Analysis of a group purchasing organization under demand and price uncertainty

Abstract

Based on an industrial case study, we present a stochastic model of a supply chain consisting of a set of buyers and suppliers and a group purchasing organization (GPO). The GPO combines orders from buyers in a two-period model. Demand and price in the second period are random. An advance selling opportunity is available to all suppliers and buyers in the first-period market. Buyers decide how much to buy through the GPO in the first period and how much to procure from the market at a lower or higher price in the second period. Suppliers determine the amount of capacity to sell through the GPO in the first period and to hold in reserve in order to meet demand in the second period. The GPO conducts a uniform-price reverse auction to select suppliers and decides on the price that will be offered to buyers to maximize its profit. By determining the optimal decisions of buyers, suppliers, and the GPO, we answer the following questions: Do suppliers and buyers benefit from working with a GPO? How do the uncertainty in demand, the share of GPO orders in the advance sales market, and the uncertainty in price influence the players’ decisions and profits? What are the characteristics of an environment that would encourage suppliers and buyers to work with a GPO? We show that a GPO helps buyers and suppliers to mitigate demand and price risks effectively while collecting a premium by serving as an intermediary between them.

Appendices

Appendix A

We first define the equations for the expected sales $S({\mathcal {A}},{\mathcal {D}})$, expected left-over inventory $I({\mathcal {A}},{\mathcal {D}})$, and expected lost sales $L({\mathcal {A}},{\mathcal {D}})$ for an on-hand inventory level of ${\mathcal {A}}$ and random demand ${\mathcal {D}}$ with a cumulative distribution function of $F_{D}(x)$ as

$$\begin{aligned} S({\mathcal {A}},{\mathcal {D}})= & \, E[\min ({\mathcal {A}},{\mathcal {D}})] ={\mathcal {A}}-\int \limits _{0}^{{\mathcal {A}}}F_{{\mathcal {D}}}(x)dx \end{aligned}$$

(21)

$$\begin{aligned} I({\mathcal {A}},{\mathcal {D}})= & \, E[({\mathcal {A}}-{\mathcal {D}})^{+}] ={\mathcal {A}}-S({\mathcal {A}},{\mathcal {D}}) \end{aligned}$$

(22)

$$\begin{aligned} L({\mathcal {A}},{\mathcal {D}})= & \, E[({\mathcal {D}}-{\mathcal {A}})^{+}] =E[{\mathcal {D}}]-S({\mathcal {A}},{\mathcal {D}}). \end{aligned}$$

(23)

Proof of Proposition 1

If the buyer does not choose to work with the GPO, its expected profit is written as in Eq. (1). The expected profit can also be expressed as

$$\begin{aligned} E[\varPi _{r}^{NG}(q_{NG})]=\left( p-w_{sp}-e_{r}\right) E\left[ \beta D_{T}\right] -\left( w_{fp}+e_{r}\right) q_{NG}+\left( w_{sp}+e_{r}\right) S(q_{NG},\beta D_{T}).\nonumber \end{aligned}$$

The expected profit $E[\varPi _{r}^{NG}(q_{NG})]$ is a concave function of $q_{NG}$, that is,

$$\begin{aligned} \dfrac{\partial ^{2} E\left[ \varPi _{r}^{NG}(q_{NG})\right] }{\partial q_{NG}^{2}}=-\left( e_{r}+w_{sp}\right) f_{\mathrm{B}}(q_{NG})<0. \end{aligned}$$

Under this setting, the first-order condition yields the solution of the buyer’s problem:

$$\begin{aligned} \dfrac{\partial E\left[ \varPi _{r}^{NG}(q_{NG})\right] }{\partial q_{NG}}= & \, -\left( w_{fp}+e_{r}\right) +\left( w_{sp}+e_{r}\right) \dfrac{\partial S(q_{NG},\beta D_{T})}{\partial q_{NG}}=0\\= & \, -\left( w_{fp}+e_{r}\right) +\left( w_{sp}+e_{r}\right) \left( 1-F_{\mathrm{B}}(q_{NG})\right) =0. \end{aligned}$$

Rearranging the terms in the above equation gives

$$\begin{aligned} F_{\mathrm{B}}(q_{NG})=1-\dfrac{w_{fp}+e_{r}}{w_{sp}+e_{r}}. \end{aligned}$$

As a result, the optimal advance order quantity for a buyer who uses the market option in the first period is defined as in Eq. (4).

When the buyer chooses to work with the GPO, its expected profit function is given as in Eq. (2). The expected profit can be rewritten as

$$\begin{aligned} E[\hat{\varPi }_{r}^{G}(q_{G})]= & \, \theta \left( \left( p-w_{sp}-e_{r}\right) E\left[ \beta D_{T}\right] -w_{r}q_{G}+\left( w_{sp}+e_{r}\right) S(q_{G},\beta D_{T})\right) \\&+\,(1-\theta )E[\varPi _{r}^{NG}(q_{NG})] \end{aligned}$$

that is also a concave function of $q_{G}$, that is,

$$\begin{aligned} \dfrac{\partial ^{2} E[\hat{\varPi }_{r}^{G}(q_{G})]}{\partial q_{G}^{2}}=-\theta \left( e_{r}+w_{sp}\right) f_{\mathrm{B}}(q_{G})<0. \end{aligned}$$

The optimal quantity that maximizes the expected profit is determined from the first-order condition as

$$\begin{aligned} F_{\mathrm{B}}(q_{G})=1-\dfrac{w_{r}}{w_{sp}+e_{r}}. \end{aligned}$$

As a result, the optimal advance order quantity for a case where the buyer works with the GPO is given as in Eq. (3). $\square$

Proof of Proposition 2

The buyer prefers working with GPO in the first period if the expected profit for this option is greater than or equal to the profit obtained when it does not work with the GPO. The difference between the two expected profit functions, $E[\hat{\varPi }_{r}^{G}(q_{G}^{*})]-E[\varPi _{r}^{NG}(q_{NG}^{*})]\ge 0$, is written as

$$\begin{aligned} \theta \left[ (w_{sp}+e_{r})S(q_{G}^{*},\beta D_{T})-w_{r}q_{G}^{*}-(w_{sp}+e_{r})S(q_{NG}^{*},\beta D_{T})+(w_{fp}+e_{r})q_{NG}^{*}\right] \ge 0. \end{aligned}$$

Dividing both sides of the above equation with $\theta (w_{sp}+e_{r})$ and writing $S(q_{G}^{*},\beta D_{T})$ and $S(q_{NG}^{*},\beta D_{T})$ explicitly give

$$\begin{aligned} q_{G}^{*}\left( 1-\dfrac{w_{r}}{w_{sp}+e_{r}} \right) -\int \limits _{0}^{q_{G}^{*}}F_{\mathrm{B}}(y)dy-q_{NG}^{*}\left( 1-\dfrac{w_{fp}+e_{r}}{w_{sp}+e_{r}}\right) +\int \limits _{0}^{q_{NG}^{*}}F_{\mathrm{B}}(y)dy \ge 0. \end{aligned}$$

From the proof of Proposition 1, $\left( 1-\dfrac{w_{r}}{w_{sp}+e_{r}}\right)$ and $\left( 1-\dfrac{w_{fp}+e_{r}}{w_{sp}+e_{r}}\right)$ can be replaced with $F_{\mathrm{B}}(q_{G}^{*})$ and $F_{\mathrm{B}}(q_{NG}^{*})$, respectively. Thus, the above inequality can be simplified as

$$\begin{aligned} q_{G}^{*}F_{\mathrm{B}}(q_{G}^{*})-q_{NG}^{*}F_{\mathrm{B}}(q_{NG}^{*})-\int \limits _{q_{NG}^{*}}^{q_{G}^{*}}F_{\mathrm{B}}(y)dy \ge 0. \end{aligned}$$

It can be shown that the above inequality holds only if $q_{G}^{*}\ge q_{NG}^{*}$. This result gives the condition in Proposition 2. $\square$

Proof of Proposition 3

Suppliers whose production costs are greater than or equal to the estimated auction price do not work with the GPO, that is, they do not allocate any capacity to the GPO.

Suppliers whose production costs are less than $\hat{w}_{s}$ prefer using the GPO option in the first period. In this case, the expected profit function of a supplier using the GPO option is given in Eq. (6). In terms of Eq. (21), the expected profit function is restated as

$$\begin{aligned} E\left[ \hat{\varPi }_{s_{j}}^{G}\left( \varDelta _{G}, \varDelta _{NG}\right) |c<\hat{w}_{s}\right]= & \, \theta \left( \hat{w}_{s}\varDelta _{G}+\left( w_{sp}-e_{s}\right) S(M-\varDelta _{G}, D_{s}^{G})\right) -cM\\\nonumber&+\,(1-\theta )\left( \left( w_{fp}-e_{s}\right) \varDelta _{NG}+\left( w_{sp}-e_{s}\right) S(M-\varDelta _{NG}, D_{s}^{NG})\right) . \end{aligned}$$

This is a concave function of $\varDelta _{G}$, namely,

$$\begin{aligned} \dfrac{\partial ^{2} E\left[ \hat{\varPi }_{s_{j}}^{G}\left( \varDelta _{G}, \varDelta _{NG}\right) |c<\hat{w}_{s}\right] }{\partial \varDelta _{G}^{2}}= & \, -\theta (w_{sp}-e_{s})f_{D_{s}^{G}}(M-\varDelta _{G})<0. \end{aligned}$$

The optimal value of $\varDelta _{G}$ that maximizes the expected profit subject to the constraint $0\le \varDelta _{G}^{*}\le \min \left\{ M, Q_{G}^{*} \right\}$ is determined in accordance with the unconstrained solution for the problem, denoted as $\delta _{G}^{*}$. The unconstrained solution for the problem is obtained from the first-order condition as

$$\begin{aligned} \delta _{G}^{*}=M-F^{-1}_{D_{s}^{G}}\left( 1-\dfrac{\hat{w}_{s}}{w_{sp}-e_{s}}\right) . \end{aligned}$$

Equation (8) gives the solution of the constrained problem, using the concavity of the profit function.

If the supplier does not choose to work with the GPO, the expected profit can be written as in Eq. (7) regardless of whether or not its production cost is less than $\hat{w}_{s}$. With Eq. (21), the expected profit function is rewritten as

$$\begin{aligned} E\left[ \varPi _{s_{j}}^{NG}\left( \varDelta _{NG}^{\prime }, \varDelta _{NG}\right) |c\right]= & \, \theta \left( \left( w_{fp}-e_{s}\right) \varDelta _{NG}^{\prime }+\left( w_{sp}-e_{s}\right) S(M-\varDelta _{NG}^{\prime }, D_{s}^{G})\right) -cM\\&+\,(1-\theta )\left( \left( w_{fp}-e_{s}\right) \varDelta _{NG}+\left( w_{sp}-e_{s}\right) S(M-\varDelta _{NG}, D_{s}^{NG})\right) . \end{aligned}$$

$E\left[ \varPi _{s_{j}}^{NG}\left( \varDelta _{NG}^{\prime }, \varDelta _{NG}\right) |c\right]$ is a concave function with respect to $\varDelta _{NG}$ and $\varDelta _{NG}^{\prime }$. That is,

$$\begin{aligned} \dfrac{\partial ^{2} E\left[ \varPi _{s_{j}}^{NG}\left( \varDelta _{NG}^{\prime }, \varDelta _{NG}\right) |c\right] }{\partial \varDelta _{NG}^{2}}= & \, -\left( 1-\theta \right) \left( w_{sp}-e_{s}\right) f_{D_{s}^{NG}}(M-\varDelta _{NG})<0,\\ \dfrac{\partial ^{2} E\left[ \varPi _{s_{j}}^{NG}\left( \varDelta _{NG}^{\prime }, \varDelta _{NG}\right) |c\right] }{\partial \varDelta _{NG}^{\prime 2}}= & \, -\theta \left( w_{sp}-e_{s}\right) f_{D_{s}^{G}}(M-\varDelta _{NG}^{\prime }) <0. \end{aligned}$$

With the same steps that yield the optimal solution for the previous case, the unconstrained optimization of $E\left[ \varPi _{s_{j}}^{NG}\left( \varDelta _{NG}^{\prime }, \varDelta _{NG}\right) |c\right]$, together with the concavity of the function, gives the solution of the optimal values of $\varDelta _{NG}^{*}$ and $\varDelta _{NG}^{\prime {*}}$ as given in Proposition 3. $\square$

Proof of Proposition 4

Supplier j, with $c\ge \hat{w}_{s}$, joins the GPO if its expected profit is higher with participation than non-participation. The difference in expected profits between these two options can be written as

$$\begin{aligned}&E\left[ \hat{\varPi }_{s_{j}^{G}}\left( \varDelta _{NG}^{*}\right) |c\ge \hat{w}_{s}\right] - E\left[ \varPi _{s_{j}}^{NG}\left( \varDelta _{NG}^{\prime {*}}, \varDelta _{NG}^{*}\right) |c\right] \\&\quad =\left( w_{sp}-e_{s}\right) \left( M-\int \limits _{0}^{M}F_{D_{s}^{G}}(u)du \right) \\&\qquad -\left( w_{fp}-e_{s}\right) \varDelta _{NG}^{\prime {*}} -\left( w_{sp}-e_{s}\right) \left( M-\varDelta _{NG}^{\prime {*}}-\int \limits _{0}^{M-\varDelta _{NG}^{\prime {*}}}F_{D_{s}^{G}}(u)du \right) . \end{aligned}$$

Dividing both sides of the equation by $(w_{sp}-e_{s})$ and using $F_{D_{s}^{G}}(M-\delta _{NG}^{\prime })$, defined in the proof of Proposition 3, simplify the above equation as

$$\begin{aligned} \varDelta _{NG}^{\prime {*}}\left( \dfrac{w_{sp}-w_{fp}}{w_{sp}-e_{s}}\right) -\int \limits _{M-\varDelta _{NG}^{\prime {*}}}^M F_{D_{s}^{G}}(u)du=\varDelta _{NG}^{\prime {*}}F_{D_{s}^{G}}\left( M-\delta _{NG}^{\prime }\right) -\int \limits _{M-\varDelta _{NG}^{\prime {*}}}^M F_{D_{s}^{G}}(u)du. \end{aligned}$$

When $\delta _{NG}^{\prime }$ is smaller than zero, the above expression yields zero considering the definition of $\varDelta _{NG}^{\prime {*}}$. Otherwise, when $\delta _{NG}^{\prime }$ is greater than or equal to zero, it will definitely be greater than or equal to $\varDelta _{NG}^{\prime {*}}$. With this argument and the basic properties of the cumulative distribution function, we can show that the above expression is less than or equal to zero. Therefore, given that supplier j’s production cost is greater than or equal to $\hat{w}_{s}$, choosing the market option is always at least as good as or better than choosing the GPO option. Accordingly, it never allocates a portion of its capacity to the GPO.

For suppliers whose production cost is less than $\hat{w}_{s}$, the GPO will be beneficial if and only if

$$\begin{aligned} E\left[ \hat{\varPi }_{s_{j}}^{G}\left( \varDelta _{NG}^{*}\right) |c<\hat{w}_{s}\right] - E\left[ \varPi _{s_{j}}^{NG}\left( \varDelta _{NG}^{\prime {*}}, \varDelta _{NG}^{*}\right) |c\right] >0. \end{aligned}$$

The above difference can be restated more explicitly as

$$\begin{aligned}&E\left[ \hat{\varPi }_{s_{j}}^{G}\left( \varDelta _{NG}^{*}\right) |c<\hat{w}_{s}\right] - E\left[ \varPi _{s_{j}}^{NG}\left( \varDelta _{NG}^{\prime {*}}, \varDelta _{NG}^{*}\right) |c\right] \\&\quad =\hat{w}_{s}\varDelta _{G}^{*} +\left( w_{sp}-e_{s}\right) S\left( M-\varDelta _{G}^{*}, D_{s}^{G}\right) \\&\qquad -\left( w_{fp}-e_{s}\right) \varDelta _{NG}^{\prime {*}}-\left( w_{sp}-e_{s}\right) S\left( M-\varDelta _{NG}^{\prime {*}}, D_{s}^{G}\right) . \end{aligned}$$

The above function can be rewritten as

$$\begin{aligned} \varDelta _{NG}^{\prime {*}}\left( 1-\dfrac{w_{fp}-e_{s}}{w_{sp}-e_{s}}\right) -\varDelta _{G}^{*}\left( 1-\dfrac{\hat{w}_{s}}{w_{sp}-e_{s}}\right) +\int \limits _{0}^{M-\varDelta _{NG}^{\prime {*}}}F_{D_{s}^{G}}(u)du-\int \limits _{0}^{M-\varDelta _{G}^{*}}F_{D_{s}^{G}}(u)du. \end{aligned}$$

If the above expression is positive, supplier j prefers the GPO option given $c<\hat{w}_{s}$.

If $\hat{w}_{s}\ge w_{fp}-e_{s}$ and $w_{r}\le w_{fp}+e_{r}$ hold, then $\varDelta _{G}^{*}$ will be greater than or equal to $\varDelta _{NG}^{\prime {*}}$ due to their definitions. Therefore, the above expression can be simplified as

$$\begin{aligned} \varDelta _{NG}^{\prime {*}}\left( \dfrac{\hat{w}_{s}-w_{fp}+e_{s}}{w_{sp}-e_{s}}\right) -\left( \varDelta _{G}^{*}-\varDelta _{NG}^{\prime {*}}\right) \left( 1-\dfrac{\hat{w}_{s}}{w_{sp}-e_{s}}\right) +\int \limits _{M-\varDelta _{G}^{*}}^{M-\varDelta _{NG}^{\prime {*}}}F_{D_{s}^{G}}(u)du. \end{aligned}$$

When $\varDelta _{G}^{*}\ge \varDelta _{NG}^{\prime {*}}$, we can show that the above equation is positive. As a result, we conclude that given $c<\hat{w}_{s}$, supplier j definitely prefers working with the GPO when $\hat{w}_{s}\ge w_{fp}-e_{s}$ and $w_{r}\le w_{fp}+e_{r}$. $\square$

Proof of Proposition 5

The capacity allocated to the GPO is exactly the same for all participating suppliers with $c<\hat{w}_{s}$. In addition, the allocated capacities are assumed to be sold in batches. Therefore, $k\varDelta _{G}^{*}$ items are procured at the end of the auction. If the allocated capacities and the total amount of procured items at the end of the auction are normalized to $\varDelta _{G}^{*}$, the mechanism analyzed corresponds to an auction in which bidders bid for only one of k objects.

Milgrom (2004) proposes that if every bidder has a demand for just a single item and can bid for only a single unit, then the uniform-price auction would be a Vickrey auction. In this setting, truthful bidding is an optimal strategy for all bidders. Accordingly, we conclude, following (Milgrom 2004), that truthful bidding is an optimal strategy for all bidders in our setting. $\square$

Proof of Proposition 6

Raghunandanan and Patil (1972) derive the probability density and the moment generating functions of the order statistics where the sample size is random. The authors also propose a special case where the variables follow a uniform distribution over the interval [0, 1] and the sample size follows a binomial distribution.

In our model, the sample size follows a binomial distribution as stated in Eq. (14), and the production cost of a supplier participating in the auction is uniformly distributed between $\underline{c}$ and $\hat{w}_{s}$. Therefore, we can directly use the approach given in Raghunandanan and Patil (1972) to obtain the expression. $\square$

Appendix B

Proposition 7

For a given $w_{r}$ quoted by the GPO, the buyer will work with the GPO in the first period when $E[\varPi _{r}^{G}(q_{G}^{*})]-E[\varPi _{r}^{NG}(q^{*}_{NG})] \ge 0$. Where $D_{T}$ follows a uniform distribution between $\underline{d}_{T}$ and $\overline{d}_{T}$ , the optimal advance order quantity, $q_{G}^{*}$ , is defined as

$$\begin{aligned} q_{G}^{*} ={\left\{ \begin{array}{ll} \beta \underline{d}_{T}+\beta \left( \overline{d}_{T} - \underline{d}_{T} \right) \left( 1-\dfrac{w_{r}}{w_{sp}+e_{R}}\right) &{} {\text {if }}\; w_{sp}+e_{R}\ge w_{r}\ge 0 \\ 0 &{} {\text {otherwise }}\end{array}\right. }. \end{aligned}$$

In addition, the total advance order quantity consolidated in the GPO is $Q_{G}^{*}=nq_{G}^{*}$.

On the other hand, if $E[\varPi _{r}^{G}(q_{G}^{*})]- E[\varPi _{r}^{NG}(q_{NG}^{*})]<0$ , the buyer will use the market option in the first period. Under this setting, when $D_{T}$ is uniformly distributed between $\underline{d}_{T}$ and $\overline{d}_{T}$ , the optimal advance order quantity, $q_{NG}^{*}$ , is defined as

$$\begin{aligned} q_{NG}^{*} ={\left\{ \begin{array}{ll} \beta \underline{d}_{T} +\beta \left( \overline{d}_{T} - \underline{d}_{T} \right) \left( 1-\dfrac{w_{fp}+e_{R}}{w_{sp}+e_{R}}\right) &{} {\text {if }}\; w_{sp}+e_{R}\ge w_{fp}+e_{R}\ge 0 \\ 0 &{} {\text {otherwise }}\end{array}\right. }. \end{aligned}$$

Proposition 8

Given that supplier j’s production cost is lower than the estimated auction price, the supplier will be eager to work with the GPO when the GPO option is more beneficial to him compared to the market option. Where $D_{T}$ is uniformly distributed between $\underline{d}_{T}$ and $\overline{d}_{T}$ , it allocates

$$\begin{aligned} \varDelta _{G}^{*} = {\left\{ \begin{array}{ll} \min \left\{ \delta _{G}, Q_{G}^{*}\right\} &{}{\text {if }}\; w_{sp}-e_{s}\ge \hat{w}_{s} >0\\ 0 &{}{\text {otherwise}} \end{array}\right. } \end{aligned}$$

where $\delta _{G}=M-\left( \overline{d}_{T} -Q_{T}^{G}\right) \psi \left( 1-\dfrac{\hat{w}_{s}}{w_{sp}-e_{S}} \right)$.

Supplier j never assigns a portion of its capacity for the GPO if its production cost c is greater than or equal to the estimated auction price, that is, $c\ge \hat{w}_{s}$.

Where $D_{T}$ follows a uniform distribution between $\underline{d}_{T}$ and $\overline{d}_{T}$ and a GPO is operational and the supplier considers the market option more beneficial, the portion of capacity it will allocate to the buyers in the market is given as

$$\begin{aligned} \varDelta _{NG}^{\prime {*}} ={\left\{ \begin{array}{ll} \min \left\{ \delta _{NG}^{\prime {*}}, \psi \dfrac{nq_{NG}(1-\tau )}{\tau }\right\} &{} {\text {if }}\; w_{sp}-e_{s}\ge w_{sp}-w_{fp}>0\\ 0 &{} {\text {otherwise}} \end{array}\right. } \end{aligned}$$

where $\delta _{NG}^{\prime }=M-\left( \overline{d}_{T} -Q_{T}^{G}\right) \psi \left( \dfrac{w_{sp}-w_{fp}}{w_{sp}-e_{S}} \right)$.

Where $D_{T}$ is uniformly distributed between $\underline{d}_{T}$ and $\overline{d}_{T}$ and no GPO is operational, the portion allocated to the buyers in the market is defined as

$$\begin{aligned} \varDelta _{NG}^{*} = {\left\{ \begin{array}{ll} \min \left\{ \delta _{NG},\psi Q_{T}^{NG}\right\} &{}{\text {if }}\; w_{sp}-e_{s}\ge w_{sp}-w_{fp}>0\\ 0 &{} {\text {otherwise }} \end{array}\right. } \end{aligned}$$

where $\delta _{NG}= M-\left( \overline{d}_{T} -Q_{T}^{NG}\right) \psi \left( \dfrac{w_{sp}-w_{fp}}{w_{sp}-e_{S}} \right)$.

Proposition 9

When the supplier’s production cost is lower than the estimated auction price and $D_{T}$ is uniformly distributed between $\underline{d}_{T}$ and $\overline{d}_{T}$ , the supplier considers the GPO beneficial to him if and only if the following inequality holds.

$$\begin{aligned}\nonumber \varDelta _{G}^{*}\delta _{G}-\varDelta _{NG}^{\prime {*}}\delta _{NG}^{\prime }-\dfrac{\left( \varDelta _{G}^{*}\right) ^{2} -\left( \varDelta _{NG}^{\prime {*}}\right) ^{2}}{2}\ge 0 \end{aligned}$$

On the other hand, suppliers whose production costs are greater than or equal to the estimated auction price would never consider the GPO beneficial to them. Therefore, they would never participate in the auction.