A Benders’ Decomposition Method to Solve a Multi-period, Multi-echelon, and Multi-product Integrated Petroleum Supply Chain

Abstract

Problems of large dimensions (linear, non-linear, integer, and continuous) are the major principles in the modeling of important natural and social phenomena. The larger are the extent and scope of the application of the phenomenon or the vastness of its applications; the larger is the dimension confronted in modeling. Integrated Petroleum Supply Chain (IPSC), which has been proposed by Nasab and Amin-Naseri (Energy 114:708–733, 2016), is one such problem. Specific solutions have been proposed over the years for problems of this kind. One of the most important methods in this regard is Benders’ decomposition method, proposed in 1962 by Benders for combinatorial optimization problems. In this paper, Benders’ decomposition method has been used to solve the IPSC model and the gap criteria are then used to evaluate the performance of this method. Here, for analyzing the performance of Benders’ decomposition method, the results of this method and those of Branch & Bound algorithm, which has been proposed by Nasab and Amin-Naseri (Energy 114:708–733, 2016), have been compared. Based on the results, while Branch & Bound algorithm method is not able to solve large size problems, the generated gap in Benders’ method is very small. This indicates the capability of Benders’ method to achieve a response close to the optimal response. Therefore, Benders’ method possesses appropriate efficiency.

Appendices

Appendix 1. Nomenclature

Sets		Indices
I	Set of crude oil fields	i	Crude oil field
J	Set of existing crude oil storage tanks	j	Crude oil storage tank
J′	Set of new crude oil storage tanks	k	Refinery
K	Set of existing refineries	l	DC
K′	Set of new refineries	m	Costumer zone
L	Set of existing DCs	v	Transportation mode
L′	Set of the new DCs	lcv	Capacity of transportation mode
M	Set of costumer zone	p	Product
V	Set of transportation modes	ej	Capacity of new crude oil storage tank
LCV	Set of capacity of transportation modes	ek	Capacity of new refinery
P	Set of products	el	Capacity of new DC
EJ	Set of capacity of new crude oil storage tanks	uj	Capacity expansion of existing crude oil storage tank
EK	Set of capacity of new refineries	uk	Capacity expansion of existing refinery
EL	Set of capacity of new DCs	ul	Capacity expansion of existing DC
UJ	Set of capacity expansion of existing crude oil storage tanks	ev	Capacity expansion of existing pipeline
UK	Set of capacity expansion of existing refineries	ez	Capacity of the storage tank in DC
UL	Set of capacity expansion of existing DCs	lv	Capacity of new pipeline
EV	Set of capacity expansion of existing pipelines	rv	New pipeline route
EZ	Set of capacity of storage tanks in DC	t	Time period
LV	Set of capacity of new pipeline
RV	Set of new pipeline routes
T	Set of time periods

Parameters
${d_p^t}_{\mathrm{m}}$	Demand of costumer zone m for product p during t(m ∈ M, t ∈ T, p ∈ P)
${capl}_{\mathrm{l}}^{\mathrm{ul}}$	Capacity expansion for existing DC l with level ul (l ∈ L, ul ∈ Ul)
${capk}_j^{\mathrm{uj}}$	Capacity expansion for existing crude oil storage tank j with level uj (uj ∈ UJ, j ∈ J)
${capk}_k^{\mathrm{uk}}$	Capacity expansion for existing refinery k with level uk (k ∈ K, uk ∈ UK)
icj j	Initial capacity of existing crude oil storage tank j (j ∈ J)
ick k	Initial capacity of existing refinery k (k ∈ K)
icl p l	Initial capacity of existing DC l for product p (l ∈ L, p ∈ P)
${Mj}_{j\hbox{'}}^{ej}$	Minimum capacity coefficient of new crude oil storage tank j with level ej (j ∈ J', ej ∈ EJ)
${Mk}_{k\hbox{'}}^{ek}$	Minimum capacity coefficient of new refinery k with level ek (k ∈ K', ek ∈ EK)
${Ml}_{p\ l\hbox{'}}^{ez}$	Minimum capacity coefficient of new DC l for product p with level el (l ∈ L', p ∈ P, ez ∈ EZ)
${ivj}_j^0$	Initial inventory level in crude oil storage j (j ∈ (J ∪ J'))
${ivk}_k^0$	Initial inventory level in refinery k (k ∈ (K ∪ K'))
${ivl}_l^0$	Initial inventory level in DC l (l ∈ (L ∪ L'))
${Ncj}_j^{e\mathrm{j}}$	Capacity of new crude oil storage tank j with level ej (j ∈ J', ej ∈ EJ)
${Nct}_{l\hbox{'}}^{ez}$	Capacity of storage tank ez in new DC l (l ∈ L', ez ∈ EZ)
${Ncl}_l^{\mathrm{el}}$	Capacity of new DC l with level el (l ∈ L', el ∈ EL)
${Nck}_{k\hbox{'}}^{\mathrm{ek}}$	Capacity of new refinery k with level ek (k ∈ K', ek ∈ EK)
icap_iji j	Capacity of existing route pipeline route between crude oil field i and existing crude oil storage tank j (i ∈ I, j ∈ J)
icap_jkj k	Capacity of existing route pipeline route between existing crude oil storage tank j and existing refinery k (j ∈ J, k ∈ K)
icap_iki k	Capacity of existing route pipeline route between crude oil field i and refinery k (i ∈ l, k ∈ K)
icap_klkl	Capacity of existing route pipeline route between refinery k and DC l (k ∈ K, l ∈ L)
icap_lpllp l	Capacity of existing route pipeline route between DC l and DC lp (lp, l ∈ L and lp ≠ 1)
${trc}_v^{lcv}$	Capacity of transportation mode v with level lcv (v ∈ V, lcv ∈ LCV)
$cap\_{ij}_{i\ j}^{ev}$	Capacity expansion of existing pipeline route between crude oil field i and existing crude oil storage tank j with level ev (i ∈ I, j ∈ J, ev ∈ EV)
$cap\_{jk}_{\mathrm{j}\ \mathrm{k}}^{ev}$	Capacity expansion of existing route between existing crude oil storage tank j and existing refinery k with level ev (j ∈ J, k ∈ K, ev ∈ EV)
$cap\_{ik}_{\mathrm{i}\ \mathrm{k}}^{ev}$	Capacity expansion of existing route between crude oil field i and existing refinery k with level ev (i ∈ I, k ∈ K, ev ∈ EV)
$cap\_{kl}_{\mathrm{k}\ \mathrm{l}}^{ev}$	Capacity expansion of existing route between existing refinery k and existing DC l with level ev (k ∈ K, l ∈ L, ev ∈ EV)
$cap\_{lpl}_{\mathrm{lp}\ \mathrm{l}}^{ev}$	The Capacity expansion of existing route between existing DC l and existing DC lp with level ev (lp, l ∈ L, lp ≠ 1, ev ∈ EV)
TPP	Amount of time periodic period
clv lv	Capacity of pipeline transportation mode with level lv (lv = LV)
Rij i j	Zero-one matrix representing the existing routes between crude oil field i and existing crude oil storage tank j i ∈ I, j ∈ J
Rik i k	Zero-one matrix representing the existing routes between crude oil field i and existing refinery k (i ∈ I, k ∈ K)
Rjk j k	Zero-one matrix representing the existing routes between existing crude oil storage tank j and existing refinery k (j ∈ J, k ∈ K)
Rkl k l	Zero-one matrix representing the existing routes between existing refinery k and existing DC l (k ∈ K, l ∈ L)
Rlpl lp l	Zero-one matrix representing the existing routes between existing DC l and existing DC l (lp, l ∈ L & lp ≠ 1)
$n\_{\max}_v^{lcv\ t}$	Maximum number of available transportation mode v with level lcv in period t (lcv ∈ LCV, v ∈ V & t ∈ T)
w i	Maximum production of crude oil field i (i ∈ I)
μ p	Product ratio of one barrel crude oil for p (p ∈ P)
lj j	Minimum inventory level of crude oil storage tank j (j ∈ (J ∪ J'))
lk k	Minimum inventory level of refinery k (k ∈ (K ∪ K'))
ll l	Minimum inventory level of DC l (l ∈ (L ∪ L'))
M	A big number
$x\cos {tj}_j^{ej\ \mathrm{t}}$	Installation cost of new crude oil storage tank j with level ej in period t (t ∈ T, ej ∈ EJ, j ∈ J')
$x\cos {tk}_k^{\mathrm{ek}\ \mathrm{t}}$	Installation cost of new refinery k with level ek in period t (t ∈ T, ek ∈ EK, k ∈ K')
$x\cos {tl}_l^{\mathrm{el}\ \mathrm{t}}$	Installation cost of new DC l with level el in period t (t ∈ T, l ∈ L')
$u\cos {tj}_j^{uj\ \mathrm{t}}$	Expansion cost of existing crude oil storage tank j with level uj in period t (t ∈ T, p ∈ P, j ∈ J, uj ∈ UJ)
$u\cos {tk}_k^{\mathrm{uk}\ \mathrm{t}}$	Expansion cost of existing refinery k with level uk in period t (t ∈ T, p ∈ P, k ∈ K, uk ∈ UK)
$ucos{tl}_l^{\mathrm{ul}\ \mathrm{p}\ \mathrm{t}}$	Expansion cost of existing DC l with level ul for product p in period t (ev ∈ EV, t ∈ T, i ∈ I, j ∈ J)
$y\cos t\_{ij}_{i\ j}^{\mathrm{ev}\ \mathrm{t}}$	Expansion cost of existing pipeline route between crude oil field i and existing crude oil storage tank j with level lv in period t (ev ∈ EV, T ∈ T, i ∈ I, j ∈ J)
$ycost\_{jk}_{\mathrm{j}\ k}^{ev\ \mathrm{t}}$	Expansion cost of existing pipeline route between existing crude oil storage tank j and existing refinery k with level lv in period t (ev ∈ EV, t ∈ T, j ∈ J, k ∈ K)
$ycost\_{ik}_{\mathrm{i}\ k}^{ev\ \mathrm{t}}$	Expansion cost of existing pipeline route between oil field i and existing refinery k with level lv in period t (ev ∈ EV, t ∈ T, i ∈ I, k ∈ K)
$ycost\_{kl}_{k\ \mathrm{l}}^{\mathrm{ev}\ \mathrm{t}}$	Expansion cost of existing pipeline route between existing refinery k and existing DC l with level lv in period t (ev ∈ EV, t ∈ T, k ∈ K, l ∈ L)
$ycost\_{lpl}_{\mathrm{lpl}}^{\mathrm{ev}\ \mathrm{t}}$	Expansion cost of existing pipeline route between existing DC l and existing DC lp with level lv in period t (ev ∈ EV, t ∈ T, lp ∈ LP, l ∈ L & lp ≠ 1)
hcostj j	Holding cost of crude oil in crude oil storage tank j in period t (t ∈ T, j ∈ (J ∪ J'))
hcostk k	Holding cost of crude oil in refinery k at period t (t ∈ T, k ∈ (K ∪ K'))
hcostl p l	Holding cost of product p in DC l at period t (t ∈ T, p ∈ P, l ∈ (L ∪ L'))
$qcost\_{ij}_{\mathrm{i}\ j}^{\mathrm{t}}$	Flow rate cost of crude oil between crude oil field i and crude oil storage tank j in period t (t ∈ T, i ∈ I, j ∈ (J ∪ J'))
$qcost\_{jk}_{\mathrm{j}\ \mathrm{k}}^t$	Flow rate cost of crude oil between crude oil storage tank j and refinery k in period t (t ∈ T, j ∈ (J ∪ J'), k ∈ (K ∪ K'))
$qcost\_{ik}_{i\ \mathrm{k}}^{\mathrm{t}}$	Flow rate cost of crude oil between crude oil field i and refinery k in period t (t ∈ T, i ∈ I, k ∈ (K ∪ K'))
$qcost\_{kl}_{\mathrm{k}\ \mathrm{l}}^{\mathrm{t}}$	Flow rate cost of crude oil between refinery k and DC l in period t (t ∈ T, k ∈ (K ∪ K'), 1 ∈ (L ∪ L'))
$qcost\_{lp l}_{lp\ l}^{\mathrm{t}}$	Flow rate cost to DC l from other DCs lp in period t (t ∈ T, lp ∈ L and l ∈ (L ∪ L') OR t ∈ T, lp ∈ L' and l ∈ L')
$rcost\_{ij}_{i\ \mathrm{j}}^{\mathrm{lv}\ \mathrm{rv}\ \mathrm{t}}$	Route installation cost of new pipeline route rv between crude oil field i and crude oil storage tank j with level lv in period t (lv ∈ LV, rv ∈ RV, t ∈ T, i ∈ I, j ∈ (J ∪ J'))
$rcost\_{jk}_{\mathrm{j}\ \mathrm{k}}^{\mathrm{lv}\ \mathrm{rv}\ \mathrm{t}}$	Route installation cost of new pipeline route rv between crude oil storage tank j and refinery k with level lv in period t (lv ∈ LV, rv ∈ RV, t ∈ T, j ∈ (J ∪ J'), k ∈ (K ∪ K'))
$rcost\_{ik}_{\mathrm{i}\ \mathrm{k}}^{\mathrm{lv}\ \mathrm{rv}\ \mathrm{t}}$	Route installation cost of new pipeline route rv between crude oil field i and refinery k with level lv in period t (lv ∈ LV, rv ∈ RV, t ∈ T, i ∈ I, k ∈ (K ∪ K'))
$rcost\_{kl}_{k\ \mathrm{lv}}^{\mathrm{lv}\ \mathrm{rv}\ \mathrm{t}}$	Route installation cost of new pipeline route rv between refinery k and DC l with level lv in period t (lv ∈ LV, rv ∈ RV, t ∈ T, k ∈ (K ∪ K'), l ∈ (L ∪ L'), v = pipeline mode)
$rcost\_{lpl}_{\mathrm{lp}\ \mathrm{l}\ \mathrm{p}}^{\mathrm{lv}\ \mathrm{rv}\ \mathrm{t}}$	Installation cost of new pipeline route rv to DC l from other DCs lp with level lv in period t (t ∈ T, lp ∈ L and l ∈ (L ∪ L') OR t ∈ T, lp ∈ L' and l ∈ L')
$ncost\_{kl}_{\mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lcv}\ \mathrm{t}}$	Transportation cost for mode v between refinery k and DC l with capacity level lcv in period t (lcv ∈ LCV, v ∈ V, t ∈ T, k ∈ (K ∪ K'), l ∈ (L ∪ L'))
$ncost\_{lm}_{\mathrm{l}\ \mathrm{m}\ \mathrm{v}}^{\mathrm{l}\mathrm{cv}\ \mathrm{t}}$	Transportation cost for mode v between DC l and customer zone m with capacity level lcv in period t (lcv ∈ LCV, v ∈ V, t ∈ T, l ∈ (L ∪ L'), m ∈ M)
$ncost\_{lpl}_{\mathrm{lp}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lcv}\ \mathrm{t}}$	Transportation cost mode v between DC lp and DC l with capacity level lcv in period t (lcv ∈ LCV, v ∈ V, t ∈ T, l ∈ (L ∪ L'), m ∈ M)
${icost}_p^{\mathrm{t}}$	Importation cost of product p in period t (t ∈ T, p ∈ P)
${ncostl}_{\mathrm{l}}^{\mathrm{ez}\ \mathrm{t}}$	Installation cost of storage tank ez in new DC l in period t (ez ∈ EZ, p ∈ P, t ∈ T, l ∈ L')
${pcostj}_{\mathrm{j}}^{\mathrm{t}}$	Variable cost of crude oil storage tank j in period t (t ∈ T, ∈(J ∪ J'))
${pcostk}_k^{\mathrm{t}}$	Variable cost of refinery k in period t (t ∈ T, k ∈ (K ∪ K'))
${pcostl}_{\mathrm{p}\ \mathrm{l}}^{\mathrm{t}}$	Variable cost of DC l for product p in period t (t ∈ T, p ∈ P, l ∈ (L ∪ L'))
${Fcostj}_j^{\mathrm{t}}$	Fixed cost of crude oil storage tank j in period t (t ∈ T, (j ∈ J ∪ J'))
${Fcostk}_k^{\mathrm{t}}$	Fixed cost of refinery k in period t (t ∈ T, (k ∈ K ∪ K'))
${Fcostl}_l^{\mathrm{t}}$	Fixed cost of DC l in period t (t ∈ T, (l ∈ L ∪ L'))
${RPP}_P^{\mathrm{t}}$	Price of product p in period t (t ∈ T, p ∈ P)
OP t	Price of crude oil in period t (t ∈ T)
${ERPP}_p^{\mathrm{t}}$	Exportation price of product p in period t (t ∈ T, p ∈ P)
EOP t	Exportation price of crude oil in period t (t ∈ T)

Variables of the model are divided into three categories, including Positive, Binary, and Positive integers as follows:

Positive variables
${qij}_{i\ \mathrm{j}}^t$	Flow rate of crude oil from field I to storage tank j in period t (t ∈ T, i ∈ I, j ∈ (J ∪ J'))
${qjk}_{\mathrm{j}\ \mathrm{k}}^t$	Flow rate of crude oil from storage tank j to refinery k in period t (t ∈ T, j ∈ (J ∪ J'), k ∈ (K ∪ K'))
${qik}_{\mathrm{i}\ \mathrm{k}}^t$	Flow rate of crude oil from field i to refinery k in period t (t ∈ T, j ∈ (J ∪ J'), k ∈ (K ∪ K'))
${qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t$	Flow rate of product p from refinery k to DC l by transportation mode v in period t (p ∈ P, v∈, t ∈ T, k ∈ (K ∪ K'), l ∈ (L ∪ L'))
${qlm}_{\mathrm{p}\ \mathrm{l}\ \mathrm{m}\ \mathrm{v}}^t$	Flow rate of product p from DC l to customer zone m by transportation mode v in period t (t ∈ T, p ∈ P, l ∈ (L ∪ L'), m ∈ M, v ∈ V)
${qlpl}_{\mathrm{plp}\ \mathrm{l}\ \mathrm{v}}^t$	Flow rate of product p to DC l from other DCs lp by transportation mode v in period t (t ∈ T, lp ∈ L and l ∈ (L ∪ L') OR t ∈ T, lp ∈ L' and l ∈ L')
${i}_{\mathrm{p}\ \mathrm{l}}^t$	Volume of importation for product p from DC l in period t (t ∈ T, p ∈ P, 1 ∈ (L ∪ L'))
${vl}_{\mathrm{p}\ \mathrm{l}}^t$	Inventory level of product p in DC l in period t (t ∈ T, p ∈ P, l ∈ (L ∪ L'))
${vk}_k^t$	Inventory level of crude oil in refinery k in period t (t ∈ T, k ∈ (K ∪ K'))
${vj}_j^t$	Inventory level of crude oil in crude oil storage tank j in period t (t ∈ T, j ∈ (J ∪ J'))
${Ep}_{\mathrm{p}\ \mathrm{l}}^t$	Volume of exportation for product p from DC l in period t (t ∈ T, p ∈ P, 1 ∈ (L ∪ L'))
${Eco}_j^t$	Volume of exportation for crude oil from storage tank j in period t (t ∈ T, j ∈ (J ∪ J'))

Binary variables

${xl}_{\mathrm{l}}^{\mathrm{el}\ t}=\left\{\begin{array}{lll}1& \mathrm{if}\ \mathrm{DC}\ l\ \mathrm{is}\ \mathrm{installed}\ \mathrm{with}\ \mathrm{level}\ el\ \mathrm{in}\ \mathrm{period}\ t& \\ {}0& \mathrm{otherwise}& \left(t\in T,\mathrm{el}\in EL,l\in {L}^{\hbox{'}}\right)\end{array}\right.$

${xk}_{\mathrm{k}}^{\mathrm{ek}\ t}=\left\{\begin{array}{lll}1& \mathrm{if}\ \mathrm{refinery}\ k\ \mathrm{is}\ \mathrm{installed}\ \mathrm{with}\ \mathrm{level}\ ek\ \mathrm{in}\ \mathrm{period}\ t& \\ {}0& \mathrm{otherwise}\kern8.5em \left(t\in T,\mathrm{ek}\in EK,k\in {K}^{\hbox{'}}\right)& \end{array}\right.$

${xj}_{\mathrm{j}}^{\mathrm{ej}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{crude}\ \mathrm{oil}\ \mathrm{storage}\ \mathrm{tank}\ j\ \mathrm{is}\ \mathrm{installed}\ \mathrm{with}\ \mathrm{level}\ ej\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern13.25em \left(t\in T,\mathrm{ej}\in EJ,j\in {J}^{\hbox{'}}\right)\end{array}\right.$

$\tau {l}_{\mathrm{p}\ \mathrm{l}}^{\mathrm{ul}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{DC}\ l\ \mathrm{is}\ \mathrm{expanded}\ \mathrm{for}\ \mathrm{product}\ p\ \mathrm{with}\ \mathrm{level}\ ul\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern12em \left(t\in T,\mathrm{p}\in P,\mathrm{l}\in L, ul\in UL\right)\end{array}\right.$

$\tau {k}_k^{\mathrm{uk}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{refinery}\ k\ \mathrm{is}\ \mathrm{expanded}\ \mathrm{with}\ \mathrm{level}\ uk\ \mathrm{in}\ \mathrm{period}\ \mathrm{t}\\ {}0& \mathrm{otherwise}\kern8.75em \left(t\in T,\mathrm{k}\in K, uk\in UK\right)\end{array}\right.$

$\tau {J}_J^{\mathrm{uj}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{crude}\ \mathrm{oil}\ \mathrm{storage}\ \mathrm{tank}\ j\ \mathrm{is}\ \mathrm{expanded}\ \mathrm{with}\ \mathrm{level}\ uj\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern9.5em \left(t\in T,\mathrm{p}\in P,\mathrm{j}\in J, uj\in UJ\right)\end{array}\right.$

${yij}_{i\ \mathrm{j}}^{\mathrm{ev}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{existing}\ \mathrm{route}\ \mathrm{between}\ \mathrm{field}\ i\ \mathrm{and}\ \mathrm{existing}\ \mathrm{crude}\ \mathrm{oil}\ \mathrm{storage}\ \mathrm{tank}\ j\ \mathrm{is}\ \mathrm{expanded}\ \mathrm{with}\\ {}& \mathrm{leve}\ ev\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern9.5em \left(\mathrm{ev}\in EV,\mathrm{t}\in T,i\in I,j\in J\right)\end{array}\right.$

${yjk}_{\mathrm{j}\ \mathrm{k}}^{\mathrm{ev}\ t}-\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{existing}\ \mathrm{route}\ \mathrm{between}\ \mathrm{existing}\ \mathrm{crude}\ \mathrm{oil}\ \mathrm{storage}\ \mathrm{tank}\ j\ \mathrm{and}\ \mathrm{existing}\ \mathrm{refinery}\ k\ \mathrm{is}\\ {}& \mathrm{expanded}\ \mathrm{with}\ \mathrm{level}\ ev\ \mathrm{in}\ \mathrm{period}\ \mathrm{t}\\ {}0& \mathrm{otherwise}\kern9.5em \left(\mathrm{ev}\in EV,t\in T,j\in J,k\in K\right)\end{array}\right.$ ${yik}_{\mathrm{i}\ \mathrm{k}}^{\mathrm{ev}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{existing}\ \mathrm{route}\ \mathrm{between}\ \mathrm{field}\ i\ \mathrm{and}\ \mathrm{existing}\ \mathrm{refinery}\ k\ \mathrm{is}\ \mathrm{expanded}\ \mathrm{with}\ \mathrm{level}\ ev\ \mathrm{in}\\ {}& \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern9em \left(\mathrm{ev}\in EV,\mathrm{t}\in T,i\in I,k\in K\right)\end{array}\right.$ ${ykl}_{\mathrm{kl}}^{\mathrm{ev}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{exsiting}\ \mathrm{route}\ \mathrm{between}\ \mathrm{existing}\ \mathrm{refinery}\ k\ \mathrm{and}\ \mathrm{DC}\ l\ \mathrm{is}\ \mathrm{expanded}\\ {}& \mathrm{with}\ \mathrm{level}\ ev\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern9.25em \left(\mathrm{ev}\in EV,t\in T,k\in K,l\in L\right)\end{array}\right.$ ${ylpl}_{\mathrm{lp}\ \mathrm{l}}^{\mathrm{ev}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{existing}\ \mathrm{route}\ \mathrm{between}\ \mathrm{exisitng}\ \mathrm{distribution}\ \mathrm{center}\ lp\ \mathrm{and}\ \mathrm{exisitng}\ \mathrm{DC}\ l\ \mathrm{is}\\ {}& \mathrm{expanded}\ \mathrm{with}\ \mathrm{level}\ ev\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern5em \left(\mathrm{ev}\in EV,t\in T, lp\ \mathrm{and}\ l\in L, lp\ne l\right)\end{array}\right.$

${zl}_l^{\mathrm{ez}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{the}\ \mathrm{number}\ \mathrm{of}\ \mathrm{storage}\ \mathrm{tanks}\ \mathrm{is}\ \mathrm{fixed}\ \mathrm{with}\ \mathrm{level}\ ez\ \mathrm{in}\ \mathrm{new}\ \mathrm{DC}\ l\\ {}& \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern8.5em \left(\mathrm{ez}\in EZ,t\in T,l\in {L}^{\hbox{'}}\right)\end{array}\right.$

${rij}_{\mathrm{i}\ \mathrm{j}}^{\mathrm{lv}\ \mathrm{rv}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{route}\ rv\ \mathrm{between}\ \mathrm{field}\ i\ \mathrm{and}\ \mathrm{crude}\ \mathrm{oil}\ \mathrm{storage}\ \mathrm{tank}\ j\ \mathrm{is}\ \mathrm{installed}\ \mathrm{with}\ \mathrm{level}\ lv\ \mathrm{in}\\ {}& \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern4.5em \left( lv\in LV,\mathrm{rv}\in RV,\mathrm{t}\in T,i\in I,j\in \left(J\cup {J}^{\hbox{'}}\right)\right)\end{array}\right.$

${rjk}_{\mathrm{j}\ \mathrm{k}}^{\mathrm{lv}\ \mathrm{rv}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{route}\ rv\ \mathrm{between}\ \mathrm{crude}\ \mathrm{oil}\ \mathrm{storage}\ \mathrm{tank}\ j\ \mathrm{and}\ \mathrm{refinery}\ k\ \mathrm{is}\ \mathrm{installed}\ \mathrm{with}\ \mathrm{level}\ lv\ \mathrm{in}\\ {}& \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern5.75em \left(\mathrm{ev}\in EV,\mathrm{t}\in T,j\in \left(J\cup {J}^{\hbox{'}}\right),k\in \left(K\cup {K}^{\hbox{'}}\right)\right)\end{array}\right.$ ${rik}_{\mathrm{i}\ \mathrm{k}}^{\mathrm{lv}\ \mathrm{rv}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{route}\ rv\ \mathrm{between}\ \mathrm{oil}\ \mathrm{field}\ i\ \mathrm{and}\ \mathrm{refinery}\ k\ \mathrm{is}\ \mathrm{installed}\ \mathrm{with}\ \mathrm{level}\ lv\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern9.5em \left(\mathrm{ev}\in EV,t\in T,i\in I,k\in \left(K\cup {K}^{\hbox{'}}\right)\right)\end{array}\right.$ ${rkl}_{\mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lv}\ \mathrm{rv}\ \mathrm{t}}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{route}\ rv\ \mathrm{between}\ \mathrm{refinery}\ k\ \mathrm{and}\ \mathrm{DC}\ l\ \mathrm{is}\ \mathrm{installed}\ \mathrm{with}\ \mathrm{level}\ lv\ \mathrm{in}\\ {}& \\ {}0& \mathrm{otherwise}\kern7.25em \left(\mathrm{ev}\in EV,t\in T,k\in \left(K\cup {K}^{\hbox{'}}\right),l\in \left(L\cup {L}^{\hbox{'}}\right)\right)\end{array}\right.$ ${rlpl}_{\mathrm{lp}\ \mathrm{l}}^{\mathrm{lv}\ \mathrm{rv}\ t}=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{route}\ rv\ \mathrm{between}\ \mathrm{distribution}\ \mathrm{center}\ lp\ \mathrm{and}\ \mathrm{DC}\ l\ \mathrm{is}\ \mathrm{installed}\\ {}& \mathrm{with}\ \mathrm{level}\ lv\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern5.5em \left(\mathrm{ev}\in EV,\mathrm{t}\in T, lp\ \mathrm{and}\ l\in \left(L\cup {L}^{\hbox{'}}\right), lp\ne l\right)\end{array}\right.$ $rkl{\hbox{'}}_{\mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t\left(\mathrm{Auxiliary}\ \mathrm{variable}\right)=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{rout}\ rv\ \mathrm{between}\ \mathrm{refinery}\ k\ \mathrm{and}\ \mathrm{DC}\ l\ \mathrm{is}\ \mathrm{not}\\ {}& \mathrm{in}\mathrm{stalled}\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern6.75em \left(\mathrm{t}\in T,k\in \left(K\cup {K}^{\hbox{'}}\right),l\in \left(L\cup {L}^{\hbox{'}}\right)\right)\end{array}\right.$ $rlpl{\hbox{'}}_{\mathrm{lp}\ \mathrm{l}\ \mathrm{v}}^t\left(\mathrm{Auxiliary}\ \mathrm{variable}\right)=\left\{\begin{array}{ll}1& \mathrm{if}\ \mathrm{route}\ rv\ \mathrm{between}\ \mathrm{DC}\ lp\ \mathrm{and}\ \mathrm{distribution}\ \mathrm{center}\ l\ \mathrm{is}\ \mathrm{not}\\ {}& \mathrm{in}\mathrm{stalled}\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern5.75em \left(\mathrm{t}\in T, lp\in \left(L\cup {L}^{\hbox{'}}\right),l\in \left(L\cup {L}^{\hbox{'}}\right)\right)\end{array}\right.$ ${s}_{\mathrm{lp}\ \mathrm{l}}^t\left(\mathrm{Auxiliary}\ \mathrm{variable}\right)=\left\{\begin{array}{ll}1& \mathrm{If}\ \mathrm{there}\ \mathrm{is}\ \mathrm{a}\ \mathrm{flow}\ \mathrm{from}\ \mathrm{DC}\ lp\ \mathrm{to}\ \mathrm{DC}\ l\ \mathrm{in}\ \mathrm{period}\ t\\ {}0& \mathrm{otherwise}\kern8em \left(\mathrm{t}\in T, lp\in L\ and\ l\in \left(L\cup {L}^{\hbox{'}}\right)\ \mathrm{OR}\right.\\ {}& \kern12.25em \left.\mathrm{t}\in T, lp\in {L}^{\hbox{'}}\ and\ l\in {L}^{\hbox{'}}\right)\end{array}\right.$

Positive integer
${nkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lcv}\ t}$	Number of fleets for transportation mode v with capacity level lcv between refinery k to DC l at period t (lcv ∈ LCV, v ∈ V, t ∈ T, k ∈ (K ∪ K'), l ∈ (L ∪ L'))
${nlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lcv}\ t}$	Number of fleets for transportation mode v with capacity level lcv between DC l to DC lp at period t (lcv ∈ LCV, v ∈ V, t ∈ T, k ∈ (K ∪ K'), l ∈ (L ∪ L'))
${nlm}_{\mathrm{p}\ \mathrm{l}\ \mathrm{m}\ \mathrm{v}}^{\mathrm{lcv}\ t}$	Number of fleets for transportation mode v with capacity level lcv between DC l and customer zone m at period t (lcv ∈ LCV, v ∈ V, t ∈ T, l ∈ (L ∪ L'), m ∈ M)
${n}_{p\ l}^{\mathrm{ez}\ t}$	The number of storage tanks with capacity level ez in product p in distribution center l in period t (lcv ∈ LCV, v ∈ V, t ∈ T, l ∈ (L ∪ L'), m ∈ M)

Appendix 2

(73)

According to the objective function, the first term shows the revenue from the exportation of the product p from DC l. The inventory costs of the existing and new crude oil storage tanks, refineries, and DCs are presented by the second to fourth terms. The fifth term depicts the exportation of crude oil from crude oil storage tank j. The cost of these flow rates is calculated by the sixth to tenth terms. The cost of products’ importation to DCs is determined by the eleventh term. By terms twelve to seven, the transportation cost can be evaluated. Constraints 1 to 2 set an upper bound on capacities for each of the existing crude oil storage tanks and refineries:

$$ \sum \limits_{i\in I}{qik}_{\mathrm{i}\ \mathrm{k}}^t+\sum \limits_{j\in \left(J\cup {J}^{\prime}\right)}{qjk}_{\mathrm{j}\ \mathrm{k}}^t+{vk}_k^{t-1}\le {ick}_k+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{uk\in UK}{capk}_k^{\mathrm{uk}}\tau {k}_k^{\mathrm{uk}\ t\hbox{'}}\forall k\in K,t\in T $$

(74)

$$ \sum \limits_{i\in I}{qij}_{i\ \mathrm{j}}^t+{vj}_j^{t-1}\le {icj}_j+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{uj\in UJ}{capj}_j^{\mathrm{uj}}\tau {j}_j^{\mathrm{uj}\ t\hbox{'}}\kern0.75em \forall j\in J,t\in T $$

(75)

According to Eqs. 76 to 80, the new facility including crude oil storage tanks, refineries, and DCs can be installed if and only if the flow rate to them is greater than a minimum flow rate. Also, the summation of the all flows to each facility should be less than its capacity.

$$ \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ej\in EJ}{Mj}_j^{ej}{Ncj}_j^{\mathrm{ej}}{xj}_{\mathrm{j}}^{\mathrm{ej}\ t\hbox{'}}\le \sum \limits_{i\in I}{qij}_{i\ \mathrm{j}}^t\kern1.75em \forall j\in {J}^{\prime },t\in T $$

(76)

$$ \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{Mk}_k^{ek}{Nck}_k^{\mathrm{ek}}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\le \sum \limits_{i\in I}{qik}_{\mathrm{i}\ \mathrm{k}}^t+\sum \limits_{j\in \left(J\cup {J}^{\prime}\right)}{qjk}_{\mathrm{j}\ \mathrm{k}}^t\kern0.75em \forall k\in {K}^{\prime },t\in T $$

(77)

$$ \sum \limits_{i\in I}{qik}_{\mathrm{i}\ \mathrm{k}}^t+\sum \limits_{j\in \left(J\cup {J}^{\prime}\right)}{qjk}_{\mathrm{j}\ \mathrm{k}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{Nck}_k^{\mathrm{ek}}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\kern1em \forall k\in {K}^{\prime },t\in T $$

(78)

$$ {\displaystyle \begin{array}{l}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ez\in EZ}{Ml}_{p\ l}^{ez}{Nct}_l^{\mathrm{ez}}{n}_{p\ l}^{\mathrm{ez}\ t\hbox{'}}\le \sum \limits_v\sum \limits_{k\in \left(K\cup {K}^{\prime}\right)}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t+{i}_{\mathrm{l}}^{p\ t}\sum \limits_{v\in V}\sum \limits_{lp\in \left(L\cup {L}^{\prime}\right)}{qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t-\sum \limits_{v\in V}\sum \limits_{lp\in {L}^{\prime }}{qlpl}_{\mathrm{p}\ \mathrm{l}\ \mathrm{l}\mathrm{p}\ \mathrm{v}}^t\\ {}\forall l\in {L}^{\prime },t\in T,p\in P\kern1.25em \mathrm{i}f\ \mathrm{l}\&\mathrm{lp}\in {L}^{\prime}\to \mathrm{l}\mathrm{p}\ne \mathrm{l}\kern0.5em 29\end{array}} $$

(79)

$$ {\displaystyle \begin{array}{l}\sum \limits_v\sum \limits_{k\in \left(K\cup {K}^{\prime}\right)}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t+{i}_{\mathrm{l}}^{p\ t}\sum \limits_{v\in V}\sum \limits_{lp\in \left(L\cup {L}^{\prime}\right)}{qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t-\sum \limits_{v\in V}\sum \limits_{lp\in {L}^{\prime }}{qlpl}_{\mathrm{p}\ \mathrm{l}\ \mathrm{l}\mathrm{p}\ \mathrm{v}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ez\in EZ}{Nct}_l^{\mathrm{ez}}{n}_{p\ l}^{\mathrm{ez}\ t\hbox{'}}\\ {}\forall l\in {L}^{\prime },t\in T,p\in P\kern1.25em \mathrm{i}f\ \mathrm{l}\&\mathrm{lp}\in {L}^{\prime}\to \mathrm{l}\mathrm{p}\ne \mathrm{l}\kern0.5em 29\end{array}} $$

(80)

Equations 81 to 83 show that the volume of crude oil in each pipeline route is less than their capacities.

$$ {\displaystyle \begin{array}{l}{qij}_{i\ \mathrm{j}}^t\le {Rij}_{i\ j}\left( icap\_{ij}_{i\ \mathrm{j}}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ev\in EV} cap\_{ij}_{i\ \mathrm{j}}^{ev}{yij}_{i\ \mathrm{j}}^{\mathrm{ev}\ t\hbox{'}}\right)+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rij}_{i\ \mathrm{j}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\\ {}\forall i\in I,j\in \left(J\cup {J}^{\prime}\right),t\in T\end{array}} $$

(81)

$$ {\displaystyle \begin{array}{l}{qjk}_{\mathrm{j}\ \mathrm{k}}^t\le {Rjk}_{j\ k}\left( icap\_{jk}_{\mathrm{j}\ \mathrm{k}}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ev\in EV} cap\_{jk}_{\mathrm{j}\ \mathrm{k}}^{ev}{yjk}_{\mathrm{j}\ \mathrm{k}}^{\mathrm{ev}\ t\hbox{'}}\right)+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rjk}_{\mathrm{j}\ \mathrm{k}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\\ {}\forall j\in \left(J\cup {J}^{\prime}\right),k\in \left(K\cup {K}^{\prime}\right),t\in T\end{array}} $$

(82)

$$ {\displaystyle \begin{array}{l}{qik}_{\mathrm{i}\ \mathrm{k}}^t\le {Rik}_{\mathrm{i}\ \mathrm{k}}\left( icap\_{ik}_{i\ \mathrm{k}}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ev\in EV} cap\_{ik}_{\mathrm{i}\ \mathrm{k}}^{ev}{yik}_{\mathrm{i}\ \mathrm{k}}^{\mathrm{ev}\ t\hbox{'}}\right)+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rik}_{\mathrm{i}\ \mathrm{k}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\\ {}\forall i\in I,k\in \left(K\cup {K}^{\prime}\right),t\in T\end{array}} $$

(83)

According to Eqs. 84 to 90, the flow rates of the refined products between two facilities are less that the capacity of the means or routes which are used.

$$ {\displaystyle \begin{array}{l}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{lcv\in LCV}{trc}_v^{lcv}{nkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lcv}\ t}\mathrm{TPP}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rkl}_{\mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}+\\ {}{Rkl}_{\mathrm{k}\ \mathrm{l}}\left( icap\_{kl}_{\mathrm{k}\ \mathrm{l}}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ev\in EV} cap\_{kl}_{\mathrm{k}\ \mathrm{l}}^{ev}{ykl}_{\mathrm{k}\ \mathrm{l}}^{\mathrm{ev}\ t\hbox{'}}\right)\forall k\in K,l\in L,p\in P,v\in V,t\in T\end{array}} $$

(84)

$$ {\displaystyle \begin{array}{l}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{lcv\in LCV}{trc}_v^{lcv}{nkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lcv}\ t}\mathrm{TPP}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rkl}_{\mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\forall k\in {K}^{\prime },l\in \left(L\cup {L}^{\prime}\right),p\in P,v\in V,t\in T\\ {}\forall k\in K,l\in {L}^{\prime },p\in P,v\in V,t\in T\end{array}} $$

(85)

$$ {\displaystyle \begin{array}{l}\sum \limits_{p\in P}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rkl}_{\mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lv}\ \mathrm{rv}\ t}+{Rkl}_{\mathrm{k}\ \mathrm{l}}\left( icap\_{kl}_{\mathrm{k}\ \mathrm{l}}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ev\in EV} cap\_{kl}_{\mathrm{k}\ \mathrm{l}}^{ev}{ykl}_{\mathrm{k}\ \mathrm{l}}^{\mathrm{ev}\ t}\right)\kern1.25em \forall k\in \left(K\cup {K}^{\prime}\right),\\ {}l\in \left(L\cup {L}^{\prime}\right),p\in P,v\in V,t\in T\ \mathrm{v}=3\end{array}} $$

(86)

$$ {\displaystyle \begin{array}{l}{qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{lcv\in LCV}{trc}_v^{lcv}{nlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lcv}\ t}\mathrm{TPP}+{Rlpl}_{\mathrm{lp}\ l}\left( icap\_{lpl}_{\mathrm{lp}\ \mathrm{l}}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ev\in EV} cap\_{lpl}_{\mathrm{lp}\ \mathrm{l}}^{ev}{ylpl}_{\mathrm{lp}\ \mathrm{l}}^{\mathrm{ev}\ t\hbox{'}}\right)\\ {}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rlpl}_{\mathrm{lp}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\kern1.25em \mathrm{t}\in T, lp\in L\ and\ l\in L\end{array}} $$

(87)

$$ {qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{lcv\in LCV}{trc}_v^{lcv}{nlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lcv}\ t}\mathrm{TPP}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rlpl}_{\mathrm{lp}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\kern1.25em \mathrm{t}\in T, lp\in L\ and\ l\in L $$

(88)

$$ {\displaystyle \begin{array}{l}\sum \limits_p{qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t\le {Rlpl}_{\mathrm{lp}\ l}\left( icap\_{lpl}_{\mathrm{lp}\ \mathrm{l}}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ev\in EV} cap\_{lpl}_{\mathrm{lp}\ \mathrm{l}}^{ev}{ylpl}_{\mathrm{lp}\ \mathrm{l}}^{\mathrm{ev}\ t\hbox{'}}\right)\\ {}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rlpl}_{\mathrm{lp}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\kern1.25em \mathrm{t}\in T, lp\&l\in \left(L\cup {L}^{\prime}\right)\end{array}} $$

(89)

$$ {qlm}_{\mathrm{p}\ \mathrm{l}\ \mathrm{m}\ \mathrm{v}}^{\kern0.5em t}\le \sum \limits_{lcv\in LCV}{trc}_v^{lcv}{nlm}_{\mathrm{p}\ \mathrm{l}\ \mathrm{m}\ \mathrm{v}}^{\mathrm{lcv}\kern0.5em t}\mathrm{TPP}\kern2.5em \forall l\in \left(L\cup {L}^{\prime}\right),m\in M,p\in P,v\in V,t\in T $$

(90)

Based on Eqs. 91 to 95, the amount of the crude oil to be extracted is determined. Also, these constraints guarantee that the sum of the coming in and out flow rates to each facility must be equal.

$$ \sum \limits_{v\in V}\sum \limits_{l\in \left(L\cup {L}^{\prime}\right)}{qlm}_{\mathrm{p}\ \mathrm{l}\ \mathrm{m}\ \mathrm{v}}^{\kern0.5em t}\ge {d}_{p\ \mathrm{m}}^t\kern0.75em \forall m\in M,t\in T,p\in P $$

(91)

$$ \sum \limits_{i\in I}{qij}_{i\ \mathrm{j}}^t+{vj}_j^{t-1}=\sum \limits_{k\in \left(K\cup {K}^{\prime}\right)}{qjk}_{\mathrm{j}\ \mathrm{k}}^t+{vj}_j^t+{Eco}_j^t\kern1.75em \forall j\in \left(J\cup {J}^{\prime}\right),t\in T $$

(92)

$$ {\displaystyle \begin{array}{l}\sum \limits_{v\in V}\sum \limits_{lp\in \left(L\cup {L}^{\prime}\right)}{qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t-\sum \limits_{v\in V}\sum \limits_{lp\in {L}^{\prime }}{qlpl}_{\mathrm{p}\ \mathrm{l}\ \mathrm{l}\mathrm{p}\ \mathrm{v}}^t+\sum \limits_{v\in V}\sum \limits_{k\in \left(K\cup {K}^{\prime}\right)}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t+{vl}_{\mathrm{p}\ \mathrm{l}}^{t-1}+{i}_{\mathrm{p}\ \mathrm{l}}^t=\sum \limits_{v\in V}\sum \limits_{m\in M}{qlm}_{\mathrm{p}\ \mathrm{l}\ \mathrm{m}\ \mathrm{v}}^{\kern0.5em t}+{Ep}_{\mathrm{p}\ \mathrm{l}}^t+{v}_{\mathrm{p}\ \mathrm{l}}^t\\ {}\forall l\in {L}^{\prime },p\in P,t\in T\end{array}} $$

(93)

$$ {\displaystyle \begin{array}{l}\sum \limits_{v\in V}\sum \limits_{lp\in L}{qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t-\sum \limits_{v\in V}\sum \limits_{lp\in \left(L\cup {L}^{\prime}\right)}{qlpl}_{\mathrm{p}\ \mathrm{l}\ \mathrm{l}\mathrm{p}\ \mathrm{v}}^t+\sum \limits_{v\in V}\sum \limits_{k\in \left(K\cup {K}^{\prime}\right)}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t+{v}_{\mathrm{p}\ \mathrm{l}}^{t-1}+{i}_{\mathrm{l}}^{p\ t}=\sum \limits_{v\in V}\sum \limits_{m\in M}{qlm}_{\mathrm{p}\ \mathrm{l}\ \mathrm{m}\ \mathrm{v}}^{\kern0.5em t}+{Ep}_{\mathrm{p}\ \mathrm{l}}^t+{vl}_{\mathrm{p}\ \mathrm{l}}^t\\ {}\kern25.25em \forall l\in L,p\in P,t\in T\end{array}} $$

(94)

$$ {\mu}_p\left(\sum \limits_{i\in I}{qik}_{\mathrm{i}\ \mathrm{k}}^t+\sum \limits_{j\in \left(J\cup {J}^{\prime}\right)}{qjk}_{\mathrm{j}\ \mathrm{k}}^t+{vk}_k^{t-1}\right)=\sum \limits_{v\in V}\sum \limits_{l\in \left(L\cup {L}^{\prime}\right)}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t+{\mu}_p\left({vk}_k^t\right)\forall k\in \left(K\cup {K}^{\prime}\right),p\in P,t\in T $$

(95)

Constraints 96 to 107 show that the amount of available inventory in each facility should be between the capacity of each facility and the minimum level determined before.

$$ {vj}_j^t\le {icj}_j+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{uj\in UJ}{capj}_j^{\mathrm{uj}}\tau {j}_j^{\mathrm{uj}\ t\hbox{'}}\kern3.75em \forall j\in J,t\in T $$

(96)

$$ {vk}_k^t\le {ick}_k+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{uk\in UK}{capk}_k^{\mathrm{uk}}\tau {k}_k^{\mathrm{uk}\ t\hbox{'}}\kern3.25em \forall k\in K,t\in T $$

(97)

$$ {vl}_{\mathrm{p}\ \mathrm{l}}^{\mathrm{t}}\le {ic}_l^p+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ul\in UL}{capl}_l^{\mathrm{ul}}\tau {l}_{\mathrm{p}\ \mathrm{l}}^{\mathrm{ul}\ t\hbox{'}}\kern2.75em \forall l\in L,p\in P,t\in T $$

(98)

$$ {vj}_j^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ej\in EJ}{Ncj}_j^{\mathrm{ej}}{xj}_{\mathrm{j}}^{\mathrm{ej}\kern0.5em {t}^{\prime }}\kern0.5em \forall j\in {J}^{\prime },t\in T $$

(99)

$$ {vk}_k^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{Nck}_k^{\mathrm{ek}}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\kern5.25em \forall k\in {K}^{\prime },t\in T $$

(100)

$$ {vl}_{\mathrm{p}\ \mathrm{l}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ez\in EZ}{Nct}_{l^{\prime}}^{\mathrm{ez}}{n}_{p\ l}^{\mathrm{ez}\ t\hbox{'}}\kern4.25em \forall l\in {L}^{\prime },p\in P,t\in T $$

(101)

$$ {vj}_j^t\ge {lj}_j\left({icj}_j+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{uj\in UJ}{capj}_j^{\mathrm{uj}}\tau {j}_j^{\mathrm{uj}\ t\hbox{'}}\right)\kern2.5em \forall j\in J,t\in T $$

(102)

$$ {vk}_k^t\ge {lk}_k\left({ick}_k+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{uk\in UK}{capk}_k^{\mathrm{uk}}\tau {k}_k^{\mathrm{uk}\ t\hbox{'}}\right)\kern2em \forall k\in K,t\in T $$

(103)

$$ {vl}_{\mathrm{p}\ \mathrm{l}}^{\mathrm{t}}\ge {l}_l\left({ic}_l^p+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ul\in UL}{capl}_l^{\mathrm{ul}}\tau {l}_{\mathrm{p}\ \mathrm{l}}^{\mathrm{ul}\ t\hbox{'}}\right)\kern1.5em \forall l\in L,p\in P,t\in T $$

(104)

$$ {vj}_j^t\ge {l}_j\left(\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ej\in EJ}{Ncj}_k^{\mathrm{ej}}{xj}_{\mathrm{j}}^{\mathrm{ej}\ t\hbox{'}}\right)\kern5.25em \forall j\in {J}^{\prime },t\in T $$

(105)

$$ {vk}_k^t\ge {lk}_k\left(\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{Nck}_k^{\mathrm{ek}}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\right)\kern5.25em \forall k\in {K}^{\prime },t\in T $$

(106)

$$ {vl}_{\mathrm{p}\ \mathrm{l}}^t\ge {l}_l\left(\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ez\in EZ}{Nct}_{l^{\prime}}^{\mathrm{ez}}{n}_{p\ l}^{\mathrm{ez}\ t\hbox{'}}\right)\kern4.25em \forall l\in {L}^{\prime },p\in P,t\in T $$

(107)

Equations 108 to 116 show that the flow rate to new facilities exists if and only if they are installed.

$$ {qij}_{i\ \mathrm{j}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ej\in EJ}{Ncj}_j^{\mathrm{ej}}{xj}_{\mathrm{j}}^{\mathrm{ej}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ej\in EJ}{xj}_{\mathrm{j}}^{\mathrm{ej}\ t\hbox{'}}\kern3.25em \forall i\in I,j\in {J}^{\prime },t\in T $$

(108)

$$ {qik}_{\mathrm{i}\ \mathrm{k}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{Nck}_k^{\mathrm{ek}}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\kern1.75em \forall i\in I,k\in {K}^{\prime },t\in T $$

(109)

$$ {qjk}_{\mathrm{j}\ \mathrm{k}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{Nck}_k^{\mathrm{ek}}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\kern1.75em \forall j\in \left(J\cup {J}^{\prime}\right),k\in {K}^{\prime },t\in T $$

(110)

$$ {qjk}_{\mathrm{j}\ \mathrm{k}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ej\in EJ}{Ncj}_j^{\mathrm{ej}}{xj}_{\mathrm{j}}^{\mathrm{ej}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ej\in EJ}{xj}_{\mathrm{j}}^{\mathrm{ej}\ t\hbox{'}}\kern1em \forall j\in {J}^{\prime },k\in \left(K\cup {K}^{\prime}\right),t\in T $$

(111)

$$ {qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ez\in EZ}{Nct}_{l^{\prime}}^{\mathrm{ez}}{n}_{p\ l}^{\mathrm{ez}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{el\in EL}{xl}_{\mathrm{l}}^{\mathrm{el}\ t\hbox{'}}\kern1.75em \forall k\in \left(K\cup {K}^{\prime}\right),l\in {L}^{\prime },t\in T $$

(112)

$$ {qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{Nck}_k^{\mathrm{ek}}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ek\in EK}{xk}_{\mathrm{k}}^{\mathrm{ek}\ t\hbox{'}}\kern2.5em \forall k\in {K}^{\prime },l\in \left(L\cup {L}^{\prime}\right),t\in T $$

(113)

$$ {qlm}_{\mathrm{p}\ \mathrm{l}\ \mathrm{m}\ \mathrm{v}}^{\kern0.5em t}\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ez\in EZ}{Nct}_{l^{\prime}}^{\mathrm{ez}}{n}_{p\ l}^{\mathrm{ez}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{el\in EL}{xl}_{\mathrm{l}}^{\mathrm{el}\ t\hbox{'}}\kern2.75em \forall l\in {L}^{\prime },m\in M,t\in T $$

(114)

$$ {qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ez\in EZ}{Nct}_{l^{\prime}}^{\mathrm{ez}}{n}_{p\ lp}^{\mathrm{ez}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{el\in EL}{xl}_{\mathrm{lp}}^{\mathrm{el}\ t\hbox{'}}\kern2.75em \forall \mathrm{t}\in T, lp\in {L}^{\prime }\ and\ l\in {L}^{\prime } $$

(115)

$$ {\displaystyle \begin{array}{l}{qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ez\in EZ}{Nct}_{l^{\prime}}^{\mathrm{ez}}{n}_{p\ lp}^{\mathrm{ez}\ t\hbox{'}}\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{el\in EL}{xl}_{\mathrm{l}}^{\mathrm{el}\ t\hbox{'}}\kern1.25em \forall \mathrm{t}\in T, lp\in L\ and\ l\in {L}^{\prime}\\ {}\ \mathrm{OR}\kern1em \forall \mathrm{t}\in T, lp\in {L}^{\prime }\ and\ l\in {L}^{\prime}\end{array}} $$

(116)

The following equations guarantee that the volume of crude oil in the new pipeline routes is less than their capacities:

$$ {qij}_{i\ \mathrm{j}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rij}_{i\ \mathrm{j}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\kern2.5em \forall i\in I,j\in {J}^{\prime },t\in T $$

(117)

$$ {qjk}_{\mathrm{j}\ \mathrm{k}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rjk}_{\mathrm{j}\ \mathrm{k}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\kern2.5em \forall j\in \left(J\cup {J}^{\prime}\right),k\in \left(K\cup {K}^{\prime}\right),t\in T $$

(118)

$$ {qik}_{\mathrm{i}\ \mathrm{k}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rik}_{\mathrm{i}\ \mathrm{k}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\kern2.25em \forall i\in I,k\in \left(K\cup {K}^{\prime}\right),t\in T $$

(119)

$$ {qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{lv\in LV}\sum \limits_{rv\in RV}{clv}_{lv}{rkl}_{\mathrm{k}\ \mathrm{l}\ \mathrm{v}}^{\mathrm{lv}\ \mathrm{rv}\ t\hbox{'}}\kern1.75em \forall k\in \left(K\cup {K}^{\prime}\right),l\in \left(L\cup {L}^{\prime}\right),t\in T $$

(120)

Equation 121 shows that the output amount of the crude oil from the fields must be less than their maximum flow rate.

$$ \sum \limits_{j\in \left(J\cup {J}^{\prime}\right)}{qij}_{i\ \mathrm{j}}^t+\sum \limits_{k\in \left(K\cup {K}^{\prime}\right)}{qik}_{\mathrm{i}\ \mathrm{k}}^t\le {w}_i\kern4.25em \forall i\in I,t\in T $$

(121)

Constraints 122 and 123 set an upper bound on capacities for each existing crude oil storage tanks and DCs:

$$ {\displaystyle \begin{array}{l}\sum \limits_{v\in V}\sum \limits_{lp\in L}{qlpl}_{\mathrm{p}\ \mathrm{l}\mathrm{p}\ \mathrm{l}\ \mathrm{v}}^t+\sum \limits_{v\in V}\sum \limits_{k\in \left(K\cup {K}^{\prime}\right)}{qkl}_{\mathrm{p}\ \mathrm{k}\ \mathrm{l}\ \mathrm{v}}^t-\sum \limits_{v\in V}\sum \limits_{lp\in \left(L\cup {L}^{\prime}\right)}{qlpl}_{\mathrm{p}\ \mathrm{l}\ \mathrm{l}\mathrm{p}\ \mathrm{v}}^t+{i}_{\mathrm{p}\ \mathrm{l}}^t+{vl}_{\mathrm{p}\ \mathrm{l}}^{t-1}\le {icl}_{p\ l}+\sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ul\in UL}{capl}_l^{\mathrm{ul}}\tau {l}_{\mathrm{p}\ \mathrm{l}}^{\mathrm{ul}\ t\hbox{'}}\\ {}\kern21.75em \forall l\in L,t\in T,p\in P\kern0.75em \mathrm{i}f\ \mathrm{l}\&\mathrm{lp}\in L\to \mathrm{l}\mathrm{p}\ne \mathrm{l}\end{array}} $$

(122)

$$ \sum \limits_{i\in I}{qij}_{i\ \mathrm{j}}^t\le \sum \limits_{t^{\prime}\le t\in T}\sum \limits_{ej\in EJ}{Ncj}_j^{\mathrm{ej}}{xj}_{\mathrm{j}}^{\mathrm{ej}\ t\hbox{'}}\kern1.75em \forall j\in {J}^{\prime },t\in T $$

(123)