According to the assumptions (i) and (ii), the supplier’s expected net revenue with default risk is \(WD(M)\left( {1  F(M)} \right) = WKe^{{\left( {a  b} \right)M}}\). Additionally, he or she will burden an additional capital opportunity cost, i.e., \(CKe^{aM} I_{s} M\), for offering trade credit period \(M\) to the retailer. Meanwhile, for the retailer, he or she can save an additional capital opportunity cost \(WKe^{aM} I_{r} M\). Therefore, the retailer’s and the supplier’s total annual profits can be expressed as
$$\Pi_{r}^{2} (Q) = \Pi_{r}^{3} (Q) = \left( {P  W} \right)Ke^{aM}  S_{r} Ke^{aM} /Q  Qh_{r} /2 + WKe^{aM} I_{r} M,$$
(10)
$$\Pi_{s}^{2} (M) = \Pi_{s}^{3} (M) = WKe^{{\left( {a  b} \right)M}}  CKe^{aM}  \frac{{Ke^{aM} S_{s} }}{Q}  \frac{{Ke^{aM} h_{s} Q}}{2A}  CKe^{aM} I_{s} M,$$
(11)
respectively.
Two parties’ decision making in Nash game
In this subsection, we assume that the supplier and the retailer have the same bargaining power, i.e., neither side has the monopoly strength. Under this background, the optimal equilibrium solution is Nash equilibrium.
From the view of Nash game, the first derivative condition of \(\Pi_{r}^{2} (Q)\) with respect to \(Q\) and the first derivative condition of \(\Pi_{s}^{2} (M)\) with respect to \(M\) should be established simultaneously. Therefore, the first derivative \({{d\Pi_{r}^{2} (Q)} \mathord{\left/ {\vphantom {{d\Pi_{r}^{2} (Q)} {dQ}}} \right. \kern0pt} {dQ}}\) and the first derivative \(d\Pi_{s}^{2} (M)/dM\) will be given by
$$d\Pi_{r}^{2} (Q)/dQ = S_{r} Ke^{aM} /Q^{2}  h_{r} /2,$$
(12)
$$\frac{{d\Pi_{s}^{2} (M)}}{dM} = \left( {a  b} \right)WKe^{{\left( {a  b} \right)M}}  aCKe^{aM}  \frac{{aKe^{aM} S_{s} }}{Q}  \frac{{aKe^{aM} h_{s} Q}}{2A}  CKe^{aM} I_{s}  aCKe^{aM} I_{s} M.$$
(13)
First, by the first derivative condition \(d\Pi_{r}^{2} (Q)/dQ = 0\), the optimal ordering lot size in Nash game is given by
$$Q^{2*} = \sqrt {2S_{r} Ke^{aM} /h_{r} } .$$
(14)
Next, substituting \(Q^{2*} = \sqrt {2S_{r} Ke^{aM} /h_{r} }\) into Eq. (13), the \(d\Pi_{s}^{2} (M)/dM\) may be reduced to
$$\frac{{d\Pi_{s}^{2} (M)}}{dM} = \left( {a  b} \right)WKe^{{\left( {a  b} \right)M}}  aCKe^{aM}  aS_{s} \sqrt {\frac{{h_{r} Ke^{aM} }}{{2S_{r} }}}  \frac{{aKe^{aM} h_{s} }}{A}\sqrt {\frac{{S_{r} Ke^{aM} }}{{2h_{r} }}}  CKe^{aM} I_{s}  aCKe^{aM} I_{s} M.$$
(15)
It includes a single decision variable \(M\).
Theorem 1
The supplier’s optimal trade credit period is zero (i.e.,
\(M^{2*} = 0\)
) if (i)
\(a \le b\)
, or (ii)
\((a  b)W \le aC\)
, or (iii)
\((a  b)W \le aC + CI_{s}\).
Proof
From Eq. (15), if \(a \le b\), \(\frac{{d\Pi_{s}^{2} (M)}}{dM} < 0\). Consequently, the optimal trade credit period is zero, i.e., \(M^{2*} = 0\). Similarly, if \((a  b)W \le aC\), or \((a  b)W \le aC + CI_{s}\), we have the same results \(\frac{{d\Pi_{s}^{2} (M)}}{dM} < 0\), and \(M^{2*} = 0\). This completes the proof.
Consequently, the retailer’s and the supplier’s total annual profits are given by
$$\Pi_{r}^{2} (\text{M}^{{\text{2*}}} = 0) = \left( {P  W} \right)K  \sqrt {2KS_{r} h_{r} } = \Pi_{r}^{0} ,$$
(16)
$$\Pi_{s}^{2} (\text{M}^{{\text{2*}}} = 0) = \left( {W  C} \right)K  S_{s} \sqrt {{{Kh_{r} } \mathord{\left/ {\vphantom {{Kh_{r} } {2S_{r} }}} \right. \kern0pt} {2S_{r} }}}  {{Kh_{s} \sqrt {{{KS_{r} } \mathord{\left/ {\vphantom {{KS_{r} } {2h_{r} }}} \right. \kern0pt} {2h_{r} }}} } \mathord{\left/ {\vphantom {{Kh_{s} \sqrt {{{KS_{r} } \mathord{\left/ {\vphantom {{KS_{r} } {2h_{r} }}} \right. \kern0pt} {2h_{r} }}} } A}} \right. \kern0pt} A} = \Pi_{s}^{0} .$$
(17)
That is to say, the two parties don’t achieve any coordination or improvement in Theorem 1.
Next, we discuss the another condition, i.e., \((a  b)W > aC + CI_{s}\). By the first derivative condition \({{d\Pi_{s}^{2} (M)} \mathord{\left/ {\vphantom {{d\Pi_{s}^{2} (M)} {dM = 0}}} \right. \kern0pt} {dM = 0}}\), we can obtain
$$\left( {a  b} \right)We^{  bM}  aC  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} Ke^{aM} }}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} Ke^{aM} }}{{2h_{r} }}}  CI_{s}  aCI_{s} M = 0.$$
(18)
From Eq. (18), the optimal trade credit period function is given by
$$M^{{2\bar{*}}} = {{\left\{ {\left( {a  b} \right)We^{{  bM^{{2\bar{*}}} }}  aC  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} Ke^{{aM^{{2\bar{*}}} }} }}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} Ke^{{aM^{{2\bar{*}}} }} }}{{2h_{r} }}}  CI_{s} } \right\}} \mathord{\left/ {\vphantom {{\left\{ {\left( {a  b} \right)We^{{  bM^{{2\bar{*}}} }}  aC  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} Ke^{{aM^{{2\bar{*}}} }} }}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} Ke^{{aM^{{2\bar{*}}} }} }}{{2h_{r} }}}  CI_{s} } \right\}} {aCI_{s} }}} \right. \kern0pt} {aCI_{s} }},$$
(19)
$${\text{when}}\;\left( {a  b} \right)We^{{  bM^{{2\bar{*}}} }}  aC  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} Ke^{{aM^{{2\bar{*}}} }} }}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} Ke^{{aM^{{2\bar{*}}} }} }}{{2h_{r} }}}  CI_{s} > 0.$$
Note that the left and right sides of Eq. (19) are functions of \(M\). Due to the complexity of the problem, it seems difficult to derive a closedform expression. Additionally, in Eq. (19), we obviously observe that the left side increases and the right side decreases as \(M\) increases. There is only one intersection point when the two sides of Eq. (19) intersect, i.e., unique optimal positive solution \(M^{{2\bar{*}}}\).
Theorem 2
When
\(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\)
, (i) if
\(M^{{2\bar{*}}} < M_{\rm max}\)
, the final optimal trade credit period is
\(M^{2*} = M^{{2\bar{*}}}\)
; (ii) if
\(M^{{2\bar{*}}} \ge M_{\rm max}\)
, the final optimal trade credit period is
\(M^{2*} = M_{\rm max}\).
Proof
Firstly, according to \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\), we can obtain that
$$\frac{{d\Pi_{s}^{2} (M)}}{dM}\left {_{M = 0}^{{}} } \right. = (a  b)WK  aCK  aS_{s} \sqrt {\frac{{h_{r} K}}{{2S_{r} }}}  \frac{{aKh_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  CKI_{s} > 0,$$
(20)
$$\frac{{d\Pi_{s}^{2} (M)}}{dM}\left {_{M \to \infty }^{{}} } \right. =  \infty .$$
(21)
Additionally, applying the second derivative of \(\Pi_{s}^{2} (M)\) with respect to \(M\), we have
$$\frac{{d^{2} \Pi_{s}^{2} (M)}}{{dM^{2} }} = \left[ {(a  b)^{2} We^{  bM}  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} Ke^{aM} }}{{2h_{r} }}}  2aCI_{s} } \right]Ke^{aM}  a^{2} S_{s} \sqrt {\frac{{h_{r} Ke^{aM} }}{{8S_{r} }}}  a^{2} CKe^{aM} I_{s} M.$$
(22)
Next, we have two alternative cases: (i) \((a  b)^{2} W  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  2aCI_{s} \le 0\) and (ii) \((a  b)^{2} W  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  2aCI_{s} > 0\).
Case 1
\((a  b)^{2} W  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  2aCI_{s} \le 0\).
In this case, we know that \((a  b)^{2} We^{  bM}  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} Ke^{aM} }}{{2h_{r} }}}  2aCI_{s} < 0\), further, \(\frac{{d^{2} \Pi_{s}^{2} (M)}}{{dM^{2} }} < 0\). Therefore, \(\Pi_{s}^{2} (M)\) is a strictly concave function in \(\left[ {0,\infty } \right)\). Therefore, combining with Eq. (20) and Eq. (21), we know that there exists a unique positive optimal solution such that \(\frac{{d\Pi_{s}^{2} (M)}}{dM} = 0\), denoted as \(M^{{2\bar{*}}}\).
Case 2
\((a  b)^{2} W  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  2aCI_{s} > 0\)
In this case, we know that the value of \(\frac{{d^{2} \Pi_{s}^{2} (M)}}{{dM^{2} }}\) moves from positive to negative as \(M\) increases, that is to say, \(\Pi_{s}^{2} (M)\) is a convexconcave function of \(M\). Therefore, combining with Eq. (20) and Eq. (21), we know that \(\Pi_{s}^{2} (M)\) is a unimodal function in \(\left[ {0,\infty } \right)\). There also exists a unique positive optimal solution such that \(\frac{{d\Pi_{s}^{2} (M)}}{dM} = 0\), denoted as \(M^{{2\bar{*}}}\).
In a word, if \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\), the solution of Eq. (19) is a unique optimal positive solution \(M^{{2\bar{*}}}\) for \(\Pi_{s}^{2} (M)\). Then, we consider the upper bound of \(M\), i.e., \(M_{\rm max}\). If \(M^{{2\bar{*}}} < M_{\rm max}\), the final optimal trade credit period is \(M^{2*} = M^{{2\bar{*}}}\). If \(M^{{2\bar{*}}} \ge M_{\rm max}\), the final optimal trade credit period is \(M^{2*} = M_{\rm max}\). This completes the proof.
From Eq. (19) and Theorem 2, we obtain the following results.
Corollary 1
(i) A higher value of
\(a\), \(W\), \(A\)
and a lower value of
\(b\), \(C\), \(S_{s}\), \(h_{s}\), \(I_{s}\)
cause a higher value of
\(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}\), and
\(M^{{2\bar{*}}}\).
(ii) The change of
\(P\)
and
\(I_{r}\), i.e., the retailer’s profit parameters, do not affect the supplier as to whether to offer trade credit to the retailer.
Proof
The above is apparent from \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\), Eq. (19) and Theorem 2.
A simple economic interpretation is as follows. A higher value of \(a\) (i.e., increasing demand coefficient) leads to a higher demand, and higher values of \(W\) and \(A\) lead to higher revenue. Hence, the supplier is willing to offer a longer trade credit period. On the other hand, lower values of \(b\) (i.e., default risk coefficient) and \(C\) lead to a higher expected revenue for supplier, and lower values of \(S_{s}\), \(h_{s}\), and \(I_{s}\) lead to a lower ordering and inventory cost. Hence, the supplier willing to offer a longer trade credit period to the retailer.
Furthermore, according to Theorem 2, Theorem 1 can be modified to Theorem 3.
Theorem 3
The supplier’s optimal trade credit period is zero (i.e.,
\(M^{2*} = 0\)
) if (i)
\(a \le b\)
, or (ii)
\((a  b)W \le aC\)
, or (iii)
\((a  b)W \le aC + CI_{s}\)
, or (iv)
\(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} \le 0\).
Corollary 2
The supplier’s optimal trade credit period is zero (i.e.,
\(M^{2*} = 0\)
) if
\(\left( {a  b} \right)W  aC  CI_{s}  a\sqrt {\frac{{S_{s} h_{s} }}{A}} \le 0\).
Proof
We use the theorem that the arithmetic mean is not always less than the geometric mean. It is omitted.
In a word, the retailer’s and the supplier’s final total annual profits in Nash game are given by
$$\Pi_{r}^{2} (M^{2*} ) = \left( {P  W} \right)Ke^{{aM^{2*} }}  \sqrt {2S_{r} h_{r} Ke^{{aM^{2*} }} } + WKe^{{aM^{2*} }} I_{r} M^{2*} ,$$
(23)
$$\Pi_{s}^{2} (M^{2*} ) = WKe^{{\left( {a  b} \right)M^{2*} }}  CKe^{{aM^{2*} }}  S_{s} \sqrt {\frac{{Ke^{{aM^{2*} }} h_{r} }}{{2S_{r} }}}  \frac{{Ke^{{aM^{2*} }} h_{s} }}{A}\sqrt {\frac{{Ke^{{aM^{2*} }} S_{r} }}{{2h_{r} }}}  CKe^{{aM^{2*} }} I_{s} M^{2*} ,$$
(24)
respectively.
Note that \(\Pi_{r}^{2} (M^{2*} )\) is an increasing function of \(M^{2*}\) only if \(\Pi_{r}^{0} \ge 0\), which is a reasonable assumption. That is to say, as long as the trade credit period is offered by the supplier, \(\Pi_{r}^{2} (M^{2*} )\) is greater than \(\Pi_{r}^{0}\), i.e., \(\Pi_{r}^{2} (M^{2*} ) \ge \Pi_{r}^{0}\). Additionally, it is obvious that \(\Pi_{s}^{2} (M^{2*} ) \ge \Pi_{s}^{0}\). Proof is omitted.
Two parties’ decision making in a supplierStackelberg game
In this subsection, we suppose that the supplier is the dominating company over the retailer. For example, a supplier, such as Siwin Foods, (a famous food manufacturer in China) has a dominate power over its downstream small store. Consequently, the dominating company (e.g., Siwin Foods) acts as a leader, its downstream small store acts as a follower, which call a supplierStackelberg game. In a supplierStackelberg game, the supplier offers a trade credit period \(M\), and then the retailer maximizes his or her own profit to find optimal ordering lot size, next, the supplier observes the retailer’s optimal solution as a function of \(M\), finally, he or she find the optimal \(M\).
Firstly, we should know how the retailer responds to any trade credit period \(M\) offered by the supplier. By the first derivative necessary condition \({{d\Pi_{r}^{3} (Q)} \mathord{\left/ {\vphantom {{d\Pi_{r}^{3} (Q)} {dQ = 0}}} \right. \kern0pt} {dQ = 0}}\), the optimal ordering lot size in a supplierStackelberg game is given by
$$Q^{3*} = \sqrt {{{2S_{r} Ke^{aM} } \mathord{\left/ {\vphantom {{2S_{r} Ke^{aM} } {h_{r} }}} \right. \kern0pt} {h_{r} }}},$$
(25)
which is a function of \(M\).
After observing the optimal response of the retailer (given by Eq. (25)), the supplier selects optimal \(M\) so that his or her total annual profit is maximized.
Therefore, substituting \(Q^{3*} = \sqrt {{{2S_{r} Ke^{aM} } \mathord{\left/ {\vphantom {{2S_{r} Ke^{aM} } {h_{r} }}} \right. \kern0pt} {h_{r} }}}\) into Eq. (11), the \(\Pi_{s}^{3} (M)\) can be modified to a new function of \(M\) will be given by
$$\Pi_{s}^{3} (M) = WKe^{{\left( {a  b} \right)M}}  CKe^{aM}  S_{s} \sqrt {\frac{{h_{r} Ke^{aM} }}{{2S_{r} }}}  \frac{{Ke^{aM} h_{s} }}{2A}\sqrt {\frac{{2Ke^{aM} S_{r} }}{{h_{r} }}}  CKe^{aM} I_{s} M.$$
(26)
In order to maximize \(\Pi_{s}^{3} (M)\) in Eq. (26), we obtain
$$\frac{{d\Pi_{s}^{3} (M)}}{dM} = \left( {a  b} \right)WKe^{{\left( {a  b} \right)M}}  aCKe^{aM}  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} Ke^{aM} }}{{2S_{r} }}}  \frac{{3ah_{s} Ke^{aM} }}{2A}\sqrt {\frac{{S_{r} Ke^{aM} }}{{2h_{r} }}}  CKe^{aM} I_{s}  aCKe^{aM} I_{s} M$$
(27)
Theorem 4
The supplier’s optimal trade credit period is zero (i.e.,
\(M^{3*} = 0\)
) if (i)
\(a \le b\)
, or (ii)
\((a  b)W \le aC\)
, or (iii)
\((a  b)W \le aC + CI_{s}\).
Proof
We omit the proof of Theorem 4 since it mimics that of Theorem 1.
Consequently, the retailer’s and the supplier’s total annual profits are given by
$$\Pi_{r}^{3} (M^{3*} = 0) = \left( {P  W} \right)K  \sqrt {2KS_{r} h_{r} } = \Pi_{r}^{0} ,$$
(28)
$$\Pi_{s}^{3} (M^{3*} = 0) = \left( {W  C} \right)K  S_{s} \sqrt {{{Kh_{r} } \mathord{\left/ {\vphantom {{Kh_{r} } {2S_{r} }}} \right. \kern0pt} {2S_{r} }}}  {{Kh_{s} \sqrt {{{KS_{r} } \mathord{\left/ {\vphantom {{KS_{r} } {2h_{r} }}} \right. \kern0pt} {2h_{r} }}} } \mathord{\left/ {\vphantom {{Kh_{s} \sqrt {{{KS_{r} } \mathord{\left/ {\vphantom {{KS_{r} } {2h_{r} }}} \right. \kern0pt} {2h_{r} }}} } A}} \right. \kern0pt} A} = \Pi_{s}^{0} .$$
(29)
That is to say, the two parties don’t achieve any coordination or improvement in Theorem 4.
Next, we discuss the another condition, i.e., \((a  b)W > aC + CI_{s}\). By the first derivative condition \({{d\Pi_{s}^{3} (M)} \mathord{\left/ {\vphantom {{d\Pi_{s}^{3} (M)} {dM = 0}}} \right. \kern0pt} {dM = 0}}\), we obtain
$$\left( {a  b} \right)We^{  bM}  aC  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} Ke^{aM} }}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} Ke^{aM} }}{{2h_{r} }}}  CI_{s}  aCI_{s} M = 0.$$
(30)
From Eq. (30), the optimal trade credit period function is given by
$$M^{{3\bar{*}}} = {{\left\{ {\left( {a  b} \right)We^{{  bM^{{3\bar{*}}} }}  aC  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} Ke^{{aM^{{3\bar{*}}} }} }}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} Ke^{{aM^{{3\bar{*}}} }} }}{{2h_{r} }}}  CI_{s} } \right\}} \mathord{\left/ {\vphantom {{\left\{ {\left( {a  b} \right)We^{{  bM^{{3\bar{*}}} }}  aC  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} Ke^{{aM^{{3\bar{*}}} }} }}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} Ke^{{aM^{{3\bar{*}}} }} }}{{2h_{r} }}}  CI_{s} } \right\}} {aCI_{s} }}} \right. \kern0pt} {aCI_{s} }},$$
(31)
$${\text{when}}\;\left( {a  b} \right)We^{{  bM^{{3\bar{*}}} }}  aC  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} Ke^{{aM^{{3\bar{*}}} }} }}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} Ke^{{aM^{{3\bar{*}}} }} }}{{2h_{r} }}}  CI_{s} > 0.$$
Theorem 5
When
\(\left( {a  b} \right)W  aC  CI_{s}  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\)
, (i) if
\(M^{{3\bar{*}}} < M_{\rm max}\)
, the final optimal trade credit period is
\(M^{3*} = M^{{3\bar{*}}}\)
; (ii) if
\(M^{{3\bar{*}}} \ge M_{\rm max}\)
, the final optimal trade credit period is
\(M^{3*} = M_{\rm max}\).
Proof
We omit the proof of Theorem 5 since it mimics that of Theorem 2.
From Eq. (31) and Theorem 5, we can obtain the following results.
Corollary 3

(i) A higher value of
\(a\), \(W\), \(A\)
and a lower value of
\(b\), \(C\), \(S_{s}\), \(h_{s}\), \(I_{s}\)
cause a higher value of
\(\left( {a  b} \right)W  aC  CI_{s}  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}\)
and
\(M^{{3\bar{*}}}\).

(ii) The change of
\(P\)
and
\(I_{r}\), i.e., the retailer’s profit parameters, do not affect the supplier as to whether to offer trade credit to the retailer.
Proof
It is omitted.
Likewise, according to Theorem 5, Theorem 4 can be modified to Theorem 6.
Theorem 6
The supplier’s optimal trade credit period is zero (i.e.,
\(\Pi_{r}^{3} (M^{3*} ) \ge \Pi_{r}^{0}\)
) if (i)
\(a \le b\)
, or (ii)
\((a  b)W \le aC\)
, or (iii)
\((a  b)W \le aC + CI_{s}\)
, or (iv)
\(\left( {a  b} \right)W  aC  CI_{s}  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} \le 0\).
Corollary 4
The supplier’s optimal trade credit period is zero (i.e.,
\(M^{3*} = 0\)
) if
\(\left( {a  b} \right)W  aC  CI_{s}  a\sqrt {\frac{{3S_{s} h_{s} }}{4A}} \le 0\).
Proof
It is omitted.
Consequently, the retailer’s and the supplier’s final total annual profits in a supplierStackelberg game are given by
$$\Pi_{r}^{3} (M^{3*} ) = \left( {P  W} \right)Ke^{{aM^{3*} }}  \sqrt {2S_{r} h_{r} Ke^{{aM^{3*} }} } + WKe^{{aM^{3*} }} I_{r} M^{3*} ,$$
(32)
$$\Pi_{s}^{3} (M^{3*} ) = WKe^{{\left( {a  b} \right)M^{3*} }}  CKe^{{aM^{3*} }}  S_{s} \sqrt {\frac{{Ke^{{aM^{3*} }} h_{r} }}{{2S_{r} }}}  \frac{{Ke^{{aM^{3*} }} h_{s} }}{A}\sqrt {\frac{{Ke^{{aM^{3*} }} S_{r} }}{{2h_{r} }}}  CKe^{{aM^{3*} }} I_{s} M^{3*} ,$$
(33)
respectively.
Likewize, \(\Pi_{r}^{3} (M^{3*} )\) is an increasing function of \(M^{3*}\) only if \(\Pi_{r}^{0} \ge 0\). Therefore, as long as the trade credit period is offered by the supplier, \(\Pi_{r}^{3} (M^{3*} )\) is greater than \(\Pi_{r}^{0}\), i.e., \(\Pi_{r}^{3} (M^{3*} ) \ge \Pi_{r}^{0}\). Additionally, it is obvious that \(\Pi_{s}^{3} (M^{3*} ) \ge \Pi_{s}^{0}\). Proof is omitted.