Dynamic mixed-item inventory control with limited capital and short-term financing

Abstract

For most companies, operational decisions and financing decisions jointly affect the earnings and efficiency. Although these effects have been studied in various contexts, there is little literature on the dynamic multi-item inventory control problem, especially the case of multi-item product firms which have limited capital and cannot get easy access to external financing. A serious problem is: What are the optimal procurement strategy and financing strategy for a company with a shortage of funds? In this article, we study a dynamic mixed-item inventory control problem where a capital-constrained firm periodically purchases items from suppliers and assembles some items to meet product orders from customers. In the single period, building the traditional model to obtain three inventory strategies. When the capital is sufficient, all items are purchased. When the capital is insufficient, perishable items are purchased. When the capital is appropriate, both perishable items and part of durable items are purchased. In the multi-period, by constructing new variable value to shape two types of functions and sequences, the procurement strategies of manufacturer at different levels of value are obtained. In the financing decision-making, the manufacturer limits the financing due to the limitation of the financing interest rate, so as to further define the concept of the joint bankruptcy point.

Appendix

The proof of Lemma 1

Finding the first derivatice of formula 1.

$$\begin{aligned} \begin{aligned}&\frac{\partial V}{\partial q_{jg}}\\&\quad = {\left\{ \begin{array}{ll} p_{jg}-w_{g}c_{g}-(1+d)w_{j}c_{g}{} p_{jg}-w_{g}\gamma _{g})F_{jg}(q_{jg}) &{}{\omega >\sum \nolimits _{j=1}^{m}{\sum \nolimits _{g=m+1}^{k}}q_{jg}w_{j}c_{j}}\\ \\ p_{jg}-w_{g}c_{g}-(1+\rho )w_{j}c_{g}{} p_{jg}-w_{g}\gamma _{g})F_{jg}(q_{jg}) &{}{ \omega < \sum \nolimits _{j=1}^{m}{\sum \nolimits _{g=m+1}^{k}}q_{jg}w_{j}c_{j}}, \end{array}\right. } \end{aligned} \end{aligned}$$

(27)

As a total of $c_{m}^{1}c_{k-m}^{1}$ species of products, for any two products $q_{jg}$ and $q_{j_{1}g_{1}}$, there are $\frac{\partial ^{2}V}{\partial q_{jg}\partial q_{j_{1}g_{1}}}=0$. The Hessian matrix is$\begin{Bmatrix} \frac{\partial ^{2} V_{1}}{\partial q^{2}_{11}}&0&\cdots&0 \\ 0&\ddots&\vdots&\vdots \\ \vdots&\vdots&\ddots&0 \\ 0&0&\cdots&\frac{\partial ^{2} V_{1}}{\partial q^{2}_{c^{1}_{m}c^{1}_{k-m}}}\end{Bmatrix}<0$, so there is only $q^{*}$ so that formula(1) is optimal. Let $q_{jg}^{1*}=F_{jg}^{-}(\frac{p_{jg}-w_{g}c_{g}-(1+d)w_{j}c_{j}}{p_{jg}-w_{g}\gamma _{g}}$ and $q_{jg}^{2*}=F_{jg}^{-}(\frac{p_{jg}-w_{g}c_{g}-(1+\rho )w_{j}c_{j}}{p_{jg}-w_{g}\gamma _{g}})$, then we get $q^{*}= {\left\{ \begin{array}{ll} \sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jgn}^{1*}w_{j} &{}\omega \ge \sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jgn}^{1*}w_{j}{} \\ \\ \frac{\omega \sum _{j=1}^{m}w_{j}}{\sum _{j=1}^{m}w_{j}c_{j}} &{}\sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jgn}^{2*}w_{j}\le \omega < \\ &{}\quad \sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jgn}^{1*}w_{j}\\ \\ \sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jg}^{2*}w_{j} &{}\omega >\sum _{j=1}^{m}{\sum _ {g=m+1}^{k}}q_{jgn}^{2*}w_{j}{} \end{array}\right. }$.

The proof of $q^{*}$ is shown below.

Let $V^{'}(\omega ,\mathbf {x})=\frac{\partial V}{\partial q_{11}}+\cdots +\frac{\partial v}{\partial q_{jg}}+ \cdots +\frac{\partial v}{\partial q_{c_{m}^{1}c_{k-m}^{1}}}$, then $V^{'}(\omega ,\mathbf {x})<0$.

• (A) When $\omega <\sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jg}^{2*}w_{j}c_{j}$,$V(\omega ,\mathbf {x})$ strictly monotonically increasing, as long as $q\le q_{jg}^{2*}$, then monotonically. When $q=q_{jg}^{2*}$, as like Fig. 4, the optimal solution is $q_{jg}^{2*}$.

• (B) When $\sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jgn}^{2*}w_{j}\le \omega < \sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jgn}^{1*}w_{j}$, $V(\omega ,\mathbf {x})$ increases monotonically with q, unitl $q=\frac{\omega }{\sum _{j=1}^{m}{\sum _{g=m+1}^{k}}w_{j}c_{j}}$, then monotonically decreasing, from Fig. 5, we can get optimal solution is $q=\frac{\omega }{\sum _{j=1}^{m}{\sum _{g=m+1}^{k}}w_{j}c_{j}}$.

• (C) When $\omega >\sum _{j=1}^{m}{\sum _{g=m+1}^{k}}q_{jgn}^{2*}w_{j}$,$V(\omega ,\mathbf {x})$ increases monotonically, unitl $q=q_{jg}^{1*}$, then monotonically decreasing, as like Fig. 6, the optimal solution is $q_{jg}^{1*}$. $\square$

Fig. 4

The value function graph of small initial capital

Fig. 5

The value function graph of moderate initial capital

Fig. 6

The value function graph of large initial capital

The proof of Lemma 5

Let $Q_{jg}^{1}=q_{jgn}w_{j}$,$Q_{jgn}^{2}=(q_{jgn}-\frac{x_{gn}}{w_{g}})^{+}w_{g}$. According to formula (12) and formula (13) available:

$$\begin{aligned} G^{d}_{n}(Q_{jgn}(q_{jgn}))= & {} (1+d)^{N-n}((p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))E[min(q_{jgn},D_{jgn})] \nonumber \\&-\,d\left( c_{j}Q^{1}_{jgn}+Q^{2}_{jgn}c_{g}+\frac{x_{jgn}}{w_{g}}w_{g}c_{g}\right) \nonumber \\&+\,E\left[ G^{d}_{n+1}\left( A^{d}_{jgn+1}w_{j}+ \left( a^{d}_{jgn+1}-\frac{x_{gn+1}}{w_{g}}\right) ^{+}w_{g}+x_{gn}c_{g}\right. \right] \end{aligned}$$

(28)

$$\begin{aligned} G^{br}{n}(Q_{jgn}(q_{jgn}))= & {} G^{d}_{n}(q_{jgn}) \nonumber \\&-\,(1+d)^{N-n}(\rho (0)-d)(c_{j}Q^{1}_{jgn}+Q^{2}_{jgn}c_{g}+x_{gn}c_{g}) \end{aligned}$$

(29)

$$\begin{aligned} G^{d}_{N}(Q_{jgN}(q_{jgN}))= & {} (p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))E[ming(q_{jgN},D_{jgN})] \nonumber \\&-\,d\left( c_{j}Q^{1}_{jgN}+Q^{2}_{jgN}c_{g}+\frac{x_{jgN}}{w_{g}}w_{g}c_{g}\right) +\,E(G^{d}_{N+1}(A^{d}_{N+1} \nonumber \\= & {} \left. \left. a^{d}_{jgN+1}w_{j}+\left( a^{d}_{jgN+1}-\frac{x_{jgN+1}}{w_{g}}\right) ^{+}w_{g}+\frac{x_{gN}}{w_{g}}w_{g} \right) \right] \end{aligned}$$

(30)

$$\begin{aligned} E\left( G^{d}_{N+1}\left( A^{d}_{N+1}\right. \right.= & {} \left. \left. a^{d}_{jgN+1}w_{j} +\left( a^{d}_{jgN+1}-\frac{x_{jgN+1}}{w_{g}}\right) ^{+}w_{g} +\frac{x_{gN}}{w_{g}}w_{g}\right) \right) \nonumber \\= & {} E\left( \frac{x_{gN+1}}{w_{g}}(w_{g}\gamma _{g}-w_{g}c_{g})-(q_{jgN}-D_{jgN})^{+}w_{j}c_{j}\right) \end{aligned}$$

(31)

$$\begin{aligned} \frac{dG^{d}_{N}}{dq_{jgN}}= & {} (p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))(1-F_{jg}(q_{jgN}))-d\left( c_{j}\frac{dQ^{1}_{jgN}}{dq_{jgN}}+c_{g}\frac{dQ^{2}_{jgN}}{dq_{jgN}}\right) \nonumber \\&+\,F_{jg}(q_{jgN})(w_{g}\gamma _{g}-w_{g}c_{g}-w_{j}c_{j})+ \frac{d\left( \frac{x_{gN}}{w_{g}}-q_{jgN}\right) ^{+}}{dq_{jgN}}(w_{g}\gamma _{g}-w_{g}c_{g})\nonumber \\ \end{aligned}$$

(32)

• 1. When $\frac{x_{gn}}{w_{g}}>q_{jgn}$, for any $j\in {\{1,\ldots ,m\}}$,$g\in \{m+1,\ldots ,k\}$,the result of the derivation of Eq. (32) is $p_{jg}-(1+d)w_{j}c_{j}-w_{g}\gamma _{g}-(p_{jg}-w_{g}\gamma _{g})F_{jg}(q_{jgN})$. Let it be equal to 0, find the euqation $F_{jg}(q_{jgN})=\frac{p_{jg}-(1+d)w_{j}c_{j}-w_{g}\gamma _{g}}{p_{jg}-w_{g}\gamma _{g}}$. The use of mathematical induction assumes that we have obtained the relevant sequence $G_{n+1}^{d},\ldots , G_{N}^{d}$ at the time $n+1$:

  $$\begin{aligned} F^{-}_{jg}\left( \frac{p_{jg}-(1+d)w_{j}c_{j}-w_{g}c_{g}}{p_{jg}-w_{j}c_{j}-w_{g}c_{g}}\right)\ge & {} a^{d}_{jg1}\ge a^{d}_{jg2}\cdots \ge a^{d}_{jgN} \nonumber \\= & {} F^{-}_{jg}\left( \frac{p_{jg}-(1+d)w_{j}c_{j}-w_{g}{\gamma _{g}}}{p_{jg}-w_{g}\gamma _{g}}\right) \end{aligned}$$

  (33)

Then we can find the first derivative and the second derivative of formula (28) respectively:

$$\begin{aligned} \frac{G^{d}_{n}(Q_{jgn}(q_{jgn}))}{dq_{jgn}}= & {} (1+d)^{N-n}((p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))(1-F_{jg}(q_{jgn})-dw_{j}c_{j}) \nonumber \\&+\,E\left[ G^{d}_{n+1}\left( A^{d}_{jgn+1}=a^{d}_{jgn+1}w_{j}+\left( a^{d}_{jgn+1}-\frac{x_{gn+1}}{w_{g}}^{+}w_{g}+x_{gn}c_{g}\right. \right. \right] \end{aligned}$$

(34)

$$\begin{aligned} \frac{dG^{d}_{n}(Q_{jgn}(q_{jgn}))^{2}}{dq^{2}_{jgn}}= & {} -(1+d)^{N-n}((p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))f_{jg}(q_{jgn})\nonumber \\&+\, E\left[ G^{''d}_{n+1}\left( A^{d}_{jgn+1}=a^{d}_{jgn+1}w_{j}+\left( a^{d}_{jgn+1}-\frac{x_{gn+1}}{w_{g}}+x_{gN}c_{g}\right. \right. \right] \nonumber \\ \end{aligned}$$

(35)

Therefore, it can be seen from the hypothesis that (35) is less than 0, so $G_{n}^{d}(Q_{jgn})q_{jgn})$ is a convex function. Assume that $Q_{jgn}=A_{jg(n+1)}^{d}=a_{jg(n+1)}^{d}w_{j}+(a_{jg(n+1)}^{d}-\frac{x_{gn}}{w_{g}})^{+}w_{g}$ is equal to 0 in the second term on the right side of formula (35) and the first item is positive. So $\frac{dG_{n}^{d}}{dq_{jgn}}(A_{jg(n+1)}^{d}(a_{jg(n+1)}^{d}))\ge 0$,and $A_{jgn}^{d}\ge A_{jg(n+1)}^{d}(a_{jg(n+1)}^{d})$. when $q_{jgn}=F_{jg}^{-}(\frac{p_{jg}-(1+d)w_{j}c_{j}-w_{g}c_{g}}{p_{jg}-w_{j}c_{j}-w_{g}c_{g}})$, the first items on the right side of formula (35) is 0, and because $A_{jg(n+1)}^{d}(a_{jg(n+1)}^{d})$ is maximum value, so $a_{jgn}^{d}\le F_{jg}^{-}(\frac{p_{jg}-(1+d)w_{j}c_{j}-w_{g}c_{g}}{p_{jg}-w_{j}c_{j}-w_{g}c_{g}})$, and the second term on the right is always negative. At same time:

$$\begin{aligned}&G^{d}_{n}(A^{d}_{jgn}(a^{d}_{jgn}))+G^{br}_{n}(A^{br}_{jgn}(a^{br}_{jgn}))\nonumber \\&\quad -\,G^{br}_{n}(A^{d}_{jgn}(a^{d}_{jgn})) -G^{d}_{n}(A^{br}_{jgn}(a^{br}_{jgn}))\>0 \end{aligned}$$

(36)

And, $A_{jgn}^{d}(a_{jgn}^{d})\ge A_{jgn}^{br}(a_{jgn}^{br})$, we get $a_{jgn}^{d}\ge a_{jgn}^{br}$. According to (36) when $a_{jg(n+1)}^{d}\le F_{jg}^{-}(\frac{p_{jg}-(1+\rho (0))w_{j}c_{j}}{p_{jg}-w{g}c_{g}-w_{j}c_{j}}$, then $\frac{dG_{n}^{br}}{dq_{jgn}}(A_{jgn}^{d}(a_{jgn}^{d}))\ge 0$, $a_{jg(n+1)}^{d}\le a_{jgn}^{br}$. if $a_{jg(n+1)}^{d}>F_{jg}^{-}(\frac{p_{jg}-(1+\rho (0))w_{j}c_{j}-w_{g}c_{g}}{p_{jg}-w_{j}c_{j}-w_{g}c_{g}})$, then $\frac{dG_{n}^{br}}{dq_{jgn}}(F_{jg}^{-}(\frac{p_{jg}-(1+\rho (0))w_{j}c_{j}-w_{g}c_{g}}{p_{jg}-w_{j}c_{j}-w_{g}c_{g}}))=0$, so$F_{jg}^{-}(\frac{p_{jg}-(1+d)w_{j}c_{j}-w_{g}c_{g}}{p_{jg}-w_{j}c_{j}-w_{g}c_{g}})\ge a_{jg1}^d,\ldots ,\ge a_{jgN}^{d}=F_{jg}^{-}(\frac{p_{jg}-(1+d)w_{j}c_{j}-w_{g}\gamma _{g}}{p_{jg}-w_{g}\gamma _{g}})$.

• 2. When $\frac{x_{gn}}{w_{g}}<q_{jgn}$, for any $j\in \{1,\ldots ,m\}$,$g\in \{m+1,\ldots ,k\}$, we can get the similar equation $\frac{dG_{N}^{d}}{dq_{jgN}}=p_{jg}-(1+d)(w_{j}c_{j}+w_{g}c_{g})-p_{jg}F_{jg}(q_{jgN})+w_{g}\gamma _{g}F_{jg}(q_{jgN})$. Let $\frac{dG_{N}^{d}}{dq_{jgN}}=0$, we get $q_{jgN}=F_{jg}^{-}(\frac{p_{jg}-(1+d)(w_{j}c_{j}+w_{g}c_{g})}{p_{jg}-w_{g}\gamma _{g}})$. The rest proof of the case of $\frac{x_{gn}}{w_{g}}<q_{jgn}$ is similar to the case of $\frac{x_{gn}}{w_{g}}>q_{jgn}$, it it not disscuss here.

$\square$

The proof of Theorem 3

• 1. If $\frac{x_{gN}}{w_{g}}>q_{jgN}$, for any $j\in \{1,\ldots ,m\}$,$g\in \{m+1,\ldots ,k\}$, we can get:

$$\begin{aligned} V_{N+1}(\mathbf {x}_{N+1},\omega _{N+1})= & {} \omega _{N+1}+E\left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}\left( \frac{x_{gN+1}}{w_{g}}(w_{g}\gamma _{g}-w_{g}c_{g}) \right. \right. \nonumber \\&\left. \left. -\,(q_{jgN}-D_{jgN})^{+}w_{j}c_{j}\right) \right) \nonumber \\= & {} H_{N+1}(R_{N+1})+\sum _{j=1}^{m}\sum _{g=m+1}^{k}G^{d}_{N+1}(A^{d}_{N+1}) \end{aligned}$$

(37)

$$\begin{aligned} \pi _{N}(Q_{N},\omega _{N})= & {} \sum _{j=1}^{m}\sum _{g=m+1}^{k} \left( E\left( \frac{x_{gN+1}}{w_{g}}(w_{g}\gamma _{g}-w_{g}c_{g})-(q_{jgN}-D_{jgN})^{+}w_{j}c_{j} \right. \right. \nonumber \\&+\,(p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))Emin(q_{jgN},D_{jgN})\nonumber \\&\left. +\,\varphi \left( \omega _{N}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}p_{jg}q_{jgN}\right) \right) \nonumber \\&+\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g}) \end{aligned}$$

(38)

If $(1+d)(R_{N}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+x_{gN}c_{g})^{+}\>0$ then:

$$\begin{aligned} V_{N+1}(\mathbf {x}_{N+1},\omega _{N+1})= & {} \sum _{j=1}^{m}\sum _{g=m+1}^{k}(G^{br}_{N}(Q_{jgN}(q_{jgN}))+d(w_{j}c_{j}q_{jgN}+x_{gN}c_{g})) \nonumber \\&-\,(1+\rho (0))\left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+\frac{x_{gN}}{w_{g}}w_{g}c_{g})-\omega _{N}\right) \nonumber \\&+\,(\rho (0)-d)\sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g})\nonumber \\&+\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g}) \nonumber \\= & {} (1+\rho (0))R_{N}+\sum _{j=1}^{m}\sum _{g=m+1}^{k}G^{br}_{jgN}(Q_{jgN}(q_{jgN})) \end{aligned}$$

(39)

If$(1+d)(R_{N}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+x_{gN}c_{g})^{+}\>0$ then:

$$\begin{aligned}&V_{N+1}(\mathbf {x_{N+1}},\omega _{N+1})\sum _{j=1}^{m}\sum _{g=m+1}^{k}(G^{d}_{N}(Q_{jgN}(q_{jgN})) +d(w_{j}c_{j}q_{jgN}+x_{gN}c_{g}))\nonumber \\&\qquad +\,(1+d)\left( \omega _{N}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+x_{gN}c_{g}\right) ^{+}\nonumber \\&\qquad +\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}(q_{jgN}w_{j}c_{j}+x_{gN}c_{g}) \nonumber \\&\quad =\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}(G^{d}_{N}(Q_{jgN}(q_{jgN}))+(1+d)R_{N} \end{aligned}$$

(40)

If $(1+\rho )(\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+x_{gN}c_{g}-R_{N})^{+}\>0$, then

$$\begin{aligned} V_{N+1}(\mathbf {x}_{N+1},\omega _{N+1})= & {} \sum _{j=1}^{m}\sum _{g=m+1}^{k}(G^{br}_{N}(Q_{jgN}(q_{jgN}))+d(w_{j}c_{j}q_{jgN}+x_{gN}c_{g})) \nonumber \\&-\,(1+\rho (0))\left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g})-\omega _{N}\right) \nonumber \\&+\,(\rho (0)-d)\sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g})\nonumber \\&+\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g}) \nonumber \\= & {} (1+\rho (0))\omega _{N}+\sum _{j=1}^{m}\sum _{g=m+1}^{k}G^{br}_{jgN}(Q_{jgN}(q_{jgN})) \end{aligned}$$

(41)

So we get the following conclusion:

$$\begin{aligned} \begin{aligned}&\pi _{N}(x_{N},R_{N})\\&\quad = {\left\{ \begin{array}{ll} \sum \nolimits _{j=1}^{m}\sum \nolimits _{g=m+1}^{k}\\ (G^{d}_{N}(Q_{jgN}(q_{jgN}))+(1+d)R_{N}, &{}R_{N} > \sum \nolimits _{j=1}^{m}\sum \nolimits _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+x_{gN}c_{g}\\ \\ \sum \nolimits _{j=1}^{m}\sum \nolimits _{g=m+1}^{k}\\ (G^{d}_{N}(Q_{jgN}(q_{jgN}))\\ +(1+\rho (0))\omega _{N}, &{}R_{N}<\sum \nolimits _{j=1}^{m}\sum \nolimits _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+x_{gN}c_{g} \end{array}\right. } \end{aligned}\nonumber \\ \end{aligned}$$

(42)

Since $\pi _{N}(Q_{N},R_{N})$ is a convex function on $q_{jgN}$. So, when $R_{N}\ge T_N^{+}1_{(a_{jgN}^{d}<\frac{x_{gN}}{w_{g}})}$, we get $Q_{N}^{*}=\sum _{j=1}^{m}{\sum _{g=m+1}^{k}}argmax_{q_{jgN}\ge 0}\pi _{N}(Q_{N},R_{N})=\sum _{j=1}^{m}{\sum _{g=m+1}^{k}}w_{j}a_{jgN}^{d}$, when $T_N^{-}1_{(a_{jgN}^{d}<\frac{x_{gN}}{w_{g}})}< R_{N}\le T_N^{+}1_{(a_{jgN}^{br}<\frac{x_{gN}}{w_{g}})}$, we get $Q_{N}^{*}=\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgN}^{*}$. subject to $R_{N}=\sum _{j=1}^{m}\sum _{g=m+1}^{k}q^{*}_{jgN}w_{j}c_{j}+\frac{x_{gN}}{w_{g}}w_{g}c_{g}$, thus Theorem 2-case1-(1) hold for $n=N$.

• 2. If $\frac{x_{gN}}{w_{g}}<q_{jgN}$ for $\forall g\in \{m+1,\ldots ,k\}, j\in \{1,\ldots ,m\}$, then

  $$\begin{aligned} V_{N+1}(\mathbf {x}_{N+1},\omega _{N+1})= & {} \omega _{N+1}+E\left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}\left( \frac{x_{gN+1}}{w_{g}}(w_{g}\gamma _{g}-w_{g}c_{g}) \right. \right. \nonumber \\&-\,(q_{jgN}-D_{jgN})^{+}w_{j}c_{j}))\nonumber \\= & {} H_{N+1}(R_{N+1})+\sum _{j=1}^{m}\sum _{g=m+1}^{k}G^{d}_{N+1}(A^{d}_{N+1}), \end{aligned}$$

  (43)
  $$\begin{aligned} \pi _{N}(Q_{N},\omega _{N})= & {} \sum _{j=1}^{m}\sum _{g=m+1}^{k} \left( E\left( \frac{x_{gN+1}}{w_{g}}(w_{g}\gamma _{g}-w_{g}c_{g})-(q_{jgN}-D_{jgN})^{+}w_{j}c_{j} \right. \right. \nonumber \\&+\,(p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))Emin(q_{jgN},D_{jgN})\nonumber \\&\left. +\,\varphi \left( \omega _{N}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}p_{jg}q_{jgN}\right) \right) \nonumber \\&+\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+(q_{jgN}-x_{gN}c_{g})^{+}+x_{gN}c_{g}) \end{aligned}$$

  (44)

If $(1+d)(R_{N}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+x_{gN}c_{g})^{+}\>0$, then:

$$\begin{aligned} V_{N+1}(\mathbf {x}_{N+1},R_{N+1})= & {} \sum _{j=1}^{m}\sum _{g=m+1}^{k}(G^{d}_{N}(Q_{jgN}(q_{jgN})) \nonumber \\&\left. +\,d\left( w_{j}c_{j}q_{jgN}+\left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gN}c_{g}\right) \right) \nonumber \\&+\,(1+d)\left( R_{N}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+\left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g}+\frac{x_{gN}}{w_{g}}\right) ^{+} \nonumber \\&+\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}\left( q_{jgN}w_{j}c_{j}+\left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gN}c_{g}\right) \nonumber \\= & {} \sum _{j=1}^{m}\sum _{g=m+1}^{k}(G^{d}_{N}(Q_{N}(q_{jgN}))+(1+d)R_{N} \end{aligned}$$

(45)

If $(1+\rho )(\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgN}w_{j}c_{j}+x_{gN}c_{g}-R_{N})^{+}\ge 0$, then

$$\begin{aligned} V_{N+1}(\mathbf {x_{N+1}},R_{N+1})= & {} \sum _{j=1}^{m}\sum _{g=m+1}^{k}(G^{d}_{N}(Q_{N}(q_{jgN})) \nonumber \\&\left. +\,d\left( w_{j}c_{j}q_{jgN}+\left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gN}c_{g}\right) \right) \nonumber \\&-\,(1+\rho (0))\left( \sum _{j=1}^{m}\sum _{g=m+1}^{k} \left( w_{j}c_{j}q_{jgN}\right. \right. \nonumber \\&\left. \left. +\,\left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+} w_{g}c_{g}+x_{gN}c_{g}\right) -R_{N}\right) \nonumber \\&+\,(\rho (0)-d)\sum _{j=1}^{m}\sum _{g=m+1}^{k}\left( w_{j}c_{j}q_{jgN} +\left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gN}c_{g}\right) \nonumber \\&+\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}\left( w_{j}c_{j}q_{jgN}+ \left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gn} c_{g}\right) \nonumber \\= & {} \sum _{j=1}^{m}\sum _{g=m+1}^{k}(G^{d}_{N}(Q_{jgN}(q_{jgN}))+(1+\rho (0))R_{N} \end{aligned}$$

(46)

When $R_{N}\ge T^{+}_{N}$, we have $Q^{*}_{N}=\sum _{j=1}^{m}\sum _{g=m+1}^{k}arg max_{q_{jgN}\ge 0}\pi _{N}(Q_{N},\omega _{N})=T^{+}_{N}$. When $T^{-}_{N}\le R_{N}\le T^{+}_{N}$, we have $Q^{*}_{N}=\sum _{j=1}^{m}\sum _{g=m+1}^{k}q^{*}_{jgN}$ subject to

$$\begin{aligned} R_{N}=\sum _{j=1}^{m}\sum _{g=m+1}^{k}q^{*}_{jgN}w_{j}c_{j} \left( q^{*}_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gN}c_{g} \end{aligned}$$

(47)

Also, when $R_{N}\ge \sum _{j=1}^{m}\sum _{g=m+1}^{k}x_{gN}c_{g}$, we have

$$\begin{aligned}&\widetilde{\pi _{N}}(x_{N},R_{N})\nonumber \\&\quad = {\left\{ \begin{array}{ll} (1+d)R_{N}+\sum \nolimits _{j=1}^{m}\sum \nolimits _{g=m+1}^{k}G^{d}_{N}\left( a^{d}_{jgN}<\frac{x_{gN}}{w_{g}}\right) , &{}R_{N}\ge T^{+}_{N}1_{\left( a^{d}_{jgN}<\frac{x_{gN}}{w_{g}}\right) }\\ (1+\rho (0))R_{N}+\sum \nolimits _{j=1}^{m}\sum \nolimits _{g=m+1}^{k}G^{br}_{N} \left( a^{br}_{jgN}<\frac{x_{gN}}{w_{g}}\right) , &{}T^{-}_{N}1_{ \left( a^{br}_{jgN}<\frac{x_{gN}}{w_{g}}\right) }\le R_{N}\\ &{} \le T^{+}_{N}1_{\left( a^{d}_{jgN}<\frac{x_{gN}}{w_{g}}\right) }\\ (1+d)R_{N}+\sum \nolimits _{j=1}^{m}\sum \nolimits _{g=m+1}^{k}G^{d}_{N} \left( a^{d}_{jgN}>\frac{x_{gN}}{w_{g}}\right) , &{} R_{N}\ge T^{+}_{N}1_{\left( a^{d}_{jgN}<\frac{x_{gN}}{w_{g}}\right) }\\ (1+\rho (0)))R_{N}+\sum \nolimits _{j=1}^{m}\sum \nolimits _{g=m+1}^{k}G^{br}_{N} \left( a^{br}_{jgN}> \frac{x_{gN}}{w_{g}}\right) , &{}T^{-}_{N}1_{\left( a^{br}_{jgN}> \frac{x_{gN}}{w_{g}}\right) } \le R_{N}\\ &{}\le T^{+}_{N}1_{\left( a^{d}_{jgN} >\frac{x_{gN}}{w_{g}}\right) } \end{array}\right. }\nonumber \\ \end{aligned}$$

(48)

If $\frac{x_{gn}}{w_{g}}\>q_{jgn}$, for all $g\in \{m+1,\ldots ,k\}, j\in \{1,\ldots ,m\}$, first we prove Theorem 2-case1-(2) holds for n, suppose $R_{n}\ge T^{+}_{n}1_{(q_{jgn}<\frac{x_{gn}}{w_{g}})}$ and $R_{n}\ge \sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}q^{d}_{jgn}+x_{gn}c_{g}$. Then, $R_{n}\ge T^{-}_{n}1_{(a^{br}_{jgn}<\frac{x_{gn}}{w_{g}})}$.

By inductive assumption on Theorem 2-case1-(2) for $n+1$, we have

$$\begin{aligned} \pi _{n}(Q_{n}(q_{jgn}),R_{n})= & {} E\left( V_{n+1}\left( \mathbf {x_{n+1}},\sum _{j=1}^{m}\sum _{g=m+1}^{k}(p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))min(q_{jgn},D_{jgn}) \right. \right. \nonumber \\&+\,\phi \left( R_{n}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j}+x_{gn}c_{g}\right) \nonumber \\&\left. \left. +\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j}+x_{gn}c_{g}\right) \right) \nonumber \\= & {} E\left( V_{n+1}\left( \mathbf {x_{n+1}},\sum _{j=1}^{m}\sum _{g=m+1}^{k}(p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))min(q_{jgn},D_{jgn}) \right. \right. \nonumber \\&+\,(1+d)\left( R_{n}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j}+\frac{x_{gn}}{w_{g}}\right) \nonumber \\&\left. \left. +\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j}+\frac{x_{gn}}{w_{g}}\right) \right) \nonumber \\= & {} (1+d)^{N-n+1}R_{n}+\sum _{j=1}^{m}\sum _{g=m+1}^{k}G^{d}_{n}(Q_{N}(q_{jgn})) \end{aligned}$$

(49)

When $R_{n}\ge T^{+}_{n}1_{(a^{br}_{jgn}<\frac{x_{gn}}{w_{g}})}$, since $a^{d}_{jgn}\ge a^{d}_{jgn}$, we know $\pi _{n}(Q_{n}(q_{jgn})),\omega _{n})$ is increasing in $q_{jgn} j\in [1,m] g\in [m+1,k]$, when $q_{jgn}\le a^{d}_{jgn}$, and it decreasing in $q_{jgn} j\in \{1,\ldots ,m\}, g\in \{m+1,\ldots ,k\}$, when $T^{-}_{N}1_{(a^{d}_{jgn}<\frac{x_{gn}}{w_{g}})}\le \sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j}+x_{gn}c_{g}+x_{gn}c_{g}\le R_{n}$. Now that $\pi _{n}(Q_{n}(q_{jgn})),\omega _{n})$ is concave in $q_{jgn}$, thus, $Q^{*}_{n}(q^{*}_{jgn})=\sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}a^{d}_{jgn}+x_{gn}w_{g}$ when $R_{n}\ge T^{+}_{n}1_{(a^{d}_{jgn}<\frac{x_{gn}}{w_{g}})}$.

Now suppose $T^{+}_{n}1_{(a^{d}_{jgn}<\frac{x_{gn}}{w_{g}})}\ge R_{n}\ge T^{+}_{n}1_{(a^{br}_{jgn}<\frac{x_{gn}}{w_{g}})}$ and we shall prove that

$$\begin{aligned} Q^{*}_{N}=\left( Q^{*}_{n}(q^{*}_{jgn})|\sum _{j=1}^{m}\sum _{g=m+1}^{k}q^{*}_{jgn}w_{j}c_{j}+x_{gn}c_{g}=R_{n}\right) . \end{aligned}$$

First we prove that $\pi _{n}(Q_{n}(q_{jgn})),\omega _{n})$ is increasing in $q_{jgn}$ on $R_{n}\ge \sum _{j=1}^{m} \sum _{g=m+1}^{k}w_{j}c_{j}q^{d}_{jgn}+x_{gn}c_{g}$.

Now suppose $T^{-}_{n}1_{(a^{br}_{jgn}<\frac{x_{gn}}{w_{g}})}\ge R_{n}\ge T^{+}_{n}1_{(a^{d}_{jgn}<\frac{x_{gn}}{w_{g}})}$, and we prove that the result is true in this case.

When $R_{n}\ge \sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}q_{jgn}+x_{gn}c_{g}$, by the inductive assumption onTheorem 2-case1-(2) for $n+1$, we have

$$\begin{aligned} \begin{aligned} \pi _{n}(Q_{n}(q_{jgn})),R_{n})&=\sum _{j=1}^{m}\sum _{g=m+1}^{k}G^{d}_{n+1}(a^{d}_{jgn}) \\&\quad +\,E\left( H_{n+1}\left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}(p_{jg}-w_{j}c_{j}-w_{g}c_{g})min(q_{jgn},D_{jgn}) \right. \right. \\&\quad +\,(1+d)R_{n}-d(w_{j}c_{j}q_{jgn}x_{gn}c_{g})) \end{aligned} \end{aligned}$$

(50)

Taking the partial derivative on $q_{jgn}$ and letting $R_{n}=\sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}q^{1}_{jgn}+x_{gn}c_{g}$, to simplify the notation, let $\omega _{jgn}=w_{j}c_{j}q^{1}_{jgn}+x_{gn}c_{g}$, so we obtain

$$\begin{aligned} \frac{\partial \pi _{n}}{\partial q_{jgn}}= & {} E(((p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))1_{(\omega _{n}\le \sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}D_{jgn})}-dw_{j}c_{j})\nonumber \\&\times \, H^{'}_{n+1}\big ((p_{jg}-(w_{j}c_{j}+w_{g}c_{g})min(q_{jgn},D_{jgn})+w_{j}c_{j}q^{1}_{jgn}+x_{gn}c_{g}\big ) \nonumber \\= & {} (p_{jg}-(1+d)w_{j}c_{j}-w_{g}c_{g})\bar{F_{jg}}(q^{1}_{jgn})H^{'}_{n+1}\big (p_{jg}q^{1}_{jgn}) \nonumber \\&-\,dw_{j}c_{j}E(1_{(\omega _{n}\ge \sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}D_{jgn})}H^{'}_{n+1}((p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))D_{jgn} \nonumber \\&+\,w_{j}c_{j}q^{1}_{jgn}+x_{gn}c_{g})\big ) \end{aligned}$$

(51)

$$\begin{aligned} (51)\ge & {} (p_{jg}-(1+d)w_{j}c_{j}-w_{g}c_{g})\bar{F_{jg}}(q^{1}_{jgn})H^{'}_{n+1}(p_{jg}q^{1}_{jgn}) \nonumber \\&-\,dw_{j}c_{j}F_{jg}(q^{1}_{jgn}) H^{'}_{n+1}(w_{j}c_{j}q^{1}_{jgn}+x_{gn}c_{g}) \nonumber \\\ge & {} ((1+d)(p_{jg}-w_{j}c_{j}-w_{g}c_{g})\bar{F_{jg}}(q^{1}_{jgn})H^{'}_{n+2}(p_{jg}q^{1}_{jgn}) \nonumber \\&-\,dw_{j}c_{j}H^{'}_{n+1}(w_{j}c_{j}q^{1}_{jgn}+x_{gn}q^{1}_{jgn}))F_{jg}(q^{1}_{jgn}) \end{aligned}$$

(52)

Therefore, when $T^{-}_{n}1_{(a^{br}_{jgn}<\frac{x_{gn}}{w_{g}})}\ge R_{n}\ge T^{+}_{n}1_{(a^{d}_{jgn}<\frac{x_{gn}}{w_{g}})}$ later,the article does not describe the proof process, and as was done there, we can prove $\frac{\partial \pi _{n}(q^{1}_{jgn})}{\partial q_{jgn}}\ge 0, j\in \{1,\ldots ,m\}$, $g\in \{m+1,\ldots ,k \}$.

Here, $\frac{\partial \pi ^{2}_{n}}{\partial q_{jgn}\partial q_{\mu \nu n}}=0 j,\mu \in \{1,\ldots ,m\}, g,\nu \in \{m+1,\ldots ,k\}$ and $j\ne \mu , g\ne \nu$, hessian matrix $H_{c^{1}_{m}c^{1}_{k-m}c^{1}_{m}c^{1}_{k-m}}<0$, $\pi _{n}(Q_{n}(q_{jgn})),\omega _{n})$ is incresing in $q_{jgn}$ on $\omega _{n}\ge \sum _{j=1}^{m}\sum _{g=m+1}^{k} w_{j}c_{j}q^{d}_{jgn}+x_{gn}c_{g}$.

We next prove $\pi _{n}(\sum _{j=1}^{m}\sum _{g=m+1}^{k}q^{1}_{jgn},\omega _{n})=max \pi _{n}(Q_{n}(q_{jgn})),\omega _{n})$ when $T^{-}_{n}1_{(a^{br}_{jgn}<\frac{x_{gn}}{w_{g}})}\ge R_{n}\ge T^{+}_{n}1_{(a^{d}_{jgn}<\frac{x_{gn}}{w_{g}})}$.

If $\frac{x_{gn}}{w_{g}}<q_{jgn}$ for $\forall g\in [m+1,k] j\in \{1,\ldots ,m\}$; we also prove Theorem2-case2-(2) holds for n. Suppose $R_{n}\ge T^{+}_{n}1_{(a^{d}_{jgn}\>\frac{x_{gn}}{w_{g}})}$ and $R_{n}\ge \sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}q^{d}_{jgn}+x_{gn}c_{g}$. By the $R_{n}\ge T^{-}_{n}1_{(a^{d}_{jgn}\>\frac{x_{gn}}{w_{g}})}$ inductive assumption on Theorem  2-case2-(2)for $n+1$, we have

$$\begin{aligned} \begin{aligned} \pi _{n}(Q_{n}(q_{jgn})),R_{n})&=E\left( V_{n+1}\left( \left( x_{n+1},\sum _{j=1}^{m}\sum _{g=m+1}^{k}(p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))min(q_{jgn},D_{jgn})\right. \right. \right. \\&\quad +\,\phi \left( R_{n}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j}+ \left( q_{jgn}-\frac{x_{gn}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gn}c_{g}\right) \\&\quad \left. \left. +\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j}+ \left( q_{jgn}-\frac{x_{gn}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gn}c_{g}\right) \right) \\&=\,E\left( V_{n+1}\left( \mathbf {x}_{n+1},\sum _{j=1}^{m}\sum _{g=m+1}^{k}(p_{jg}-(w_{j}c_{j}+w_{g}c_{g}))min(q_{jgn},D_{jgn}) \right. \right. \\&\quad +\,(1+d)\left( R_{n}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j}+\left( q_{jgn}-\frac{x_{gn}}{w_{g}}w_{g}c_{g}\right) \right. \\&\quad \left. \left. +\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}q_{jgn}w_{j}c_{j} \left( q_{jgn}-\frac{x_{gn}}{w_{g}}\right) w_{g}c_{g}+\frac{x_{gn}}{w_{g}}w_{g}c_{g}\right) \right) \\&=\,(1+d)^{N-n+1}R_{n}+\sum _{j=1}^{m}\sum _{g=m+1}^{k}G^{d}_{n}(Q_{n}(q_{jgn})) \end{aligned} \end{aligned}$$

(53)

The rest proof of condition about $\frac{x_{gn}}{w_{g}}<q_{jgn}$, for all $g \in \{m+1,\ldots ,k\}$, $j\in \{1,\ldots ,m\}$ is similar to the previous proofs, it is not explained here. $\square$

The proof of Proposition 3

We first prove $Q^{*}_{N}(q^{*}_{jgn})=arg max \pi _{N}(Q^{*}_{N}(q_{jgN}),\omega _{N})$ when $\bar{K}<\omega _{N}<T^{-}_{n}1_{(a^{br}_{jgn}\>\frac{x_{gn}}{w_{g}})}$.

Now suppose $\sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}q_{jgN}+(q_{jgN}-\frac{x_{gN}}{w_{g}})^{+}w_{g}c_{g}+x_{gN}c_{g}$.

Since

$$\begin{aligned} V_{N+1}(\mathbf {x}_{N+1},R_{N+1})= & {} \omega _{N+1}+E \left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}\left( \frac{x_{gN+1}}{w_{g}}(w_{g}\gamma _{g}-w_{g}c_{g})\right. \right. \nonumber \\&\left. \left. -\,(q_{jgN}-D_{jgN})^{+}w_{j}c_{j}\right) \right) \end{aligned}$$

(54)

$$\begin{aligned} \pi _{N}(Q_{N},R_{N})= & {} E\left( V_{N+1}\left( \mathbf {x}_{N+1},\sum _{j=1}^{m}\sum _{g=m+1}^{k}((p_{jg}-w_{j}c_{j}-w_{g}c_{g})min\{q_{jgN},D_{jgN}\}) \right. \right. \nonumber \\&+\,\phi \left( R_{N}-\sum _{j=1}^{m}\sum _{g=m+1}^{k}\left( w_{j}c_{j}q_{jgN}+ \left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g}+x_{gN}c_{g} \right) \right) \nonumber \\&\left. +\,\sum _{j=1}^{m}\sum _{g=m+1}^{k}\left( w_{j}c_{j}q_{jgN}+ \left( q_{jgN}-\frac{x_{gN}}{w_{g}}\right) ^{+}w_{g}c_{g} +x_{gN}c_{g}\right) \right) \end{aligned}$$

(55)

If $\frac{x_{gN}}{w_{g}}\>q_{jgN}$ for $\forall g\in \{m+1,\ldots ,k\}, j\in \{1,\ldots ,m\}$, taking the partial derivative of $\pi _{N}(Q_{N},\omega _{N})$ on $q_{jgN}$, we obtain

$$\begin{aligned} \begin{aligned} \frac{\partial \pi _{N}(Q_{N},R_{N})}{\partial q_{jgN}}&=(p_{jg}-w_{j}c_{j}-w_{g}c_{g})(1-F_{jg}(q_{jgN})) \\&\quad - \,\rho \left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g})-R_{N}\right) w_{j}c_{j} \\&\quad + \,(w_{g}\gamma _{g}-w_{g}c_{g}-w_{j}c_{j})F_{jg}(q_{jgN})-w_{g}\gamma _{g}+w_{g}c_{g} \end{aligned} \end{aligned}$$

(56)

$$\begin{aligned} \begin{aligned} \frac{\partial \pi _N(Q_N,R_N)}{\partial q_{jgN}}&=(p_{jg}-w_jc_j-w_g\gamma _g)-(p_{jg}-w_gc_g)(F_{jg}(q_{jgN}))\\&\quad -\,\rho \left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_jc_jq_{jgN}+x_{gN}c_g)-R_N\right) w_jc_j \end{aligned} \end{aligned}$$

(57)

Since $\pi _{N}(Q_{N},R_{N})$ is concave in $q_{jgN}$, it is increasing in $q_{jgN}$ when $\frac{\partial \pi _{N}}{\partial q_{jgN}}\ge 0$, or equivalently, when $F_{jg}(q_{jgN})\le \frac{p_{jg}-w_{g}\gamma _{g}-w{j}c_{j}(1+\rho (\sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g})-R_{N})}{p_{jg}-w_{g}\gamma _{g}}$ since

$$\begin{aligned} \begin{aligned} Q^{*}_{N}(R_{N})&=\sum _{j=1}^{m}\sum _{g=m+1}^{k}sup\left( \omega ^{+}_{N}\le \sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}q_{jgN}+x_{jgN}c_{g}|F_{jg}(q_{jgN}) \right. \\&\left. \le \,\frac{p_{jg}-w_{g}\gamma _{g}-w{j}c_{j}(1+\rho \Big (\sum _{j=1}^{m} \sum _{g=m+1}^{k}(w_{j}c_{j}q_{jgN}+x_{gN}c_{g})-R_{N}\Big )}{p_{jg}-w_{g}\gamma _{g}}\right) \end{aligned}\nonumber \\ \end{aligned}$$

(58)

If $\frac{x_{gN}}{w_{g}}\le q_{jgN}$ for $\forall g\in \{m+1,\ldots ,k\}, j\in \{1,\ldots ,m\}$,

$$\begin{aligned} \begin{aligned} \frac{\partial \pi _{N}(Q_{N},R_{N})}{\partial q_{jgN}}&=(p_{jg}-w_{j}c_{j}-w_{g}c_{g})(1-F_{jg}(q_{jgN})) \\&\quad -\,\rho \left( \sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}+w_{g}c_{g})q_{jgN}-R_{N}\right) w_{j}c_{j} \\&\quad +\,(w_{g}\gamma _{g}-w_{g}c_{g}-w_{j}c_{j})F_{jg}(q_{jgN}) \end{aligned} \end{aligned}$$

(59)

$$\begin{aligned} \begin{aligned} \frac{\partial \pi _N(Q_{N},R_N)}{\partial q_{jgN}}&=(p_{jg}-w_jc_j-w_gc_g)-(p_{jg}-w_g\gamma _g)F_{jg}(q_{jgN})\\&\quad -\,\rho \left( \sum _{j=1}^{m+1}\sum _{g=m+1}^{k}(w_jc_j+w_gc_g)q_{jgN}-R_N\right) w_jc_j \end{aligned} \end{aligned}$$

(60)

Since

$$\begin{aligned} \begin{aligned} Q^{*}_{N}(R_{N})&=\sum _{j=1}^{m}\sum _{g=m+1}^{k}sup\left\{ R^{+}_{N}\le \sum _{j=1}^{m}\sum _{g=m+1}^{k}w_{j}c_{j}q_{jgN}+x_{jgN}c_{g}|F_{jg}(q_{jgN}) \right. \\&\left. \le \,\frac{p_{jg}-w_{g}c_{g}-w{j}c_{j} \Big (1+\rho \Big (R^{+}_{N}\le \sum _{j=1}^{m}\sum _{g=m+1}^{k}(w_{j}c_{j}+w_{g}c_{g})q_{jgN}\Big )-R_{N}\Big )}{p_{jg}-w_{g}\gamma _{g}}\right\} . \end{aligned}\nonumber \\ \end{aligned}$$

(61)

$\square$