q_{2,\dots ,}{Q}_n)$ , we change the shape of the RHS of (22). To ensure that the RHS stays as close as possible to the LHS, we also restrict the RHS to have at most two pieces.

Proposition 6

(Robust plan based on MAD) With MAD and $n\ge 2$ sources, the optimal capacity vector satisfies $\mathbf{Q}=(L,\dots ,L,\mu ,\dots ,\mu ,H,\dots ,H)$ , where

$$Q_i^{*}=\{\begin{array}{ll}L,& \text{if }{\alpha }_i\le \frac{d}{2(\mu -L)}\\ \mu ,& \text{if }\frac{d}{2(\mu -L)}<{\alpha }_i\le 1-\frac{d}{2(H-\mu )}\\ H,& \text{if }{\alpha }_i>1-\frac{d}{2(H-\mu )}\end{array}$$

(26)

We let $k_1$ be the integer satisfying ${\alpha }_{k_1}\le (d/2(\mu -L))<{\alpha }_{k_1+1}$ and let $k_2$ be the integer satisfying ${\alpha }_{k_2}\le 1-(d/2(H-\mu ))<{\alpha }_{k_2+1}$ . The first $k_1$ sources have $Q_i^{*}=L$ ; the middle $(k_2-k_1)$ sources have $Q_i^{*}=\mu$ ; and the remaining $(n-k_2)$ sources have $Q_i^{*}=H$ . As the extreme distribution is unique, the cumulative probabilities ${\alpha }_i$ determine the capacity levels according to Lemma 2. Unsurprisingly, the optimal $Q_i^{*}$ must be one of the three values of $L$ , $\mu$ or $H$ , which are the three realized values of the unique extreme distribution. We recall that $q_i=Q_i-Q_{i-1}$ and $Q_1=q_1$ . With no more than three different $Q_i$ ’s, at most three $q_i$ can be non-zero, suggesting that the MAD plan chooses no more than three sources. ³ We will compare the performance of MAD and the mean–variance model in the numerical section, to understand how the choice of robust model influences the solution.

5.4. Portfolio Effect

The closed-form expressions in equation (18) enable us to investigate the portfolio effect. We define $Z_{\mathrm{UB}}^{*}$ as the upper bound and $Z_{\mathrm{LB}}^{*}$ as the lower bound on the firm’s worst-case expected profit $Z^{*}$ such that $Z_{\mathrm{LB}}^{*}\le Z^{*}\le Z_{\mathrm{UB}}^{*}$ . According to equation (18), $Z_{\mathrm{LB}}^{*}$ and $Z_{\mathrm{UB}}^{*}$ depend on the bounds on the constant $\mathrm{\Delta }$ , which has a noteworthy statistical interpretation. In the equilibrium of the zero-sum game, nature chooses a distribution such that the firm attains a realized marginal benefit ${\delta }_i$ with probability ${\beta }_i$ . We regard the marginal benefit $\delta$ as a random variable and ${\delta }_i$ as the $i$ -th realization such that $Pr(\delta ={\delta }_i)={\beta }_i$ . Proposition 3 indicates that the mean of the random variable $\delta$ is zero and the standard deviation is $\mathrm{\Delta }$ . Consistent with the literature on portfolio theory, an optimal portfolio reduces the variance of the marginal benefit $\delta$ to improve the firm’s utility.

Lemma 6

The constant $\mathrm{\Delta }$ satisfies the following inequality:

$$c_1\rho \le \mathrm{\Delta }\le \sqrt{c_1(p-r_1-c_1)}.$$

(27)

Substituting equation (27) into equation (18), we obtain that

$$\{\begin{array}{l}{Z}_{\mathrm{LB}}^{*}=(p-r_1-c_1)\mu (1-\sqrt{\frac{c_1}{p-r_1-c_1}}\rho )\\ Z_{\mathrm{UB}}^{*}=(p-r_1-c_1)\mu (1-\frac{c_1}{p-r_1-c_1}{\rho }^2)\end{array}.$$

We define the portfolio effect (PE) as

$$PE=\frac{Z_{\mathrm{UB}}^{*}-Z_{\mathrm{LB}}^{*}}{Z_{\mathrm{UB}}^{*}}=\frac{\sqrt{\frac{c_1}{p-r_1-c_1}}\rho }{1+\sqrt{\frac{c_1}{p-r_1-c_1}}\rho }.$$

(28)

The definition of $PE$ is similar to that of Fu et al. (2010) except that we consider the worst-case (rather than distribution-specific) scenario. A large value of $PE$ implies that using a portfolio of option contracts, supply sources, or grades can substantially improve the firm’s expected profit due to the flexibility of the portfolio in coping with demand uncertainty. In equation (28), $c_1$ is the overstock cost and $(p-r_1-c_1)$ is the understock cost of source $1$ . We observe that the portfolio effect becomes more beneficial when (i) the understock cost decreases and (ii) the demand variability or overstock cost increases. Thus, when the most preferable source $-1$ is less flexible with a higher reservation cost $c_1$ (or lower $r_1$ ), the portfolio effect is more significant.

Identifying the portfolio that minimizes or maximizes the constant $\mathrm{\Delta }$ is important.

Proposition 7

(i) Side $AC$ shown in Figure 3 is the unique worst path ensuring that $\mathrm{\Delta }=\sqrt{c_1(p-r_1-c_1)}$ . (ii) Path $ABC$ shown in Figure 3 is one of the best paths ensuring that $c_1\rho =\mathrm{\Delta }$ .

Figure 3.

Visualizing the firm’s portfolio.

We first explain the portfolio that maximizes $\mathrm{\Delta }$ . Figure 3 visualizes the worst, the best, and a candidate portfolio Definition 2 indicates that $\delta$ is bounded between ${\delta }_1=-c_1$ and ${\delta }_{n+1}=p-r_1-c_1$ . To increase the variance of $\delta$ , we should allocate mass probabilities only to the two end points ${\delta }_1$ and ${\delta }_{n+1}$ only. Graphically, ${\delta }_1$ pertains to the vertical coordinate of point $A$ and ${\delta }_{n+1}$ pertains to the horizontal coordinate of point $C$ in Figure 3. Side $AC$ shown in Figure 3 is the firm’s worst path ensuring that $\mathrm{\Delta }=\sqrt{c_1(p-r_1-c_1)}$ . Any convex decreasing path that lies below $AC$ must divert the probability mass from points $A$ and $C$ to some other points, thus reducing the variance of $\delta$ . We observe that side $AC$ in Figure 3 is the firm’s unique worst path. If the portfolio includes source $1$ and several other sources corresponding to points on side $AC$ , the firm’s equilibrium profit $Z^{*}$ remains unchanged, suggesting that the firm’s worst portfolio could be multiple, but her worst path is unique.

Next, we explain the portfolio that minimizes $\mathrm{\Delta }$ . Graphically, when choosing a portfolio, the firm is choosing a convex decreasing path that lies below side $AC$ to reduce the variance of $\delta$ . For $n\ge 2$ , multiple best paths exist. In general, to reduce the variance of $\delta$ , we should allocate mass probability around the mean of $\delta$ as much as possible. With $r_B=r_1+c_1$ , we observe that ${\delta }_B=0$ , suggesting that point $B$ in Figure 3 corresponds to the mean of $\delta$ . To increase ${\beta }_B$ , we reduce the reservation cost $c_B$ as much as possible while keeping $r_B=r_1+c_1$ . We can verify that when $c_B\ge ((c_1\rho {)}^2)/((p-r_1))$ , the lowest realization ${\theta }_1^{*}$ in Definition 2 is nonnegative. Thus, side $A-B-C$ in Figure 3 forms the best path, creating an optimal portfolio that uses only source $1$ and source $B$ .

A useful insight of Proposition 7 is that the benefit of the portfolio effect can be fully attained by engaging with the appropriate supply sources with the appropriate cost parameters. This insight echoes the numerical examples in Fine and Freund (1990) that are based on normal distributions. However, we consider the worst-case perspective. As shown in Figure 3, path $A-B-C$ has as few as two sources, suggesting that the two most suitable supply sources can realize the full benefit of the portfolio effect.

Following the concept of lower convex hull presented by Fu et al. (2010), if the path of a portfolio lies below the path of another portfolio, it is superior. Proposition 7 indicates that path $ABC$ in Figure 3 is the lowest possible convex hull without making source $1$ out of the portfolio. As a “lower” path implies a better portfolio, any portfolio forming a convex decreasing path staying above path $ABC$ must be dominated by the portfolio associated with path $ABC$ . In this application context, both the max-min and expected value maximization criteria reach the same conclusion on what the best or worst portfolio should be.

6. Numerical Study

6.1. A Comprehensive Example

Given the multiple extensions we have presented, it is helpful to illustrate various results through a single, comprehensive example. We set the selling price of the end product at $p=27$ per unit. The mean equals $\mu =0.5$ , the standard deviation equals $\sigma =1/\sqrt{12}$ , and the MAD equals $d=0.25$ so that the standard uniform distribution is a feasible distribution. There are four available sources (indexed by $1$ , $2$ , $3$ , and $4$ ), and source $5$ is the artificial source representing lost sales. The data of these five sources are shown in Table 2.

Table 2.

Cost parameters and various capacity plans.

Source $i$	$c_i$	$r_i$	$K_i$	${\alpha }_i$	${\beta }_i$	${\delta }_i$	$Q_i^{*}$	$Q_i^d$	$Q_i^F$
1	$8$	$0$	$1.1$	$0.33$	$0.33$	$-8$	$0.3$	$0.5$	$0.33$
2	$6$	$3$	$1$	$0.57$	$0.238$	$-5$	$0.45$	$0.5$	$0.57$
3	$3$	$10$	$0.6$	$0.8$	$0.229$	$2$	$0.72$	$1$	$0.8$
4	$1$	$20$	$0.3$	$0.86$	$0.057$	$12$	$0.98$	$1$	$0.86$
5	$0$	$27$	$0$	$1$	$0.143$	$19$	$N/A$

We first solve the baseline model without considering the fixed cost. By using $(r_i,c_i)$ as the coordinates to draw $5$ points, we find that these $5$ points form a convex decreasing curve. In the context of option contracts, source $1$ represents a fixed price contract (with zero execution cost). The fifth column of Table 2 computes the sequence of $\{{\alpha }_i\}$ as ${\alpha }_0=0$ , ${\alpha }_1=0.333$ , ${\alpha }_2=0.571$ , ${\alpha }_3=0.8$ , ${\alpha }_4=0.857$ , and ${\alpha }_5=1$ . Thus, the sequence of $\{{\beta }_i\}$ is ${\beta }_1=0.333$ , ${\beta }_2=0.238$ , ${\beta }_3=0.229$ , ${\beta }_4=0.057$ , and ${\beta }_5=0.143$ , as shown in the sixth column of Table 2. The seventh column of Table 2 computes the sequence of $\{{\delta }_i\}$ as ${\delta }_1=-8$ , ${\delta }_2=-5$ , ${\delta }_3=2$ , ${\delta }_4=12$ , and ${\delta }_5=19$ . We obtain that the constant is $\mathrm{\Delta }=9.38$ based on Definition 2. According to Propositions 2 and 4, we compute the firm’s robust optimal capacity plan as ${\mathbf{Q}}^{*}=(0.3,0.45,0.72,0.98)$ , or equivalently, ${\mathbf{q}}^{*}=(0.3,0.15,0.26,0.26)$ . The firm’s optimized utility equals $Z^{*}=6.79$ according to equation (18). Additionally, we compute the MAD plan as ${\mathbf{Q}}^d=(0.5,0.5,1,1)$ via Proposition 6 or equivalently, ${\mathbf{q}}^d=(0.5,0,0.5,0)$ .

To benchmark these two plans, we find that the distribution-dependent capacity plan is ${\mathbf{Q}}_F^{*}=(0.33,0.57,0.8,0.86)$ , yielding an expected profit of $Z_F=7.08$ . We compute the VOI by using the formula $\mathrm{VOI}=(1-(Z_Q/Z_F))\times 100\mathrm{\%}$ . In general, a larger VOI indicates a less satisfactory performance of the robust capacity plan. In this example, we find that the MAD plan yields an expected profit of $6.75$ and $\mathrm{VOI}=4.7\mathrm{\%}$ but the mean–variance plan yields an expected profit of $6.95$ and $\mathrm{VOI}=1.9\mathrm{\%}$ . In this numerical example, the mean–variance plan ${\mathbf{Q}}^{*}$ outperforms the MAD plan ${\mathbf{Q}}^d$ .

We now consider the extension with a fixed cost for opening sources (see the fourth column of Table 2). A full search based on equation (20) reveals that opening only source $2$ is optimal. In contrast, the analysis based on MAD suggests that the supplier portfolio should be $\{1,4\}$ . Interestingly, the unambiguous benchmark reveals that opening only source $2$ is optimal and that the expected profit under the standard uniform prior equals $Z_F=5.75$ . We find that the MAD plan yields $\mathrm{VOI}=6.1\mathrm{\%}$ (due to the sub-optimal supplier portfolio and capacity plan), whereas the mean–variance plan yields $\mathrm{VOI}=1.4\mathrm{\%}$ (due to only the sub-optimal capacity plan). In summary, both the distribution-dependent model and the max-min model reach the same conclusion regarding the optimal supplier portfolio (while the MAD plan deviates from the optimal portfolio).

6.2. MAD Versus Mean–Variance

We conduct a numerical study using beta distributions, which involve two parameters $(a,b)$ such that

$$\mu =\frac{a}{a+b}\text{ and }{\mu }^2+{\sigma }^2=\frac{a(a+1)}{(a+b)(a+b+1)}.$$

Because the symbols of $(\alpha ,\beta )$ have been specifically defined in Definition 1, we use $(a,b)$ as the input parameters of beta distributions. Readers can refer to the link https://en.wikipedia.org/wiki/Beta_distribution to see many different shapes of beta distributions (e.g., U-shaped, inverse U-shaped, and positively or negatively skewed). We use $16$ pairs of $(a,b)$ , which cover a wide range of beta distributions, including the standard uniform distribution when $a=b=1$ .

Table 3 reports the VOI for the MAD and mean–variance plans. As a smaller VOI implies better performance, we find the mean–variance plan outperforms the MAD plan except when $\beta =0.5$ . The average VOI of the mean–variance plan is $1.8\mathrm{\%}$ while that of the MAD plan is $7.1\mathrm{\%}$ . While van Eekelen et al. (2022) advocate MAD as the preferred descriptor of dispersion, we demonstrate that the mean–variance plan can deliver better performance, as it effectively captures the essential distribution information in ambiguous environment. Moreover, the closed-form results presented in Section 4 enhance the practicality of the mean–variance framework, provide clear insights, and facilitate easier implementation for practitioners.

Table 3.

VOI for MAD and mean–variance plans.

$a$	$0.5$	$0.5$	$0.5$	$0.5$	$1$	$1$	$1$	$1$	$1.5$	$1.5$	$1.5$	$1.5$	$2$	$2$	$2$	$2$
$b$	$0.5$	$1$	$1.5$	$2$	$0.5$	$1$	$1.5$	$2$	$0.5$	$1$	$1.5$	$2$	$0.5$	$1$	$1.5$	$2$
$\mu$	$0.5$	$0.33$	$0.25$	$0.2$	$0.67$	$0.5$	$0.4$	$0.33$	$0.75$	$0.6$	$0.5$	$43$	$0.8$	$0.67$	$0.57$	$0.5$
$\rho =\sigma /\mu$	$0.71$	$0.89$	$1$	$1.07$	$0.45$	$0.58$	$0.65$	$0.71$	$0.33$	$0.44$	$0.5$	$0.54$	$0.27$	$0.35$	$0.41$	$0.45$
MV (%)	3.9	1.2	0.9	1.1	3.5	1.9	1.3	1.1	2.8	2.0	1.5	1.2	2.4	1.9	1.5	1.2
MAD (%)	2.6	13.6	18.7	17.9	1.2	4.7	10.2	13.1	0.9	2.6	6.3	8.6	0.7	1.8	4.2	6.6

6.2.1. Bound on Value of Information

The value of information (VOI), defined as $\text{VOI}=Z(q_F,F)-Z(q^{*},F)$ , quantifies the profit gain from knowing the true demand distribution $F$ . Since VOI depends on $F$ , evaluating it requires computing the expected profit of the robust capacity plan $q^{*}$ (from Proposition 4) under the known distribution $F$ via Equation (10).

The robust plan $q^{*}$ includes the term $\mathrm{\Delta }=\sqrt{\sum _{i=1}^{n+1}{\beta }_i{\delta }_i^2}$ , which encapsulates supply-side cost heterogeneity. These square root expressions typically prevent closed-form simplification of the profit integrals. Nevertheless, we can bound VOI as:

$$\begin{array}{rl}\text{VOI}& =Z(q_F,F)-Z(q^{*},F)\le \underset{F}{sup}\{Z(q_F,F)\}-\underset{F}{inf}\{Z(q^{*},F)\}\\ & =(p-r_1-c_1)\mu -[(p-r_1-c_1)\mu -\sigma \mathrm{\Delta }]=\sigma \mathrm{\Delta },\end{array}$$

where the lower bound uses the closed-form result in Equation (18).

In summary, $\mathrm{\Delta }$ serves as a comprehensive summary of the supply portfolio’s structural characteristics—capturing both the probability mass and marginal impact sequences—and provides a clean upper bound on the VOI. This result generalizes the classical bound established by Scarf (1958), where for a single-source newsvendor model, the value of information satisfies:

$$\text{VOI}\le \sigma \sqrt{c(p-c)}.$$

Our multi-sourcing extension replaces the term $\sqrt{c(p-c)}$ with $\mathrm{\Delta }$ , which reflects the dispersion in adjusted execution costs across the supplier base. This generalization preserves the key insight of Scarf’s model—that the worst-case penalty from distributional ambiguity scales linearly with demand variability $\sigma$ —while extending its applicability to modern supply chain settings involving heterogeneous supplier contracts.

6.3. A Case Study on Supply Network Design

We apply equation (20) to assist a retailer (the case company) in designing its supply network. The case company is located in Shanghai, China; it supplies $84$ customers, whose products are unique and customized. The replenishment inventories are delivered by a regional warehouse located outside of the city. Let $p_n$ be the price of product $n$ (where $n=1,2,\dots ,84$ ) that customer $n$ buys from the case company. The demand for product $n$ in each period is ${\theta }_n$ . The case company provides the mean and variance of each ${\theta }_n$ , denoted by ${\mu }_n$ and ${\sigma }_n$ . The company plans to open upfront warehouses to improve the speed of deliveries to its customers. There are four candidate locations: Chang Ning, Hong Kou, Pu Dong, and Min Hang, located in the west, north, east, and south districts of Shanghai, respectively. The case company also provides the latitude and longitude of each customer $n$ and candidate upfront warehouse $m$ (where $m=1,2,3,4$ ) so that we can compute the distance between any two points in the network. The fixed cost of opening warehouse $m$ is $K_m$ . The delivery cost $r_{mn}$ is linear with respect to the distance and is incurred only when a delivery is made. On average, the delivery cost is approximately $4.5$ CNY per unit per kilometer. After the network structure is chosen, the company prestocks $q_{mn}$ units of product $n$ at location $m$ (if location $m$ is open). The location to store the inventory for customer $n$ depends on the lower convex hull (see Remark 1). In each period, after observing the realized demand of each product $n$ , the company uses a priority rule (based on the distance between warehouse $m$ to customer $n$ ) to make the deliveries. Any unsatisfied demand is lost and unused inventory incurs the holding cost $c_{mn}$ . We formulate the optimization problem as follows:

$$Z=\underset{a_m,q_{mn}}{max}\{\sum _{n=1}^{84}{Z}_n-\sum _{m=1}^4{a}_m{K}_m\},$$

(29)

where $a_m=1$ if location $m$ is open and $a_m=0$ otherwise; $Z_n$ represents the worst-case expected profit of product $n$ ; and $q_{mn}$ represents the inventory of product $n$ stored at location $m$ .

We explain how to apply Proposition 5 to compute $Z_n$ as follows. We observe that for any given network structure $\mathbf{a}=(a_1,a_2,a_3,a_4)$ , the total fixed cost $\sum _{m=1}^4{a}_m{K}_m$ is sunk, and each product is unique. We can then decompose the analysis into $84$ capacity planning models, as presented in Section 5.2. The key step is to update the ranked index of the lower convex hull for each product $n$ . As some locations may not hold any inventory for product $n$ , the number of effective nodes on each lower convex hull could be less than $a_1+a_2+a_3+a_4$ . We use $(m)$ to indicate the updated rank of locations in each lower convex hull. The opened location is ranked such that (i) $c_{(1)n}>c_{(2)n}>\dots >c_{(5)n}=0$ ; (ii) $r_{(1)n}<r_{(2)n}<\dots <r_{(5)n}=p_n$ ; and (iii) $c_{(1)n}+r_{(2)n}<c_{(2)n}+r_{(2)n}\dots <c_{(5)n}+r_{(5)n}$ . As mentioned in Section 3, the last location of any convex lower hull is an artificial node representing lost sales. Using the updated ranking, we compute (a) the marginal impacts using ${\delta }_{(m)n}=r_{(m)n}-c_{(1)n}-r_{(1)n}$ and (b) the probabilities ${\alpha }_{(m)n}$ and ${\beta }_{(m)n}$ according to Definition 1. As soon as ${\delta }_{(m)n}=r_{(m)n}-c_{(1)n}-r_{(1)n}$ and ${\beta }_{(m)n}$ become available, we apply Proposition 5 to compute $Z_n$ as follows:

$$Z_n=(p_n-c_{(1)n}-r_{(1)n}){\mu }_n-{\sigma }_n\sqrt{\sum _{m=1}^4{\beta }_{(m)n}{\delta }_{(m)n}^2}.$$

(30)

With four candidate warehouses, we first evaluate $15$ candidate solutions to determine the optimal network structure ${\mathbf{a}}^{*}$ and then apply Proposition 4 to determine $q_{(m)n}$ as follows:

$$q_{(m)n}={\mu }_n+\frac{{\sigma }_n{\delta }_{n(m)}}{\sqrt{\sum _{m=1}^4{\beta }_{n(m)}{\delta }_{n(m)}^2}}.$$

The performance of different network structures is shown in Table 4. Based on the results, we recommend that upfront warehouses in Pu Dong and Min Hang be open. We conduct additional experiments to evaluate the performance of the robust solution under normal distribution. We denote the worst-case profit as $Z_{\mathrm{wst}}$ . We also let $Z_{\mathrm{wst}}^{\mathrm{Normal}}$ and $Z_{\mathrm{Normal}}^{*}$ be the expected profit of the robust solution and the optimal solution under normal distributions, respectively. The comparison demonstrates that the robust solution performs very well with a performance gap less than $1.0\mathrm{\%}$ compared with the optimal solution.

Table 4.

Candidate warehouse locations and resulted profit.

$a_1$	$a_2$	$a_3$	$a_4$	$Z_{\mathrm{wst}}$	$Z_{\mathrm{wst}}^{\mathrm{Normal}}$	$Z_{\mathrm{Normal}}^{*}$
(Chang Ning)	(Hong Kou)	(Pu Dong)	(Min Hang)	( $\times {10}^8$ )	( $\times {10}^8$ )	( $\times {10}^8$ )
1	0	0	0	1.6550	1.9735	1.9790
0	1	0	0	1.6807	1.9898	2.0012
0	0	1	0	1.7418	2.0481	2.0620
0	0	0	1	1.7776	2.0722	2.0926
1	1	0	0	1.6921	1.9902	2.0086
1	0	1	0	1.7383	2.0432	2.0578
1	0	0	1	1.7736	2.0682	2.0886
0	1	1	0	1.7411	2.0447	2.0597
0	1	0	1	1.7783	2.0641	2.0908
0	0	1	1	1.7824	2.0722	2.0932
1	1	1	0	1.7377	2.0398	2.0555
1	1	0	1	1.7743	2.0601	2.0868
1	0	1	1	1.7784	2.0682	2.0892
0	1	1	1	1.7814	2.0686	2.0902
1	1	1	1	1.7774	2.0646	2.0862

The network design problem in equation (29) combines the features of facility location problems and random packing problems. Notably, if location $m$ is open, then the inventory $q_{mn}$ may not be used (due to its lower priority in the lower convex hull than other locations). Whenever a new location is added to the lower convex hull, it becomes a new capacity for product $n$ and incrementally increases $Z_n$ . Therefore, the decision to open a candidate warehouse $m$ depends on whether the increased benefits outweigh the fixed cost $K_m$ . The closed-form expression of $Z_n$ substantially facilitates the computation of the total profits.

7. Concluding Remark

In this article, we characterize a distributionally robust capacity plan with multiple supply sources. Each supply source incurs source-dependent reservation and execution costs. A source with a higher reservation cost and a lower execution cost has a lower total cost, but it is less flexible; this creates a trade-off between cost and flexibility. The use of multiple supply sources inevitably complicates the capacity plan. We show that after demand is realized, the firm’s optimal fulfillment plan is a priority rule that exhausts the capacity of the more profitable source before using any capacity of any less profitable source. When the underlying demand distribution is known, the optimal capacity plan is determined by a sequence of increasing percentiles. However, the absence of information on the precise demand distribution prevents the firm from determining the relevant percentiles.

To address the new challenge posed by ambiguity in the demand distribution, we adopt the max-min decision rule and derive the robust optimal capacity plan for the procurement stage, using only the mean and variance of demand. We reformulate the robust max-min model as a joint optimization model by transforming the inner minimization problem into a linear SIP model. By identifying the binding constraints and the set of optimality conditions, we solve the problem in closed form. We develop upper and lower bounds on the firm’s worst-case expected profit, and demonstrate that the simplest portfolio to attain the full benefit of the portfolio effect contains only two sources. We conduct a numerical study to verify that the robust solution performs reasonably well compared with the optimal solution under a known demand distribution. We also discuss several extensions of our robust multi-source capacity planning model to consider other information structures, price-dependent demand case, and the fixed costs of engaging supply sources. Our closed-form solutions for the firm’s worst-case demand distribution, the robust optimal capacity plan, and the worst-case expected profit are elegant and neat, enabling us to obtain managerial insights into robust multi-sourcing capacity planning and to further examine the portfolio effect.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478251365557 - Supplemental material for Multi-Sourcing Procurement Management: A Closed-Form Mean–Variance Approach

Supplemental material, sj-pdf-1-pao-10.1177_10591478251365557 for Multi-Sourcing Procurement Management: A Closed-Form Mean–Variance Approach by Qi Fu, Zhaolin Li and Chung-Piaw Teo in Production and Operations Management