Fixed Allocation of Capacity for Multiple Retailers Under Demand Competition

Abstract

Product capacity shortages frequently occur in many industries, such as electrical goods, pharmaceuticals and automobiles. For example, in 2018 the supply of the Tesla Model 3 electric car was unable to meet the market demand (Stangel 2018). An allocation mechanism is typically implemented by suppliers in such cases to partly fill the demand of the retailers, in which a supplier makes decisions about pricing and how the available capacity can be allocated among retailers.

Appendix

Proof of Lemma 5.1

We separately prove the three properties in (i), (ii) and (iii) of Lemma 5.1 as follows.

(i) Suppose $\sum _{i=1}^n{m_i}\le K$. From Eq. (5.1), it holds that for $\forall ~i, g_i=m_i$;

Suppose $\sum _{j=1}^n{m_j}> K$. Then there exist an i satisfying $m_i>K-\sum _{j=1}^{i-1}{g_j(m)}-\sum _{j=i+1}^{n}{\min \{m_j,\alpha _j K\}}$, otherwise, we have $\sum _{j=1}^n{m_j}\le K$.

Under condition $m_i>K-\sum _{j=1}^{i-1}{g_j(m)}-\sum _{j=i+1}^{n}{\min \{m_j,\alpha _j K\}}$, from Eq. (5.1), we can obtain

$$\begin{aligned} \sum _{j=1}^i{g_j(m)}=K-\sum _{j=i+1}^{n}{\min \{m_j,\alpha _j K\}} \end{aligned}$$

and then

$$\begin{aligned} g_{i+1}(m)&=\min \{m_{i+1},K-(K-\sum _{j=i+1}^{n}{\min \{m_j,\alpha _j K\}})-\sum _{j=i+2}^{n}{\min \{m_j,\alpha _j K\}}\}\\&=\min \{m_{i+1},\alpha _{i+1}K\} \end{aligned}$$

By the same analysis, for $j=i+2,\ldots ,n$, we have $g_{j}(m)=\min \{m_{j},\alpha _{j}K\}$. Thus, it holds that $\sum _{j=1}^n{g_j(m)}=K$.

(ii) From Eq. (5.1), it is easy to see $g_i(m)$ is nondecreasing with $m_i$.

(iii) From Eq. (5.1), we have that $g_i(m)$ is no more than $m_i$. $\Box$

Proof of Lemma 5.2

Let $(m_1^*,\ldots ,m_n^*)$ be retailers’ equilibrium order quantities, and $m_{-i}^*=(m_1^*,m_{i-1}^*,\ldots ,m_{i+1}^*,m_n^*)$. From retailer i’s objective function of Eq. (5.3) and (i) of Lemma (5.1), there exists equilibrium order quantities $(m_1^*,\ldots ,m_n^*)$ satisfying $\sum _{i=1}^n {m_i^*}\le K$, if and only if for $\forall ~i$, it holds

$$\begin{aligned} \left\{ \begin{array}{l} m_i^*=arg\max _{m_i}{[M-w-m_i-(\sum _{j=1,j\ne i}^n {m_j^*})]m_i}\\ (M-w-\sum _{j=1}^n {m_j^*})m_i^*\ge max_{m_i>K-\sum _{j=1,j\ne i}^n {m_j^*}}{(M-w-K)g_i(m_i,m_{-i}^*)}\\ \sum _{j=1}^n {m_j^*}\le K \end{array} \right. \end{aligned}$$

which is equivalent to

$$\begin{aligned} \left\{ \begin{array}{l} m_i^*=\frac{M-w}{n+1}\\ (\frac{M-w}{n+1})^2\ge (M-w-K)g_{i,\max }\\ \frac{n(M-w)}{n+1}\le K \end{array} \right. \end{aligned}$$

If the above condition doesn’t hold, then in equilibrium the total order quantity would be $\sum _{i=1}^n {m_i^*}> K$, and $m_i^*=arg\max _{m_i}{(M-w-K)g_i(m_i,m_{-i}^*)}$, note that under fixed allocation, $g_{i}$ is nondecreasing with $m_i$ as (ii) of Lemma 5.1, thus, it is optimal for retailer i to order the maximum quantity K in equilibrium.

Next, we give the proof of the expression of $g_{i,\max }$ under condition $\frac{n(M-w)}{n+1}\le K$.

Given $m_{-i}=(\frac{M-w}{n+1},\ldots ,\frac{M-w}{n+1})$, from the nondecreasing property of fixed allocation in Lemma 5.1, retailer i can receive his maximum allocation when he orders the whole capacity K. Based on Eq. (5.1), retailers’ allocated quantities vary with the ideal quantity $\frac{M-w}{n+1}$ in different intervals. Specifically, there are two cases.

Case (i). $0\le \frac{M-w}{n+1} \le \alpha _n K$. Since the ideal quantity is no greater than any retailer’s guaranteed allocation, we have $g_j=\frac{M-w}{n+1}$, $j\in (1,\ldots ,n)$, $j\ne i$, and

$$\begin{aligned} g_{i,\max }= & {} K-\frac{n-1}{n+1}(M-w). \end{aligned}$$

Case (ii). There exists an integer t, $1\le t \le n-1$, such that $\alpha _{t+1} K<\frac{M-w}{n+1} \le \alpha _t K$. We have the following two subcases.

Case (ii.a). If $1\le i \le t$, then for retailer j, $j\in \{1,\ldots ,t\}$, $j\ne i$, it holds $\frac{M-w}{n+1}\le \alpha _j K$ and retailer j’s allocation quantity equals his ideal quantity. From the definition of fixed allocation in Eq. (5.1), we have that the maximum allocated quantity of retailer i is given by

$$\begin{aligned} g_{i,\max }= & {} K-\sum _{j=1}^{i-1}{g_j(m)}-\sum _{j=i+1}^{n}{\min \{m_j,\alpha _j K\}}\\= & {} K-\frac{(i-1)(M-w)}{n+1}-\sum _{j=i+1}^{n}{\min \left\{ \frac{M-w}{n+1},\alpha _j K\right\} }\\= & {} K-\left[ \frac{t-1}{n+1}(M-w)+\sum _{j=t+1}^n{\alpha _j K}\right] \end{aligned}$$

Case (ii.b). If $t<i\le n$, then for $1\le j \le t$, retailer j’s ideal quantity is less than his guaranteed allocation, so he receives his ideal quantity as he orders. From Eq. (5.1), we have

$$\begin{aligned} g_{t+1}= & {} \min \left\{ \frac{M-w}{n+1},K-\frac{t(M-w)}{n+1}-\sum _{j=t+2}^n{\alpha _j K}\right\} \\= & {} \frac{M-w}{n+1} \end{aligned}$$

By the same analysis, we can obtain that for $\forall ~t<j\le i-1$, it holds $g_{j}=\frac{M-w}{n+1}$. Thus, from Eq. (5.1), retailer i’s maximum allocated quantity is formulated as:

$$\begin{aligned} g_{i,\max }= & {} K-\sum _{j=1}^{i-1}{g_j(m)}-\sum _{j=i+1}^{n}{\min \{m_j,\alpha _j K\}}\\= & {} K-\frac{(i-1)(M-w)}{n+1}-\sum _{j=i+1}^{n}{\min \left\{ \frac{M-w}{n+1},\alpha _j K\right\} }\\= & {} K-[\frac{i-1}{n+1}(M-w)+\sum _{j=i+1}^n{\alpha _j K}] \end{aligned}$$

Combining (ii.a) and (ii.b), we can obtain retailer i’s maximum allocation by substituting the other retailers’ allocated quantity into Eq. (5.1), as follows.

$$\begin{aligned} g_{i,\max }=\left\{ \begin{array}{ll} K-[\frac{t-1}{n+1}(M-w)+\sum _{j=t+1}^n{\alpha _j K}] &{} i\le t,\\ K-[\frac{i-1}{n+1}(M-w)+\sum _{j=i+1}^n{\alpha _j K}] &{} i>t. \end{array} \right. \end{aligned}$$

Case (iii). $\frac{M-w}{n+1}>\alpha _1 K$.

For $i=1$, from Eq. (5.1), the maximum allocated quantity of retailer 1 is

$$\begin{aligned} g_{1,\max }=K-\sum _{j=2}^n{\alpha _j K}\bigg ] \end{aligned}$$

If $i>1$, then for $1\le j<i$, by the same analysis as $t\le j<i$ in Case (ii.b), we have $g_j=\frac{M-w}{n+1}$. From Eq. (5.1), we have retailer i’s maximum allocation as follows.

$$\begin{aligned} g_{i,\max }=K-[\frac{i-1}{n+1}(M-w)+\sum _{j=i+1}^n{\alpha _j K}]&i\ge 2 \end{aligned}$$

Therefore, Lemma 5.2 holds. $\Box$

Proof of Lemma 5.3

When $w\in [M-(n+1)\alpha _t K, M-(n+1)\alpha _{t+1}K]$, $1\le t \le n-1$, retailer i’s maximum allocation is $g_{i,\max }$, as Lemma 5.2 shows. Since $\frac{M-w}{n+1}>\alpha _j K$ for $j>t$, we have $\frac{t-1}{n+1}(M-w)+\sum _{j=t+1}^{n}\alpha _j K<\frac{i-1}{n+1}(M-w)+\sum _{j=i+1}^{n}\alpha _j K$, $\forall ~i>t$. Thus, it holds that $g_{i_1,\max }>g_{i_2,\max }$, $i_1 \le t$ and $i_2>t$. Therefore, we obtain that the inequality $(\frac{M-w}{n+1})^2 > (M-K-w)g_{i,\max }$ holding for all i is equivalent to

$$\begin{aligned} (\frac{M-w}{n+1})^2>(M-K-w)\{K-[\frac{t-1}{n+1}(M-w)+\sum _{j=t+1}^{n}\alpha _j K]\}, \end{aligned}$$

which is further equivalent to

$$\begin{aligned} w>&M-\big (\frac{(n+1)[(n+1)(1-\sum _{j=t+1}^n \alpha _j)+t-1]}{2[(n+1)t-n]}-~~~~~~~~~~~~~\\&\frac{(n+1)\sqrt{[(n+1)(1-\sum _{j=t+1}^n \alpha _j)-(t-1)]^2-4(1-\sum _{j=t+1}^n \alpha _j)}}{2[(n+1)t-n]}\big )K,\\ =&\beta _t \end{aligned}$$

or

$$\begin{aligned} w\le&M-\big (\frac{(n+1)[(n+1)(1-\sum _{j=t+1}^n \alpha _j)+t-1]}{2[(n+1)t-n]}+~~~~~~~~~~~~\\&\frac{(n+1)\sqrt{[(n+1)(1-\sum _{j=t+1}^n \alpha _j)+(t-1)]^2-4(1-\sum _{j=t+1}^n \alpha _j)}}{2[(n+1)t-n]}\big )K\\ =&\gamma _t, \end{aligned}$$

Recall that in Lemma 5.3, there are conditions that $w\ge M-\frac{n+1}{n}K$ and $w\ge (n+1)\alpha _t K$. Since $\gamma _t<\min \{M-\frac{n+1}{n}K,(n+1)\alpha _t K\}$, then we omits the second inequality $w\le \gamma _t$.

Above all, Lemma 5.3 holds. $\Box$

Proof of Theorem 5.1

Our proof is in two steps: first, we find retailers’ equilibrium order quantities in the scenario with the total ideal quantity greater than capacity, i.e., $\frac{n(M-w)}{n+1}>K$; second, we find retailers’ equilibrium order quantities when capacity is less than retailers’ total ideal quantity.

(I) Suppose $0\le w < M-\frac{n+1}{n}K$, i.e., $\frac{n(M-w)}{n+1}>K$. From Lemma 5.2, each retailer will order as much as possible to maximize his profit, thus, each retailer except retailer 1 orders more than his guaranteed quantity in equilibrium, and receives his guaranteed allocation, while retailer 1 orders no less than the remaining capacity, with equilibrium allocated quantities as $(g_1,\ldots ,g_n)=((1-\sum _{2}^n\alpha _j)K, \ldots , \alpha _n K)$.

(II) Suppose $M-\frac{n+1}{n}K\le w \le M$, i.e., $\frac{n(M-w)}{n+1}\le K$. We separately solve retailers’ equilibrium ordering quantities in the following two cases.

(i) When retailers have equal fixed proportions, i.e., $\alpha _1=\cdots =\alpha _n=\frac{l}{n}$.

Suppose $l=1$.

From Lemma 5.2, we have $\forall ~i$, $g_{i,\max }=K-\frac{(n-1)(M-w)}{n+1}$. Since for any retailer i, it holds $(\frac{M-w}{n+1})^2 \ge (M-w-K)g_{i,\max }=(M-w-K)[K-\frac{(n-1)(M-w)}{n+1}]$, then retailers’ equilibrium order quantities are their ideal quantities, i.e., $(m_i^*,\ldots ,m_n^*)=(\frac{M-w}{n+1},\ldots ,\frac{M-w}{n+1})$. Combining the result with part (I) of the proof, cases (i.a) and (i.b) of Theorem 5.1 hold.

Suppose $l<1$.

From Lemma 5.2, we have $\forall ~i$, $g_{i,\max }=K-[\frac{i-1}{n+1}(M-w)+\sum _{j=i+1}^n{\alpha _j K}]$. Since retailer 1 is given a highest priority, then we have that the retailers order their ideal quantities in equilibrium, if and only if $(\frac{M-w}{n+1})^2 \ge g_{1,\max }$, which is equivalent to

$$\begin{aligned} w\ge&\,M-\frac{1}{2}\left\{ (n+1)\left[ (n+1)\left( 1-\frac{(n-1)l}{n}\right) \right] \right. \\ {}&\left. - (n+1)\sqrt{(n+1)^2 (1-\frac{(n-1)l}{n})^2-4(1-\frac{(n-1)l}{n})}\right\} K \end{aligned}$$

And if the above inequality doesn’t hold, then each retailer orders as much as possible to have a favorable allocation, where each retailer except retailer 1 orders more than his guaranteed quantity in equilibrium, and receives his guaranteed allocation, while retailer 1 orders no less than the remaining capacity, with equilibrium allocated quantities as $(g_1,\ldots ,g_n)=((1-\sum _{2}^n\alpha _j)K, \ldots , \alpha _n K)$. Thus, cases (i.c) and (i.d) of Theorem 5.1 hold.

To sum up, when the retailers’ allocation proportions are the same and the supplier guarantees the whole capacity to retailers in fixed allocation, then no matter whether the supplier’s capacity can satisfy the total ideal quantities of the retailers, each retailer will order his ideal quantity in equilibrium, and hence the fixed allocation mechanism is truth-inducing, as (i.a) and (i.b) show. However, when the supplier just guarantees part of capacity with equal proportions to retailers, under the condition that the wholesale price is low ($w\in [0, M-\delta K)$), the retailers order as much as possible to receive favorable allocations, apart from their ideal quantities, even when the capacity is enough to fill the total ideal quantity of retailers, i.e., $w\in (M-\frac{(n+1)}{n}K, M-\delta K)$. And each retailer except retailer 1 orders more than his guaranteed quantity in equilibrium, and receives his guaranteed allocation; while retailer 1 orders more than the difference between capacity and the total guaranteed quantity of other retailers, as (i.c) shows. When the wholesale price is high ($w\in [M-\delta K,M)$), then the retailers reach a unique equilibrium with ordering their ideal quantities.

(ii) When there exists at least two retailers with different fixed proportions, i.e., there exist $i\ne j$ such that $\alpha _i\ne \alpha _j$. First, we establish two properties of $\beta _t$ in Lemma 5.3, which will be used in the following analysis.

(1) if $\beta _i>(n+1)\alpha _{i+1}$, then $\beta _j>(n+1)\alpha _{j}$, $\forall ~j\in \{i+1,\ldots ,n\}$;

(2) $\{s|\beta _{s}>(n+1)\alpha _{s+1}, \alpha _{s+1}<\frac{1}{n}, s\in \{1,\ldots ,n-1\}\}\ne \emptyset$.

Proof of part (1) By the definition of $\beta _i$, we have

$$\begin{aligned}&\beta _{i+1}>(n+1)\alpha _{i+1} \\ \Leftrightarrow&\alpha _{i+1}<\frac{[(n+1)(1-\sum _{j=i+1}^n{\alpha _j})+i-1]- \sqrt{[(n+1)(1-\sum _{j=i+1}^n{\alpha _j})-(i-1)]^2-4(1-\sum _{j=i+1}^n{\alpha _j})}}{2[(n+1)i-n]}\\&=\frac{\beta _i}{n+1}.\\ \text{ or }&\\&\alpha _{i+1}>\frac{[(n+1)(1-\sum _{j=i+1}^n{\alpha _j})+i-1]+ \sqrt{[(n+1)(1-\sum _{j=i+1}^n{\alpha _j})-(i-1)]^2-4(1-\sum _{j=i+1}^n{\alpha _j})}}{2[(n+1)i-n]}. \end{aligned}$$

which shows that $\beta _i>(n+1)\alpha _{i+1} \Rightarrow \beta _{i+1}>(n+1)\alpha _{i+1}$. Since $\alpha _{i+1}>\alpha _{i+2}$, it holds $\beta _{i+1}>(n+1)\alpha _{i+2}$. By similar analysis, we have $\beta _{i+2}>(n+1)\alpha _{i+2}$. Finally, we can obtain $\beta _j>(n+1)\alpha _{j}$, $\forall ~j\in \{i+1,\ldots ,n\}$.

Proof of part (2) By the expression of $\beta _i$, we have

$$\begin{aligned} \beta _{n-1}= & {} \frac{(n+1)[(n+1)(1-\alpha _n)+n-2]- (n+1)\sqrt{[(n+1)(1-\alpha _n)-(n-2)]^2-4(1-\alpha _n)}}{2[(n+1)(n-1)-n]} \end{aligned}$$

By the above equation of $\beta _{n-1}$, we have that $\beta _{n-1}>(n+1)\alpha _{n} \Leftrightarrow (n\alpha _n-1)^2>0$. Since there exit $i \ne j$ such that $\alpha _i \ne \alpha _j$, we have $\alpha _n<\frac{1}{n}$, and it always holds $\beta _{n-1}>(n+1)\alpha _{n}$. Thus, $n-1\in \{s|\beta _{s}>(n+1)\alpha _{s+1}, \alpha _{s+1}<\frac{1}{n}, s\in \{1,\ldots ,n-1\}\}$.

Next, we characterize retailers’ equilibrium ordering quantities with the wholesale price in different intervals. Let $\theta =\min \{s|\alpha _s<\frac{1}{n}, s\in \{2,\ldots ,n\}\}$, and $\theta ^*=\min \{s|\beta _{s}>(n+1)\alpha _{s+1}, \alpha _{s+1}<\frac{1}{n}, s\in \{1,\ldots ,n-1\}\}$.

(a) Consider the case with $M-(n+1)\alpha _{j-1}K<w\le M-(n+1)\alpha _j K, j\in \{\theta ,\ldots ,\theta ^*\}$. By the definition of $\theta ^*$, we have $\forall ~j \in \{\theta ,\ldots ,\theta ^*\}$, $\beta _{j-1}\le (n+1)\alpha _j$, and then $w\le M-\beta _{j-1} K$. By Lemmas 5.2 and 5.3, each retailer except retailer 1 orders more than his guaranteed quantity in equilibrium, and receives his guaranteed allocation, while retailer 1 orders no less than the remaining capacity, with equilibrium allocated quantities as $(g_1,\ldots ,g_n)=((1-\sum _{2}^n\alpha _j)K, \ldots , \alpha _n K)$.

(b) Consider the case with $M-(n+1)\alpha _{\theta ^*} K<w\le M-(n+1)\alpha _{\theta ^*+1} K$. By Lemmas 5.2 and 5.3, if $M-(n+1)\alpha _{\theta ^*} K<w\le M-\beta _{\theta ^*} K$, then each retailer except retailer 1 orders more than his guaranteed quantity in equilibrium, and receives his guaranteed allocation, while retailer 1 orders no less than the remaining capacity, with equilibrium allocated quantities as $(g_1,\ldots ,g_n)=((1-\sum _{2}^n\alpha _j)K, \ldots , \alpha _n K)$; if $M-\beta _{\theta ^*} K<w\le M-(n+1)\alpha _{\theta ^*+1} K$, then each retailer orders and receives his ideal quantity, $(m_1^*,\ldots ,m_n^*)=(\frac{M-w}{n+1},\ldots ,\frac{M-w}{n+1})$.

(c) Consider the case with $M-(n+1)\alpha _{j} K<w\le M-(n+1)\alpha _{j+1} K$, $j\in \{\theta ^*+1,\ldots ,n-1\}$. By property (1) of $\beta _j$, we have $\beta _j>(n+1)\alpha _{j}$. Together with Lemmas 5.2 and 5.3, we have that retailers have a unique vector of equilibrium order quantities, $(m_1^*,\ldots ,m_n^*)=(\frac{M-w}{n+1},\ldots ,\frac{M-w}{n+1})$.

(d) Consider the case with $M-(n+1)\alpha _{n} K<w\le M$. Similar to the proof of part (c), retailers have a unique vector of equilibrium order quantities, $(m_1^*,\ldots ,m_n^*)=(\frac{M-w}{n+1},\ldots ,\frac{M-w}{n+1})$.

Combining (a), (b), (c) and (d), part (ii) of Theorem 5.1 holds.

In sum, when there are at least two different guaranteed proportions in the fixed allocation, we have a similar analysis of equilibrium order quantities as cases (i.c) and (i.d). If the wholesale price is low, i.e., $w\in [0,M-\beta K)$, then each retailer orders no less than his guaranteed quantity, where retailer 1 orders more than the difference between capacity and the total guaranteed quantity of other retailers, with allocated quantity as $(1-\sum _{j=2}^n \alpha _j) K$, and the other retailers receive their guaranteed allocations $\alpha _i K$. If the wholesale price is relatively low ($w\in [0,M-\frac{(n+1)}{n}K]$), then the retailers’ total ideal quantity will exceed the available capacity and each retailer will order as much as possible to obtain more, and thus the retailers except retailer 1 will receive their respective guaranteed quantities in equilibrium, while retailer 1 obtains all the remaining capacity. When the wholesale price is medium ($w\in (M-\frac{(n+1)}{n}K,M-\beta K)$), although the available capacity can fulfil the total ideal quantities of all retailers, there is a retailer whose profit resulting from maximum allocation is greater than that from ordering his ideal quantity, which makes the total order quantity greater than the available capacity. Furthermore, other retailers will also order more than their respective guaranteed quantities, and consequently retailers receive their equilibrium allocations as $((1-\sum _{j=2}^n \alpha _j) K,\ldots ,\alpha _n K)$. If the wholesale price is high ($w\in [M-\beta K, M]$), then each retailer obtains his maximum profit by ordering and receiving his ideal quantity, which results in a unique ordering equilibrium $(\frac{M-w}{n+1},\ldots ,\frac{M-w}{n+1})$.$\Box$

Proof of Corollary 5.1

From Theorem 5.1, when retailers have the same allocation proportion, i.e., $\alpha _1=\cdots =\alpha _n, 0\le l \le 1$, the supplier’s optimization problem is formulated as follows:

(i) Under fixed allocation with full capacity commitment, i.e. $l=1$

$$\begin{aligned} \Pi _s^*=\max _{w}\{\max _{w\in [0,M-\frac{n+1}{n}K)}K\cdot w; \max _{w\in [M-\frac{n+1}{n}K,M]}\frac{nw(M-w)}{n+1}\} \end{aligned}$$

By solving the equation, we have that

$$\begin{aligned} \Pi _s^*=\left\{ \begin{array}{ll} K \cdot (M-\frac{n+1}{n}K)^- &{} \text{ if } M>\frac{2(n+1)}{n}K \\ \frac{nM^2}{4(n+1)} &{} \text{ if } M\le \frac{2(n+1)}{n}K \end{array} \right. \end{aligned}$$

with

$$\begin{aligned} w^*=\left\{ \begin{array}{ll} (M-\frac{n+1}{n}K)^- &{} \text{ if } M>\frac{2(n+1)}{n}K \\ \frac{M}{2} &{} \text{ if } M\le \frac{2(n+1)}{n}K \end{array} \right. \end{aligned}$$

(ii) Under fixed allocation with partial capacity commitment, i.e. $l<1$

$$\begin{aligned} \tilde{\Pi }_s^*=\max _{w}\{\max _{w\in [0,M-\delta K)}K\cdot w; \max _{w\in [M-\delta K,M]}\frac{nw(M-w)}{n+1}\} \end{aligned}$$

By solving the equation, we have that

$$\begin{aligned} \tilde{\Pi }_s^*=\left\{ \begin{array}{ll} K \cdot (M-\delta K)^- &{} \text{ if } M>\tau (\delta )K \\ \frac{nM^2}{4(n+1)} &{} \text{ if } M\le \tau (\delta )K \end{array} \right. \end{aligned}$$

with

$$\begin{aligned} w^*=\left\{ \begin{array}{ll} (M-\delta K)^- &{} \text{ if } M>\tau (\delta )K \\ \frac{M}{2} &{} \text{ if } M\le \tau (\delta )K \end{array} \right. \end{aligned}$$

Here, $\delta =\frac{1}{2}\{(n+1)[(n+1)(1-\frac{(n-1)l}{n})]- (n+1)\sqrt{(n+1)^2 (1-\frac{(n-1)l}{n})^2-4(1-\frac{(n-1)l}{n})}\}, \tau (\delta )=\frac{2(n+1)-2\sqrt{(n+1)^2-n(n+1)\delta }}{n}$.

Note that $\delta<\frac{n+1}{n}, \tau {\delta }<\frac{2(n+1)}{n}$, then comparing $\tilde{Pi}_s^*$ and $\tilde{Pi}_s^*$, we have the result of Corollary 5.1. $\Box$

Proof of Corollary 5.2

From Theorem 5.1, when either of conditions as (i) and (ii) holds, then each retailer except retailer 1 orders more than his guaranteed quantity in equilibrium, and receives his guaranteed allocation, while retailer 1 orders no less than the remaining capacity, with the total order quantity exceeds capacity. Although the total ideal quantity of retailers $\frac{n(M-w)}{n+1}$ is less than capacity K, the fixed allocation induces retailers to inflate their ideal quantities.$\Box$

Proof of Theorem 5.2

First, we verify that $\beta <\frac{n+1}{n}$, which would be used in the following proof.

By Eq. (5.4), we have

$$\begin{aligned}&\beta<\frac{n+1}{n}\\ \Leftrightarrow&n[(n+1)(1-\sum _{j=\theta ^*+1}^n{\alpha _j})+\theta ^*-1]-2[(n+1)\theta ^*-n]\\&<n\sqrt{[(n+1)(1-\sum _{j=\theta ^*+1}^n{\alpha _j})-(\theta ^*-1)]^2-4(1-\sum _{j=\theta ^*+1}^n{\alpha _j})}\\ \Leftrightarrow&\sum _{j=\theta ^*+1}^n{\alpha _j}<1-\frac{\theta ^*}{n} \end{aligned}$$

Then we have that $\beta <\frac{n+1}{n}$ always holds.

Next, we prove the supplier’s optimal wholesale price.

From Eq. (5.5), the supplier’s maximum profit is given by:

$$\begin{aligned} \Pi _s= & {} \max _w\{\Pi _s^1;\Pi _s^2\},\\ \Pi _s^1= & {} \max _{0\le w<M-\beta K}{Kw}=K(M-\beta K)^-,\\ \Pi _s^2= & {} \max _{M-\beta K \le w \le M}{\frac{n(M-w)}{n+1}w}\\= & {} \left\{ \begin{array}{ll}\frac{nM^2}{4(n+1)} &{} M\le 2\beta K\\ \frac{n\beta K}{n+1}(M-\beta K) &{} M>2\beta K. \end{array} \right. \end{aligned}$$

When $M\le 2\beta K$, $\Pi _s^1<\Pi _s^2 \Leftrightarrow \frac{nM^2}{4(n+1)}>K(M-\beta K)$, which is equivalent to $M<\frac{2(n+1)-2\sqrt{(n+1)^2-n(n+1)\beta }}{n}K$ or $M>\frac{2(n+1)+2\sqrt{(n+1)^2-n(n+1)\beta }}{n}K$. Note that $\frac{2(n+1)-2\sqrt{(n+1)^2-n(n+1)\beta }}{n}<2\beta <\frac{2(n+1)+2\sqrt{(n+1)^2-n(n+1)\beta }}{n}$ is equivalent to $\beta <\frac{n+1}{n}$, we can obtain the following results:

(a) If $0\le M \le \tau (\beta )K$, then $\Pi _s^*=\Pi _s^2=\frac{nM^2}{4(n+1)}$ and $w^*=\frac{M}{2}>M-\beta K$. From Theorem 5.1, retailers’ equilibrium order quantities are equal to their ideal quantities $(\frac{M-w^*}{n+1},\ldots ,\frac{M-w^*}{n+1})=(\frac{M}{2(n+1)},\ldots ,\frac{M}{2(n+1)})$.

(b) If $\tau (\beta )K<M\le 2\beta K$, then $\Pi _s^*=\Pi _s^1$ and $w^*=(M-\beta K)^-$. From Theorem 5.1, each retailer orders no less than his guaranteed allocation $\alpha _i K$.

(c) If $M>2\beta K$, then since $\beta <\frac{n+1}{n}$, we have $\Pi _s^1>\Pi _s^2=\frac{n\beta K}{n+1}(M-\beta K)$. Thus, $\Pi _s^*=\Pi _s^1, w^*=(M-\beta K)^-$.

Above all, Theorem 5.2 holds. $\Box$

Proof of Lemma 5.4

Define a function $l(x,y)=\frac{(n+1)[(n+1)x+y-1]-(n+1)\sqrt{[(n+1)x-(y-1)]^2-4x}}{2[(n+1)y-n]}$, where $x>y/n$ and $y\ge 1$. Let $x=1-\sum _{j=\theta ^*+1}^n{\alpha _j}$ and $y=\theta ^*$. Note that $\beta =l(x,y)$. Next, we analyze the monotony of l(x, y) over x and y, and how $\beta$ changes with $\sum _{j=\theta ^*+1}^n{\alpha _j}$ and $\theta ^*$. We have

$$\begin{aligned} \frac{\partial {l}}{\partial {x}}=n+1-\frac{(n+1)[(n+1)x-(y-1)]-2}{\sqrt{[(n+1)x-(y-1)]^2-4x}}, \end{aligned}$$

and thus

$$\begin{aligned} \frac{\partial {l}}{\partial {x}}<0\Leftrightarrow & {} (n+1)\sqrt{[(n+1)x-(y-1)]^2-4x}<(n+1)[(n+1)x-(y-1)]-2\\\Leftrightarrow & {} (n+1)^2\{[(n+1)x-(y-1)]^2-4x\}<\{(n+1)[(n+1)x-(y-1)]-2\}^2\\\Leftrightarrow & {} 4(n+1)(y-1)+4>0. \end{aligned}$$

Since $y\ge 1$, we have $\partial {l(x,y)}/\partial {x}<0$. We also have

$$\begin{aligned} \frac{\partial {l}}{\partial {y}}= & {} \frac{1}{2[(n+1)y-n]^2}\cdot \{\frac{[(n+1)x-(y-1)][(n+1)y-n]}{\sqrt{[(n+1)x-(y-1)]^2-4x}}\\{} & {} +(n+1)\sqrt{[(n+1)x-(y-1)]^2-4x}-[(n+1)^2x-1]\} \end{aligned}$$

and thus

$$\begin{aligned}&\frac{\partial {l}}{\partial {y}}>0 \\ \Leftrightarrow&\frac{[(n+1)x-(y-1)][(n+1)y-n]}{\sqrt{[(n+1)x-(y-1)]^2-4x}}+(n+1)\sqrt{[(n+1)x-(y-1)]^2-4x}>(n+1)^2x-1\\ \Leftrightarrow&\{\frac{[(n+1)x-(y-1)][(n+1)y-n]}{\sqrt{[(n+1)x-(y-1)]^2-4x}}+(n+1)\sqrt{[(n+1)x-(y-1)]^2-4x}\}^2>[(n+1)^2x-1]^2\\ \Leftrightarrow&\frac{4x[(n+1)^2y-n]^2}{\sqrt{[(n+1)x-(y-1)]^2-4x}}>0. \end{aligned}$$

Thus, we have $\partial {l(x,y)}/\partial {y}>0$.

From the monotony of l(x, y), we obtain that $\beta$ increases with $\sum _{j=\theta ^*+1}^n\alpha _j$ and $\theta ^*$. Note that $\theta ^* \in Z^+$ and $\theta ^* \ge 1$. Thus, when $\theta ^*=1$ and $\sum _{j=\theta ^*+1}^n \alpha _j=0$, we obtain the minimum value of $\beta$. Here, fixed proportions satisfy $\alpha _1\le 1$ and $\alpha _2=\cdots =\alpha _n=0$.

Plugging $\theta ^*=1$ and $\sum _{j=\theta ^*+1}^n \alpha _j=0$ into Eq. (5.4), we have $\beta _{min}=[(n+1)^2-(n+1)\sqrt{(n+1)^2-4}]/2$. In the following, we prove $\beta <(n+1)/n$. Since fixed proportions satisfy $\alpha _i\le \alpha _j$, $i\le j$, and there exist i and j such that $\alpha _i\ne \alpha _j$, we have $1-\sum _{j=\theta ^*+1}^n \alpha _j>\frac{\theta ^*}{n}$, which is equivalent to

$\frac{(n+1)[(n+1)(1-\sum _{j=\theta ^*+1}^n \alpha _j)+\theta ^*-1]- (n+1)\sqrt{[(n+1)(1-\sum _{j=\theta ^*+1}^n \alpha _j)-(\theta ^*-1)]^2-4(1-\sum _{j=\theta ^*+1}^n \alpha _j)}}{2[(n+1)\theta ^*-n]}{<}\frac{n+1}{n}$.

Thus, $\beta <(n+1)/n$ always holds. $\Box$

Proof of Corollary 5.3

Note that $v=M/K-\frac{n(M/K)^2}{4(n+1)}$, and from Theorem 5.2,

$$\begin{aligned} \tau (\beta )=\frac{2(n+1)-2\sqrt{(n+1)^2-n(n+1)\beta }}{n}<\frac{2(n+1)}{n}. \end{aligned}$$

Further, through simple computation, we can obtain the following inequations:

$$\begin{aligned}&M<[2(n+1)-(n+1)\sqrt{4-2n[(n+1)-\sqrt{(n+1)^2-4}]}]/nK\\ \Leftrightarrow&v<[(n+1)^2-(n+1)\sqrt{(n+1)^2-4}]/2;\\&M \le \tau (\beta )K \Leftrightarrow \beta \ge v. \end{aligned}$$

(i) When $M<[2(n+1)-(n+1)\sqrt{4-2n[(n+1)-\sqrt{(n+1)^2-4}]}]/nK$, we have $v<[(n+1)^2-(n+1)\sqrt{(n+1)^2-4}]/2<\beta$, which is equivalent to $M \le \tau (\beta )K$, following part (i) of Theorem 5.2. Thus, the supplier’s optimal wholesale price is $w^*=\frac{M}{2}$.

(ii) When $[2(n+1)-(n+1)\sqrt{4-2n[(n+1)-\sqrt{(n+1)^2-4}]}]/nK \le M<2(n+1)/nK$, we have $[(n+1)^2-(n+1)\sqrt{(n+1)^2-4}]/2 \le v <(n+1)/n$. If $\beta <v$, then it holds that $M>\tau (\beta )$, and from part (ii) of Theorem 5.2, the supplier’s optimal wholesale price is $w^*=(M-\beta K)^-$. If $\beta \ge v$, then it holds that $M\le \tau (\beta )K$, and from part (i) of Theorem 5.2, the supplier’s optimal wholesale price is $w^*=\frac{M}{2}$.

(iii) When $M\ge 2(n+1)/nK$, since $\tau (\beta )<\frac{2(n+1)}{n}$, we have $M>\tau (\beta )$, following part (ii) of Theorem 5.2. Thus, the supplier’s optimal wholesale price is $w^*=(M-\beta K)^-$. $\Box$

Proof of Corollary 5.4

(i) When $(\alpha _1,\ldots ,\alpha _n)=(\frac{1}{n},\ldots ,\frac{1}{n})$, from Theorem 5.1, the supplier’s maximum profit is

$$\begin{aligned} \Pi _s^*=\max _{w}\{\Pi _s^1=\max _{0\le w<M-\frac{(n+1)}{n}K} Kw;\Pi _s^2=\max _{M-\frac{(n+1)}{n}K \le w\le M} \frac{n(M-w)}{n+1}w\}. \end{aligned}$$

Solving the above equation, we obtain the supplier’s optimal wholesale price and profit: if $M\le 2(n+1)/n K$, then $w^*=\frac{M}{2}$ and $\Pi _s^*=\frac{nM^2}{4(n+1)}$; if $M> 2(n+1)/n K$, then $w^*=M-(n+1)/n K$ and $\Pi _s^*=K[M-(n+1)/n K]$.

From Theorem 5.2 and Table 5.1, if there exist $i\ne j$ such that $\alpha _i\ne \alpha _j$, then the supplier’s maximum profit is $\Pi _s^*(\beta )$. Since $\beta <(n+1)/n$, it holds that $\Pi _s^*\le \Pi _s^*(\beta )$.

(ii) When $(\alpha _1,\alpha _2,\ldots ,\alpha _n)=(1,0,\ldots ,0)$, from Theorem 5.2 and Table 5.1, we have that $\Pi _s^*(\beta )$ decreases with $\beta$. Note that $\beta$ reaches its minimum value at $(\alpha _1,\alpha _2,\ldots ,\alpha _n)=(1,0,\ldots ,0)$. Thus, for any fixed proportion, the supplier achieves her maximum profit when $(\alpha _1,\alpha _2,\ldots ,\alpha _n)=(1,0,\ldots ,0)$. $\Box$

Proof of Theorem 5.3

When $\beta =\gamma _n=\frac{(n+1)(\sqrt{n^2+4n-4}-n)}{2(n-1)}$, we have $\tau (\beta )=2(n+1)(1-\varphi _n)/n$. Comparing Theorem 5.2 and Lemma 5.5, we have that the supplier’s optimal wholesale price and the corresponding profit under fixed allocation are respectively equal to that under proportional allocation. And under two allocations, the supply chain profit satisfies: if $M\le 2(n+1)(1-\varphi _n)/nK$, then $\Pi _{sc}^*=n(n+2)M^2/[4(n+1)^2]$; if $M>2(n+1)(1-\varphi _n)/nK$, then $\Pi _{sc}^*=(M-K)K$. $\Box$

Proof of Theorem 5.4

Given retailers’ allocated quantities $(g_1,g_2,\ldots ,g_n)$, we first solve the best response function of the selling quantity $q_i(g_i)$, and then explore the equilibrium selling quantities of retailers.

Retailer i’s objective function is formulated as:

$$\begin{aligned} \max _{0\le q_i \le g_i}\{[M-(q_i+\sum _{j=1,j\ne i}^{n}q_j)-w]q_i-w(g_i-q_i)\} \end{aligned}$$

By solving the equation, we can obtain the best response function of selling quantity as follows:

$$\begin{aligned}q_i(g_i)=\left\{ \begin{array}{ll} g_i &{} \text{ if } g_i\le [M-(\sum _{j=1,j\ne i}^n q_j)]/2\\ \frac{M-(\sum _{j=1,j\ne i}^n q_j)}{2} &{} \text{ if } g_i> [M-(\sum _{j=1,j\ne i}^n q_j)]/2 \end{array} \right. \end{aligned}$$

Substituting the conditions and equilibrium selling quantities of (i) (ii) and (iii) in Theorem 5.4 into the above equation, we can verify the results of Theorem 5.4 hold. Note that retailers are labeled by their allocated quantities, where $g_i\ge g_{i+1}, i=1,\ldots ,n-1$. Then the conditions in (i) (ii) and (iii) of Theorem 5.4 are perfect space of $(g_1,g_2,\ldots ,g_n)$. $\Box$

Proof of Corollary 5.5 (i) From Theorem 5.4, we have the equilibrium selling quantities of retailers. Without capacity constraint, one retailer can freely choose his allocation, which just equals his order.

Next, we explore retailer i’s optimal allocated quantity. Given the allocated quantities of other retailers as $g_{-i}=(g_1,\ldots ,g_{n-1})$, where $g_j\ge g_{j+1}, j=1,\ldots ,n-2$. From (ii) of Theorem 5.4, if there exits a k such that $g_k> \frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1}$ and $g_{k+1}\le \frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1}$, then in equilibrium the selling quantities satisfy

$$\begin{aligned} q_j^*=\left\{ \begin{array}{ll} \frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1} &{} \text{ for } j=1,\ldots ,k;\\ g_j &{} \text{ for } j=k+1,\ldots ,n-1. \end{array} \right. \end{aligned}$$

For $g_i$, if $g_i>\frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1}$, then $q_i^*=q_j^*=\frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1}$, $j=1,\ldots ,k$, and $q_j^*=g_j$, $j=k+1,\ldots ,n-1$.

Retailer i’s objective function is

$$\begin{aligned} \max _{g_i}\{\max _{g_i> \frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1}}\Pi _{i,1};\max _{g_i\le \frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1}}\Pi _{i,2}\}, \end{aligned}$$

where $\Pi _{i,1}=(\frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1})^2-w \cdot g_i$. Since $\Pi _{i,1}$ is decreasing with $g_i$, then the optimal allocated quantity satisfies $g_i^* \le \frac{M-\sum _{j=k+1}^{n-1}{g_j}}{k+1}$, and the optimal selling quantity would be equal to the allocated quantity. If $g_{-i}$ satisfies the condition of (i) and (iii) in Theorem 5.4, we also have $q_i^*=g_i^*$ by similar analysis. Above all, part (i) of Corollary 5.5 holds.

(ii) Suppose there exists limited capacity K. Recall that retailers’ ideal quantity is $\frac{M-w}{n+1}$ when there’s no capacity limit. Next, we divide the proof into two cases based on whether the total ideal order quantity exceeds capacity, i.e., $w\le M-\frac{n+1}{n}K$ and $w> M-\frac{n+1}{n}K$.

Case 1: $w\le M-\frac{n+1}{n}K$, i.e., $\frac{n(M-w)}{n+1}\le K$.

Step 1: First, we explore the condition when order inflation happens.

In this case, capacity is sufficient to satisfy each retailer’s ideal quantity, one retailer would deviate from his ideal quantity only when he could gain more profit from inflating his order quantity by a high allocation, with the total order quantity exceeding capacity K. Next, we show retailer j’s maximum profit by inflating his order quantity.

Given other retailers’ order quantity as $g_{-j}=(\frac{M-w}{n+1},\ldots ,\frac{M-w}{n+1})$, if retailer j inflates his order quantity $m_j\ge K-\frac{(n-1)(M-w)}{n+1}$ with an allocation $g_{j}$, denote k as the index of the retailer who has lower priority than retailer j and receives an allocation quantity in the interval $(\alpha _k K, \frac{M-w}{n+1})$, then we have that when $m_j\le g_{j,\max }$, retailers’ allocated quantity would be adjusted as

$$\begin{aligned} g_i=\left\{ \begin{array}{ll} \frac{M-w}{n+1} &{} \text{ for } i=1,\ldots ,k-1, i\ne j;\\ m_j &{} \text{ for } i=j;\\ K-\frac{(k-2)(M-w)}{n+1}-m_j-(\sum _{i=k+1}^n {\alpha _i K}) &{} \text{ for } i=k;\\ \alpha _i K &{} \text{ for } i=k+1,\ldots ,n. \end{array} \right. \end{aligned}$$

When $m_j> g_{j,\max }$, retailers’ allocated quantity would be adjusted as

$$\begin{aligned} g_i=\left\{ \begin{array}{ll} \frac{M-w}{n+1} &{} \text{ for } i=1,\ldots ,j-1;\\ m_j &{} \text{ for } i=j;\\ \alpha _i K &{} \text{ for } i=j+1,\ldots ,n. \end{array} \right. \end{aligned}$$

From Theorem 5.4, we can verify that $q_i^*=g_i$ for $i\ne j$, and the total selling quantity except retailer j is $\sum _{i=1,i\ne j}^n{q_i^*}=K-g_j$. From (ii) of Theorem 5.4, we have

$$\begin{aligned} q_j^*=\left\{ \begin{array}{ll} g_j &{} \text{ if } g_j\le M-K;\\ \frac{M-(K-g_j)}{2} &{} \text{ if } g_j> M-K. \end{array} \right. \end{aligned}$$

with

$$\begin{aligned} \Pi _j(g_j)=\left\{ \begin{array}{ll} (M-K-w)g_j &{} \text{ if } g_j\le M-K;\\ \frac{[M-(K-g_j)]^2}{4}-w\cdot g_j &{} \text{ if } g_j> M-K. \end{array} \right. \end{aligned}$$

From the monotonicity of $\Pi _j$, we have that $\Pi _j^*=\max {0,\Pi _j(g_{j,\max })}$. Whether retailer j would inflate his order quantity depends on the comparison of $\max \{0,\Pi _j(g_{j,\max })\}$ and $(\frac{M-w}{n+1})^2$, which can be simplified to compare $\Pi _j(g_{j,\max })$ and $(\frac{M-w}{n+1})^2$. We have the result of (ii.a) in Corollary 5.5.

Step 2: In the following, we analyze the equilibrium allocated quantities and optimal selling quantities with order inflation, and further investigates the condition where capacity hoarding occurs.

If there exists a j, such that $\Pi (g_{j,\max })>(\frac{M-w}{n+1})^2$, then the total order quantity would be more than capacity. Anticipating the competition in capacity, each retailer would order as more quantity (e.g., K) to obtain higher allocation. Finally, based on fixed allocation, each retailer except retailer 1 receives his guaranteed quantity $\alpha _j K$, and retailer 1 acquires the whole remaining capacity $K-\sum _{j=2}^n{\alpha _j K}$.

From Theorem 5.4, we have that only when the allocation quantities satisfy that there exists an i such that $g_i>[M-(\sum _{j=1,j\ne i}^n {g_j})]/2$, there would be certain retailer sells part of his capacity. Note that under fixed allocation, retailer 1 receives the most capacity, then we have if $g_1>[M-(\sum _{j=2}^n {g_j})]/2$, which is equivalent to $M<(2+\sum _{j=2}^n {\alpha _j})K$, then at least retailer 1 would hoard capacity. Otherwise, each retailer would sell all his available quantity out.

Case 2: $w> M-\frac{n+1}{n}K$, i.e., $\frac{n(M-w)}{n+1}> K$.

In this case, retailers’ total ideal quantity is more than capacity, then they would order as more quantity (e.g., K) as possible to compete for limited quantity. The analysis of the equilibrium allocated quantities and optimal selling quantities are the same as Step 2 of Case 1, and we omit it here.

Combining Step 2 of Case 1 and Case 2, (ii.b) of Corollary 5.5 is obtained.$\Box$