Integrated Production-Inventory Supply Chain Model
Completion requirements
Read this article. An integrated production-inventory model is constructed to address supplier, manufacturer, and retailer uncertainties. According to the author, what are the three types of uncertainties in supply chain management?
Mathematical formulation of the model
Formulation of suppliers' individual average profit
Differential equation for the supplier in Figure 2 in [0,T] is given by
(3)
The total idle cost = \(id_s [TR+P_s t_s(\dfrac{1}{D_c}−\dfrac{1}{p_m})]\), purchase cost = \(c_s p_m T_s\), selling price = \(c_m p_m T_s\), and ordering cost = \(A_s\).
The total idle cost = \(id_m[\dfrac{pm_{Tm}−nD_R}{D_c}]\), purchase cost =\(c_m p_m T_s\), selling price = \(c_r p_m T_s\), and ordering cost = \(A_m\).
\(I_{em}=I_{pr}=\text{Amount of interest earned by the manufacturer in [0,T] from retailer.}\)
\(Hr = \text{Holding cost of retailer.}\)
The total idle cost = \(id_r T_R\), purchase cost = \(c_r p_m T_s\), selling price = \(c_{r1} p_m T_s\), and ordering cost = \(A_r\).
Interest earned by the retailers for (\(n + 1\)) cycle is given by
Interest payable by the retailers for the first n cycle is given by
\(I_{rp}=\text{Retailers' payable interest.}\)\(\begin{aligned}
&=c_{r} I_{p}\left[n \int_{M}^{T_{R}}\left(D_{R}-D_{c} t\right) d t\right] \\
&=\frac{n c_{r} I_{p}}{2}\left[T_{R} D_{R}+D_{c} M^{2}-2 M D_{R}\right]
\end{aligned}\)
\(ATP_1=\text{Total average profit for integrated model}\)
Case 2
=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[\left(c_{m}-c_{s}\right) p_{m} T_{s}-h_{s}\left(\frac{p_{s} t_{s}^{2}}{p_{m}}-p_{s} t_{s}^{2}\right)-\mathrm{id}_{s}\left(T_{R}+p_{s} t_{s} \frac{1}{D_{c}}-\frac{1}{p_{m}}\right)-A_{s}\right.\\
&+\left(c_{r}-c_{m}\right) p_{m} T_{s}-h_{m}\left(n p_{m} T_{s} T_{R}-\frac{n^{2}+n-2 r-2}{2} T_{R} D_{R}-\frac{p_{s}^{2} t_{s}^{2}}{2 p_{m}}\right) \\
&-\operatorname{id}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right) \\
&-A_{m}+\left(c_{r 1}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right) \\
&\left.+\frac{n c_{r_{1}} I_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} I_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)(2 M-T)-\mathrm{id}_{r} T_{R}-A_{r}\right] \\
=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[A p_{s}^{2} t_{s}^{2}-B p_{s} t_{s}+F\right]
\end{aligned}\)
(19)
where A and B are given in (16) and (17), respectively and
&+\left(c_{r}-c_{m}\right) p_{m} T_{s}-h_{m}\left(n p_{m} T_{s} T_{R}-\frac{n^{2}+n-2 r-2}{2} T_{R} D_{R}-\frac{p_{s}^{2} t_{s}^{2}}{2 p_{m}}\right) \\
&-\widetilde{i d}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right) \\
&-A_{m}+\left(c_{r 1}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right) \\
&\left.+\left\{\frac{n c_{r_{1}} D_{c} M^{2}}{2}+\frac{c_{r_{1}}}{2}\left(p_{m} T_{s}-n D_{R}\right)\left(2 M-T^{\prime}\right)\right\} \widetilde{I}_{r e}-\widetilde{i d}_{r} T_{R}-A_{r}\right]
\end{aligned}\)
Differential equation for the supplier in Figure 2 in [0,T] is given by
\( \dfrac{dq_s}{dt} =\left\{ \begin{array}{lll}
p_s−p_m, & {0 ≤ t ≤ t_s} \\
−p_m, & {t_s ≤ t ≤ T_s} \\
0, & {T_s ≤ t ≤ T}
\end{array} \right. \)
with boundary conditions \(q_s (t) = 0\) and \(t = 0,T_s\). Solving the differential equation with the boundary condition, we have
\( {q_s}(t) =\left\{ \begin{array}{lll}
(p_s−p_m)t, & 0 ≤ t ≤ t_s \\
p_m (T_s - t), & t_{s} ≤ t ≤ T_s \\
0, & T_s ≤ t ≤ T
\end{array} \right. \)
(3)
\(H_s = \text{Holding cost of supplier}\)
\( = h_s [∫^{ts}_0 (p_s−p_m) tdt + ∫^{Ts}_{t_s} p_m (T_s−t) dt ]\)
\(=h_s [\dfrac{p_s t^{2}_{s}} {p_m} −p_s t^{2}_{s}]\).
Figure 2 Inventory level of supplier.
The total idle cost = \(id_s [TR+P_s t_s(\dfrac{1}{D_c}−\dfrac{1}{p_m})]\), purchase cost = \(c_s p_m T_s\), selling price = \(c_m p_m T_s\), and ordering cost = \(A_s\).
\(APS = \text{Average profit for supplier}\)
\(=\dfrac{1}{T} [\text{revenue from sale-purchase cost-holding cost-idle cost-ordering cost}]\)
\(=\dfrac{1}{T} [(c_m−c_s) p_s t_s − h_s (\dfrac{p^{2}_{s} t^{2}_{s}} {p_m} − p_s t^{2}_{s}) − id_s (T_R + p_s t_s (\dfrac{1}{D_c} −\dfrac{1}{p_m}))−A_s]\)
(4)
Formulation of manufacturer individual average profit
Inventory level of manufacturer in Figure 3 in [0,T] is given by\( {q_m}(t) =\left\{ \begin{array}{lllllll}
p_mt, & 0 ≤ t ≤ T_R \\
p_mT - iD_R, & iT_R < t ≤ (i+1) \quad i = 1, 2, ..., (r-1)\\
p_mT - rD_R, & rT_R < t ≤ T_s \\
p_m T_s - rD_r, & T_{s} < t ≤ (r+1) T_R \\
p_mT_s - iD_R, & iT_R < t ≤ (i+1) T_R \quad i = r+1, r+2, ..., (n-1)\\
p_m T_s - nD_R, &nT_R < t ≤ (n+1) T_R\\
0, & (n+1)T_R < t ≤ T
\end{array} \right. \)
(5)
with boundary conditions \(q_m (0) = 0\) and \(q_m (i T _R + 0) = q_m (i T_R ) − D_R\).
Figure 3 Inventory level of manufacturer.
\(H_m=\text{Holding cost for manufacturer}\).
\( =hm[∫^{TR}_{0}p_mtdt+∑^{1}_{r−1}∫^{(i+1)T_R}_{iT_R}(p_mt−iD_R)dt+∫^{(T_s}_{rT_R}(p_mt−rD_R)dt \\ + ∫^{(r+1)T_R}_{T_s}(p_mT_s−rD_R)dt+∑_{r+1}^{n−1}∫^{(i+1)T_R}_{iT_R} (p_mT_s−iD_R)dt+∫^{(n+1)T_R}_{nT_R} (p_mT_s−nD_R)dt] \\ =h_m[np_mT_sT_R−\dfrac{n^2+n−2r−2}{2}T_RD_R−\dfrac{p^{2}_{s}t^{2}_{s}}{2p_m}] \).
The total idle cost = \(id_m[\dfrac{pm_{Tm}−nD_R}{D_c}]\), purchase cost =\(c_m p_m T_s\), selling price = \(c_r p_m T_s\), and ordering cost = \(A_m\).
Case 1
(When \(M ≤ T′ ≤ T R\))\(I_{em}=I_{pr}=\text{Amount of interest earned by the manufacturer in [0,T] from retailer}\).
\(= \text{Amount of interest paid by the retailer to the manufacturer in [0,T]}\).
\(=c_rI_p[n∫^{T_R}_{M}(D_R−D_ct)dt+∫^{T′}_{M}(p_mT_s−nD_R−D_ct)dt]\)
\(=\dfrac{nc_rI_p}{2}[T_RD_R+D_cM_2−2MDR]+c_rI_p[(\dfrac{(p_mT_s−nD_R)^2}{2D_c} + (p_mT_s−nD_R)M+\dfrac{D_cM^2}{2}]\)
\(APM_1=\text{Average profit of manufacturer}\).
\(=\dfrac{1}{T} \text{[revenue from sale-purchase cost-holding cost-idle cost+earned interest-ordering cost.]}\)
\(=\dfrac{1}{T}[(c_r−c_m)p_mT_h_m(np_mT_sT_R−\dfrac{n^2+n−2r−2}{2}T_RD_R−\dfrac{p^{2}_{s}t^{2}_{s}}{2pm}) \\ −id)m(\dfrac{p_mT_m−nD_R}{D_c})+\dfrac{nc_rI_p}{2}[T_RD_R+D_cM^2−2MD_R] \\ +c_rI_p(\dfrac{(p_mT_s−nD_R)^2}{2D_c}+(p_mT_s−nD_R)M+\dfrac{D_cM^2}{2})−Am]\)
(6)
(6)
Case 2
(When \(T′ ≤ M ≤ T R \))\(I_{em}=I_{pr}=\text{Amount of interest earned by the manufacturer in [0,T] from retailer.}\)
\(=\text{Amount of interest paid by the retailer to the manufacturer in [0,T.]\)</div><div style="text-align: left;">\(=c_rI_p[n∫^{T_R}_{M}(D_R−D_ct)dt+∫^{T′}_{M}(p_mT_s−nD_R−D_ct)dt]\)
\(=\dfrac{nc_rI_p}{2}[T_RD_R+D_cM^2−2MD_R]+c_rI_p[(\dfrac{(p_mT_s−nD_R)^2}{2D_c} \\
+(p_mT_s−nD_R)M+\dfrac{D_cM^2}{2}]\)
\(APR_2=\text{Average profit of retailer.}\)
\(=\dfrac{1}{T} \text{[revenue from sale-purchase cost-holding cost+earned interest-payable interest-idle cost-ordering cost].}\)
\( =\dfrac{1}{T}[(cr_1−c_r)p_mT_s−\dfrac{h_r}{2}(\dfrac{p^{2}_{m}T^{2}_{s}}{D_c}−2np_mT_sT_R−(2n+1)T_RD_R) \\ +\dfrac{nc_{r+1}i_eD_cM^2}{2}+ \dfrac{c_{r_1}{i_e}}{2} (p_mT_s−nD_R)(2M−T′) \\ −\dfrac{nc_rI_p}{2}[T_RD_R+D_cM_2−2MD_R]−id_rT_R−A_r] \)
(7)
(7)
Formulation of retailer individual average profit
Inventory level of retailer in Figure 4 in [0,T] is given by\( q_r(t)= \left\{ \begin{array}{ll} D_ct, & iT_R ≤ t ≤ (i+1)T_R \\ p_mT_s nD_r - d_ct, & (n+1)T_R ≤ t ≤ T \end{array} \right. \)
(8)
with boundary conditions \(q _r ((n + 1)T_R ) = 0 \) and \(q_r (T) = 0\).
with boundary conditions \(q _r ((n + 1)T_R ) = 0 \) and \(q_r (T) = 0\).
Figure 4 Inventory level of retailer.
\(Hr = \text{Holding cost of retailer.}\)
\(=nh_r[∫^{T_R}_{0}(D_R−D_ct)dt+∫^{T′}_{0}(p_mT_s−nD_R−D_ct)dt]\)
\(=\dfrac
{2}[\dfrac{p^{2}_{m}T^{2}_{s}}{D_c}−2np_mT_sT_R−(2n+1)T_RD_R]\).
{2}[\dfrac{p^{2}_{m}T^{2}_{s}}{D_c}−2np_mT_sT_R−(2n+1)T_RD_R]\).
The total idle cost = \(id_r T_R\), purchase cost = \(c_r p_m T_s\), selling price = \(c_{r1} p_m T_s\), and ordering cost = \(A_r\).
Case 1
(When \(M ≤ T′ ≤ T R \))Interest earned by the retailers for (\(n + 1\)) cycle is given by
\(I_{er}=\text {Amount of interest earned by the retailer from the bank in} (n+1) \text {cycle}\).
\(=(n+1)c_{r1}i_e[∫^{M}_{0}(M−t)D_cdt]\)
\(=\dfrac{(n+1)c_{r1}i_eD_cM^2}{2}\),
\(I_{pr}=\text{Amount of interest paid by the retailer to the manufacturer in [0,T]}\).
\(=c_rI_p[n∫^{T_R}{M}(D_R−D_ct)dt+∫^{T′}_{M}(p_mT_s−nD_R−D_ct)dt]\)
\(=nc_rI_p[\dfrac{T_RD_R+D_cM^2−2MD_R}{2}]+c_rI_p[\dfrac{(p_mT_s−nD_R)^2}{2D_c} \\ +(p_mT_s−nD_R)M+\dfrac{D_cM^2}{2}]\),
\(APR_1=\text{Average profit for retailer}\).
\( =\dfrac{1}{T} [\text {revenue from sale-purchase cost-holding cost +earned interest-payable interest-idle cost-ordering cost.]} \)
\( \dfrac{1}{T}[c_{r1}p_mT_s−c_rp_mT_s−\dfrac{h_r}{2}(\dfrac{p^{2}_{m}T^{2}_{s}}{D_c}−2np_mT_sT_R−(2n+1)T_RD_R) \\ +\dfrac{(n+1)c_{r1}i_eD_cM^2}{2}−\dfrac{n_{cr}I_p}{2}[T_RD_R+D_cM^2−2MD_R]−id_rT_R−A_r \\ +(p_mT_s−nD_R)M+\dfrac{D_cM^2}{2})−c_rI_p(\dfrac{(p_mT_s−nDR)^2}{2D_c}] \)
(9)
(9)
Case 2
(When \(T′ ≤ M ≤ T R \))Interest earned by the retailers for (\(n + 1\)) cycle is given by
\(I_{re}=\text{Retailers' earned interest.}\)
\(\begin{aligned}
&=c_{r_{1}} i_{e}\left[n \int_{0}^{M}(M-t) D_{c} d t+\int_{0}^{T^{\prime}}\left(T^{\prime}-t\right) D_{c} d t+\left(M-T^{\prime}\right)\left(p_{m} T_{s}-n D_{R}\right)\right] \\
&=\frac{n c_{r_{1}} i_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} i_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)\left(2 M-T^{\prime}\right)
\end{aligned}\)
&=c_{r_{1}} i_{e}\left[n \int_{0}^{M}(M-t) D_{c} d t+\int_{0}^{T^{\prime}}\left(T^{\prime}-t\right) D_{c} d t+\left(M-T^{\prime}\right)\left(p_{m} T_{s}-n D_{R}\right)\right] \\
&=\frac{n c_{r_{1}} i_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} i_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)\left(2 M-T^{\prime}\right)
\end{aligned}\)
Interest payable by the retailers for the first n cycle is given by
\(I_{rp}=\text{Retailers' payable interest.}\)
&=c_{r} I_{p}\left[n \int_{M}^{T_{R}}\left(D_{R}-D_{c} t\right) d t\right] \\
&=\frac{n c_{r} I_{p}}{2}\left[T_{R} D_{R}+D_{c} M^{2}-2 M D_{R}\right]
\end{aligned}\)
(10)
\(APR_2=\text{Average profit for retailer.}\)
\($=\frac{1}{T}[$ revenue from sale-purchase cost-holding cost
+earned interest-payable interest-idle cost-ordering cost]
\(
\begin{aligned}
=& \frac{1}{T}\left[\left(c_{r_{1}}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right)\right.\\
&+\frac{n c_{r_{1}} i_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} i_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)\left(2 M-T^{\prime}\right) \\
&\left.-\frac{n c_{r} I_{p}}{2}\left[T_{R} D_{R}+D_{c} M^{2}-2 M D_{R}\right]-\mathrm{id}_{r} T_{R}-A_{r}\right]
\end{aligned}
\)\)
+earned interest-payable interest-idle cost-ordering cost]
\(
\begin{aligned}
=& \frac{1}{T}\left[\left(c_{r_{1}}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right)\right.\\
&+\frac{n c_{r_{1}} i_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} i_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)\left(2 M-T^{\prime}\right) \\
&\left.-\frac{n c_{r} I_{p}}{2}\left[T_{R} D_{R}+D_{c} M^{2}-2 M D_{R}\right]-\mathrm{id}_{r} T_{R}-A_{r}\right]
\end{aligned}
\)\)
(11)
Crisp environment
Case 1
(\(M ≤ T′ ≤ T R \))\(ATP_1=\text{Total average profit for integrated model}\)
\(=AP_S+APM_1+APR_1\)
\(\begin{aligned}
=& \frac{D_{c}}{p_{m} T_{s}}+D_{R}\left[\left(c_{m}-c_{s}\right) p_{m} T_{s}-h_{s}\left(\frac{p_{s} t_{s}^{2}}{p_{m}}-p_{s} t_{s}^{2}\right)-\mathrm{id}_{s}\left(T_{R}+p_{s} t_{s} \frac{1}{D_{c}}-\frac{1}{p_{m}}\right)\right.\\
&+\left(c_{r}-c_{m}\right) p_{m} T_{s}-h_{m}\left(n p_{m} T_{s} T_{R}-\frac{n^{2}+n-2 r-2}{2} T_{R} D_{R}-\frac{p_{s}^{2} t_{s}^{2}}{2 p_{m}}\right) \\
&-\mathrm{id}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right) \\
&-A_{m}+\left(c_{r_{1}}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right) \\
&\left.+\frac{(n+1) c_{r_{1}} I_{e} D_{c} M^{2}}{2}-\mathrm{id}_{r} T_{R}-A_{r}\right] \\
=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[A p_{s}^{2} t_{s}^{2}-B p_{s} t_{s}+E\right]
\end{aligned}\)
=& \frac{D_{c}}{p_{m} T_{s}}+D_{R}\left[\left(c_{m}-c_{s}\right) p_{m} T_{s}-h_{s}\left(\frac{p_{s} t_{s}^{2}}{p_{m}}-p_{s} t_{s}^{2}\right)-\mathrm{id}_{s}\left(T_{R}+p_{s} t_{s} \frac{1}{D_{c}}-\frac{1}{p_{m}}\right)\right.\\
&+\left(c_{r}-c_{m}\right) p_{m} T_{s}-h_{m}\left(n p_{m} T_{s} T_{R}-\frac{n^{2}+n-2 r-2}{2} T_{R} D_{R}-\frac{p_{s}^{2} t_{s}^{2}}{2 p_{m}}\right) \\
&-\mathrm{id}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right) \\
&-A_{m}+\left(c_{r_{1}}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right) \\
&\left.+\frac{(n+1) c_{r_{1}} I_{e} D_{c} M^{2}}{2}-\mathrm{id}_{r} T_{R}-A_{r}\right] \\
=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[A p_{s}^{2} t_{s}^{2}-B p_{s} t_{s}+E\right]
\end{aligned}\)
(12)
where \(P_m T_s = P_s t_s\) , \(A=\dfrac{h_m−h_s}{2p_m}−\dfrac{h_r}{2D_c}\) (13)
where \(P_m T_s = P_s t_s\) , \(A=\dfrac{h_m−h_s}{2p_m}−\dfrac{h_r}{2D_c}\) (13)
\(B=(c_{r1}−c_s)+n(h_r−h_m)T_R−\dfrac{id_m+id_s}{D_c}\) (14)
and
and
\(E=[h_m \dfrac{n^2+n−2r−2}{2}+h_r \dfrac{2n+1}{2}]T_RD_R+\dfrac{(n+1)c_{r1}I_eD_cM^2}{2} \\ +(nid_m−id_s−id_r)T_R+id_sT_s−(A_s+A_m+A_r\)) (15)
\(\begin{aligned}
\frac{d}{d p_{s}}\left(\mathrm{ATP}_{1}\right) &=0 \\
\Rightarrow p_{s} &=\frac{2 D_{R}-B \pm \sqrt{\left(2 t_{s}-A\right)^{2}-4\left(2 t_{s}-A\right)\left(E-B D_{R}\right)}}{2\left(2 t_{s}-A\right)}
\end{aligned}\) (16)
\frac{d}{d p_{s}}\left(\mathrm{ATP}_{1}\right) &=0 \\
\Rightarrow p_{s} &=\frac{2 D_{R}-B \pm \sqrt{\left(2 t_{s}-A\right)^{2}-4\left(2 t_{s}-A\right)\left(E-B D_{R}\right)}}{2\left(2 t_{s}-A\right)}
\end{aligned}\) (16)
\(\frac{d^{2}}{d p_{s}^{2}}\left(\mathrm{ATP}_{1}\right)<0, \quad \text { if } \quad 2 A p_{s}+B<4 p_{s} t_{s}+B t_{s}+2 D_{R}\) (17)
Therefore, \(ATP_1\) is concave if \(2Ap_s+B < 4p_st_s+Bt_s+2D_R\). (18)
Therefore, \(ATP_1\) is concave if \(2Ap_s+B < 4p_st_s+Bt_s+2D_R\). (18)
Case 2
(\(T′ ≤ M ≤ T R \))
\(ATP_2= \text{Total average profit for integrated model.}\)
\(\begin{aligned}\(ATP_2= \text{Total average profit for integrated model.}\)
=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[\left(c_{m}-c_{s}\right) p_{m} T_{s}-h_{s}\left(\frac{p_{s} t_{s}^{2}}{p_{m}}-p_{s} t_{s}^{2}\right)-\mathrm{id}_{s}\left(T_{R}+p_{s} t_{s} \frac{1}{D_{c}}-\frac{1}{p_{m}}\right)-A_{s}\right.\\
&+\left(c_{r}-c_{m}\right) p_{m} T_{s}-h_{m}\left(n p_{m} T_{s} T_{R}-\frac{n^{2}+n-2 r-2}{2} T_{R} D_{R}-\frac{p_{s}^{2} t_{s}^{2}}{2 p_{m}}\right) \\
&-\operatorname{id}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right) \\
&-A_{m}+\left(c_{r 1}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right) \\
&\left.+\frac{n c_{r_{1}} I_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} I_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)(2 M-T)-\mathrm{id}_{r} T_{R}-A_{r}\right] \\
=& \frac{D_{c}}{p_{m} T_{s}+D_{R}}\left[A p_{s}^{2} t_{s}^{2}-B p_{s} t_{s}+F\right]
\end{aligned}\)
(19)
where A and B are given in (16) and (17), respectively and
F=[hmn2+n−2r−22+hr2n+12]TRDR+(n+1)cr1IeDcM22
+(nidm−ids−idr)TR+idsTs−(As+Am+Ar) (20)
\(\begin{aligned}
\frac{d}{d p_{s}}\left(\mathrm{ATP}_{2}\right) &=0 \\
\Rightarrow p_{s} &=\frac{2 D_{R}-B \pm \sqrt{\left(2 t_{s}-A\right)^{2}-4\left(2 t_{s}-A\right)\left(F-B D_{R}\right)}}{2\left(2 t_{s}-A\right)}
\end{aligned}\) (21)
\frac{d}{d p_{s}}\left(\mathrm{ATP}_{2}\right) &=0 \\
\Rightarrow p_{s} &=\frac{2 D_{R}-B \pm \sqrt{\left(2 t_{s}-A\right)^{2}-4\left(2 t_{s}-A\right)\left(F-B D_{R}\right)}}{2\left(2 t_{s}-A\right)}
\end{aligned}\) (21)
\(\frac{d^{2}}{d p_{s}^{2}}\left(\mathrm{ATP}_{1}\right)<0, \quad \text { if } \quad 2 A p_{s}+B<4 p_{s} t_{s}+B t_{s}+2 D_{R}\) (22)
Therefore, \(ATP_1\) is concave if \(2Ap_s+B < 4p_st_s+Bt_s+2D_R\) (23)
Therefore, \(ATP_1\) is concave if \(2Ap_s+B < 4p_st_s+Bt_s+2D_R\) (23)
Proposed inventory model in uncertain environment
Let us consider \(\widetilde {id}_s\), \(\widetilde{id}_r\), \(\widetilde{id}_m\), and \(\widetilde{I}_{re}\) as zigzag uncertain variables where \(\widetilde{id}_s=L(id_{s1},id_{s2},id_{s3})\), \(\widetilde{id}_r =L(id_{r1},id_{m2},id_{r3})\), \(\widetilde{id}_m= L(id_{m1},id_{m2},id_{m3})\), and \(\widetilde{I}_{re}=L(I_{re1},I_{re2},I_{re3})\). Then, the objective is reduce to the following:- For Case 1 (\(M ≤ T′ ≤ T R\) )
(24)
\begin{aligned}- For Case 2 (\(T′ ≤ M ≤ T R\))
&+\left(c_{r}-c_{m}\right) p_{m} T_{s}-h_{m}\left(n p_{m} T_{s} T_{R}-\frac{n^{2}+n-2 r-2}{2} T_{R} D_{R}-\frac{p_{s}^{2} t_{s}^{2}}{2 p_{m}}\right) \\
&-\widetilde{i d}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right) \\
&-A_{m}+\left(c_{r 1}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right) \\
&\left.+\left\{\frac{n c_{r_{1}} D_{c} M^{2}}{2}+\frac{c_{r_{1}}}{2}\left(p_{m} T_{s}-n D_{R}\right)\left(2 M-T^{\prime}\right)\right\} \widetilde{I}_{r e}-\widetilde{i d}_{r} T_{R}-A_{r}\right]
\end{aligned}\)
(25)
The equivalent crisp model
Using Lemma 1 and applying Theorem 2, the expected total average profit is given by the following:- For Case 1 \((M ≤ T′ ≤ T R )\)
\( E A [\widetilde {T} P_1] =\dfrac{D_c}{pm^{T_s+D_R}} [0(c_m−c_s)p_mT_s−h_s(\dfrac{p_{s}t^{2}_{s}} {p_m} −p_st^{2}_{s}) \\ − E[\widetilde{id}_s](T_R+p_st_s(\dfrac{1}{D_c}−\dfrac{1}{p_m})) \\ − A_s+(c_r−c_m)p_mT_s−h_m(np_mT_sT_R−\dfrac{n_2+n−2r−2}{2}T_RD_R−\dfrac{p^{2}_{s}t^{2}_{s}}{2p_m}) \\ − A_m+(cr_1−c_r)p_mT_s−\dfrac{h_r}{2}(\dfrac{p^{2}_{m}T^{2}_{s}}{D_c}−2np_mT_sT_R−(2n+1)T_RD_R)\\ − E[\widetilde{id}_m](\dfrac{p_mT_m−nD_R}{D_c})+\dfrac{(n+1)cr_1E[\widetilde{Ire}]D_cM^2}{2}−E[\widetilde{id}_r]T_R−A_r] \)
(26)
- For Case 2 (T′ ≤ M ≤ T R )
\( E A [\widetilde {T} P_2] =\dfrac{D_c}{pm^{T_s+D_R}} [(c_m−c_s)p_mT_s−h_s(\dfrac{p_{s}t^{2}_{s}} {p_m} −p_st^{2}_{s}) −E[\widetilde {id}_s](T_R+p_st_s \dfrac{1}{D_c} −\dfrac{1}{p_m})−A_s \\
+ (c_r−c_m)p_mT_s−h_m(np_mT_sT_R−\dfrac{n^2+n−2r−2}{2}T_RD_R−\dfrac{p^{2}_{s}t^{2}_{s}} {2p_m}) \\
−E[\widetilde {id}_m](\dfrac{p_mT_m−nD_R}{D_c}) −A_m+(cr_1−c_r)p_mT_s \\
−\dfrac{h_r}{2} (\dfrac{p^{2}_{m}T^{2}_s}{D_c} −2np_mT_sT_R−(2n+1)T_RD_R) \\
+{\dfrac{nc_{r1}D_cM^2}{2}+\dfrac{c_{r1}}{2}(p_mT_s−nD_R)(2M−T′)}E[\widetilde {Ire}]−E[\widetilde {id}r]T_R−A_r] \)
(27)