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

\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)


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)


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.

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].

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)


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}


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}
(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]\(\begin{aligned}=& \frac{1}{T}\left[\left(c_{r_{1}}-c_{r}\right) p_{m} T_{s}-\frac{h_{r}}{2}\left(\frac{p_{m}^{2} T_{s}^{2}}{D_{c}}-2 n p_{m} T_{s} T_{R}-(2 n+1) T_{R} D_{R}\right)\right.\\&+\frac{n c_{r_{1}} i_{e} D_{c} M^{2}}{2}+\frac{c_{r_{1}} i_{e}}{2}\left(p_{m} T_{s}-n D_{R}\right)\left(2 M-T^{\prime}\right) \\&\left.-\frac{n c_{r} I_{p}}{2}\left[T_{R} D_{R}+D_{c} M^{2}-2 M D_{R}\right]-\mathrm{id}_{r} T_{R}-A_{r}\right]\end{aligned}\)
(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}
(12)

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

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^{2}}{d p_{s}^{2}}\left(\mathrm{ATP}_{1}\right) (17)

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}=& \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^{2}}{d p_{s}^{2}}\left(\mathrm{ATP}_{1}\right) (22)

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 )
A\widetilde{T}P_1 =\dfrac{D_c}{p_mT_s+D_R}[0(c_m−c_s)p_mT_s−h_s(\dfrac{p_st^{2}_{s}}{p_m}−p_st^{2}_{s})−\widetilde{id}_s(T_R+p_st_s(\dfrac{1}{D_c}−\dfrac{1}{p_m})) \\ \begin{aligned} &-A_{s}+\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) \\ &-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.-\widetilde{i d}_{m}\left(\frac{p_{m} T_{m}-n D_{R}}{D_{c}}\right)+\frac{(n+1) c_{r_{1}} \widetilde{I}_{r e} D_{c} M^{2}}{2}-\widetilde{i d}_{r} T_{R}-A_{r}\right] \end{aligned}
(24)

  •  For Case 2 (T′ ≤ M ≤ T R)
A\widetlide{T}P_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)-\tilde{i d}_{s}\left(T_{R}+p_{s} t_{s} \frac{1}{D_{c}}-\frac{1}{p_{m}}\right)−A_s \right. \\\begin{aligned}&+\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)