A Dynamic Sustainable Competitive Petroleum Supply Chain Model for   Various Stakeholders with Shared Facilities

Nazanin Moradinasab; Hassan Jafarzadeh; M. R. Amin-Naseri; Cody H.; Fleming

arXiv:1907.11789·math.OC·July 30, 2019

A Dynamic Sustainable Competitive Petroleum Supply Chain Model for Various Stakeholders with Shared Facilities

Nazanin Moradinasab, Hassan Jafarzadeh, M. R. Amin-Naseri, Cody H., Fleming

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TL;DR

This paper introduces a multi-objective, multi-level dynamic model for the petroleum supply chain that considers various stakeholders, aiming to optimize profit, job creation, and environmental impact through detailed planning.

Contribution

It presents a novel MILP model for a dynamic, sustainable petroleum supply chain considering multiple stakeholders and objectives, validated with real data and sensitivity analysis.

Findings

01

Sensitivity analysis identified key cost factors affecting outcomes.

02

The model effectively balances economic and environmental objectives.

03

Application demonstrated using real petroleum supply chain data.

Abstract

Petroleum industry is the world's biggest energy source, and its associated industries such as production, distribution, refining and retail are considered as the largest ones in the world. Having the increasing price and governments job creation and international environmental policies, the petroleum companies try to maximize the number of created job, and their profit and minimize the air pollution simultaneously. To meet these objectives, an effective detailed and precise planning is needed. On the other hand, the dynamic environment and the presence of various stakeholders add to the complexity of planning and design of petroleum supply chain. Therefore, the multi-period, multi-objective, multi-level and multi-product dynamic sustainable competitive petroleum supply chain (DSCPSC) model taking into consideration the various stakeholders have been proposed in this paper. The proposed…

Tables25

Table 1. Table I: Papers classified according to supply chain, decision levels and problem types

	Sector			Decision level			Supply chain decisions
Author (year)	Upstream	Midstream	Downstream	Strategic	Tactical	Operational	Location allocation		Capacity expansion		Capacity reduction		Inventory	Raw material Preparation	Production	Routing	Transportation modes	Multi- Stakeholders	Sharing Facilities	Multi- Objective
							Facility	Route	Facility	Route	Facility	Route
Sear 1993[7]
Escudero, Quintana et al. 1999[8]
Dempster, Pedron et al. 2000 [9]
Pinto, Joly et al. 2000 [10]
Neiro and Pinto 2006 [11]
Mendez, Grossmann et al. 2006[12]
Pongsakdi, Rangsunvigit et al. 2006[13]
Al-Othman, Lababidi et al. 2008 [14]
Kuo and Chang 2008[15]
Pitty, Li et al. 2008 [16]
Kim, Yun et al. 2008[17]
Al-Qahtani and Elkamel 2008[18]
MirHassani 2008 [19]
Guyonnet, Grant et al. 2008[20]
Rocha, Grossmann et al. 2009[21]
Al-Qahtani and Elkamel 2011[22]
Ribas, Hamacher et al. 2010[1]
Gill 2011[23]
Khosrojerdi, Hadizadeh et al. 2012[4]
Chen, Grossmann et al. 2012[24]
Fernandes, Relvas et al. 2013[25]
Nasab and Amin-Naseri 2016[5]
Moradinasab et al. 2018[3]
Nasab et al. 2018 [6]

Table 2. Table II: Optimal values of the objective function for the whole supply chain and separate stakeholders

Beneficiary	Objective function	Nash game value
Stakeholder 1	$P_{e}$	1.08E+15
	$P u l_{e}$	5.19926E+12
	$S_{e}$	21570
Stakeholder 2	$P_{e}$	7.23E+14
	$P u l_{e}$	3.48061E+12
	$S_{e}$	17989
Stakeholder 3	$P_{e}$	8.82E+14
	$P u l_{e}$	4.24606E+12
	$S_{e}$	19173
$λ$		0.60105

Table 3. Table III: Sensitivity analysis of changes in the objective function coefficients

Parameter	change in parameter%	change in objective function value% (Stakeholder 1)	change in objective function value% (Stakeholder 2)	change in objective function value% (Stakeholder 3)	Parameter	change in parameter%	change in objective function value% (Stakeholder 1)	change in objective function value% (Stakeholder 2)	change in objective function value% (Stakeholder 3)
Pipeline transport costs	30	0.31	0.62	0.58	Costs of other transportation modes except pipeline	30	0	1.26	0
	20	0.20	0.40	0.39		20	0	0.84	0
	10	0.10	0.20	0.19		10	0	0.42	0
	-10	-0.10	-0.20	-0.19		-10	0	-0.42	0
	-20	-0.20	-0.40	-0.39		-20	0	-0.84	0
	-30	-0.31	-0.60	-0.56		-30	0	-1.26	0
Facilities’ expansion costs	30	0	0	0	Inventory costs	30	0.10	0.41	0.13
	20	0	0	0		20	0.07	0.27	0.09
	10	0	0	0		10	0.03	0.13	0.04
	-10	0	0	0		-10	-0.06	-0.13	-0.01
	-20	0	0	0		-20	-0.12	-0.28	-0.04
	-30	-0.09	0.15	0.02		-30	-0.21	-0.47	0
Variable costs	30	2.66	4.18	4.02	Routs’ expansion and installation costs	30	0.11	0.10	0.16
	20	1.80	2.69	2.67		20	0.08	0.06	0.11
	10	0.90	1.31	1.35		10	0.04	0.03	0.06
	-10	-0.92	-1.26	-1.32		-10	-0.04	-0.03	-0.05
	-20	-1.82	-2.53	-2.65		-20	-0.07	-0.06	-0.11
	-30	-2.64	-3.95	-4.02		-30	-0.11	-0.10	-0.16
Facilities’ installation costs	30	1.41	1.08	1.09	Fixed cost	30	0.03	0.06	0.06
	20	0.94	0.72	0.73		20	0.02	0.04	0.04
	10	0.47	0.36	0.37		10	0.01	0.02	0.02
	-10	-0.47	-0.35	-0.36		-10	-0.01	-0.02	-0.02
	-20	-0.94	-0.72	-0.73		-20	-0.02	-0.04	-0.03
	-30	-1.41	-1.07	-1.08		-30	-0.11	-0.10	-0.16

Table 4. A-1: Sets

$K$	Set of existing refineries
$\overset{´}{K}$	Set of new refineries
$L$	Set of existing DCs
$\overset{´}{L}$	Set of the new DCs
$M$	Set of costumer zone
$V$	Set of transportation modes
$L C V$	Set of capacity of transportation modes
$P$	Set of products
$E K$	Set of capacity of new refineries
$E L$	Set of capacity of new DCs
$U K$	Set of capacity expansion of existing refineries
$U L$	Set of capacity expansion of existing DCs
$E V$	Set of capacity expansion of existing pipelines
$E Z$	Set of capacity of storage tanks in DC
$L V$	Set of capacity of new pipeline
$R V$	Set of new pipeline routes
$E$	Set of stakeholders including government and private sectors
$E N$	Set of regions EN
$L E V$	Set of education levels LEV
$T$	Set of Time of periods

Table 5. A-2: Indices

$k$	Refinery
$l$	DC
$m$	Costumer zone
$v$	Transportation mode
$l c v$	Capacity of transportation mode
$p$	Product
$e k$	Capacity of new refinery
$e l$	Capacity of new DC
$u k$	Capacity expansion of existing refinery
$u l$	Capacity expansion of existing DC
$e v$	Capacity expansion of existing pipeline
$e z$	Capacity of the storage tank in DC
$l v$	Capacity of new pipeline
$r v$	New pipeline route
$e$	Beneficiaries including government and private sectors
$e n$	regions
$l e v$	Education levels
$P S$	Private sectors
$G$	Government
$t$	Time of periods

Table 6. A-3: Parameters

$d_{p}^{t, e}$	Demand for product p within time period t supplied by stakeholder e ( $e \in E, t \in T, p \in P$ )
$d_{p, m}^{t}$	Demand of customer m for product p within time period t ( $m \in M, t \in T, p \in P$ )
$c a p l_{l}^{u l}$	Capacity expansion for existing distribution center 1 in level ul ( $l \in L, u l \in L$ )
$c a p k_{k}^{u k}$	Capacity expansion for existing refinery k in level uk ( $k \in K, u k \in U K$ )
$i c k_{k}$	Initial capacity of existing refinery k ( $k \in K$ )
$i c l_{p, l}$	Initial capacity of existing distribution center 1 ( $l \in L, p \in P$ )

Table 7. A-3: Parameters (Continued)

$M k_{\overset{´}{k}}^{e k}$	Minimum rate of capacity utilization coefficient of new refinery k in level ek ( $e k \in E K, k \in \overset{´}{K}$ )
$M l_{p, \overset{´}{l}}^{e z}$	Minimum rate of capacity utilization coefficient of new distribution center l for product p in level el ( $e z \in E Z, p \in P, l \in \overset{´}{L}$ )
$i v k_{k}^{0}$	Initial inventory level in refinery k ( $k \in (\overset{´}{K} \cup K)$ )
$i v l_{l}^{0}$	Initial inventory level in distribution center l ( $l \in (\overset{´}{L} \cup L)$ )
$λ E_{e n}$	Coefficient of pollution significance in area en ( $e n \in E N$ )
$N c t_{\overset{´}{l}}^{e z}$	Capacity of storage tank in level ez in new distribution center l ( $l \in \overset{´}{L}, e z \in E Z$ )
$N c l_{l}^{e l}$	Capacity of new distribution center l in level el ( $l \in \overset{´}{L}, e l \in E L$ )
$N c k_{\overset{´}{k}}^{e k l}$	Capacity of new refinery k in level ek ( $k \in \overset{´}{K}, e k \in E K$ )
$i c a p k l_{k, l}$	Initial capacity of the pipeline between refinery k and distribution center ( $k \in K, l \in L$ )
$i c a p l p l_{l p, l}$	Initial capacity of the pipeline between distribution center l and distribution center ( $l p, l \in L, l p \neq l$ )
$t r c_{v}^{l c v}$	Capacity of transportation mode v in level lcv ( $v \in V, l c v \neq L C V$ )
$c a p k l_{k, l}^{e v}$	Capacity expansion of the route between existing refinery k and distribution center l in level ev ( $k \in K, l \in L, e v \in E V$ )
$c a p l p l_{l p, l}^{e v}$	Capacity expansion of the route between existing distribution center lp and existing distribution center l in level ev ( $l p, l \in L, l p \neq l, e v \in E V$ )
$T P P$	Time period
$c l v_{l v}$	Capacity of pipeline transportation mode in level lv ( $l v \in L V$ )
$R k l_{k, l}$	Zero and one matrix indicating the existing routes between existing refinery k and existing distribution center l ( $l \in L, k \in K$ )
$R l p l_{l p, l}$	Zero and one matrix indicating the existing routes between existing distribution center lp and existing distribution center l ( $l, l p \in L, l p \neq l$ )
$n m a x_{v}^{l c v, t}$	Maximum number of available transportation mode v in level lcv in time t ( $l c v \in L C V, v \in V, t \in T$ )
$w_{i}$	Maximum crude oil production from oil field i ( $i \in I$ )
$μ_{p}$	Production rate of product p from one crude oil barrel ( $p \in P$ )
$l k_{k}$	Minimum inventory level of refinery k ( $k \in (K \cup \overset{´}{K}) P$ )
$l l_{l}$	Minimum inventory level of distribution center l ( $l \in (L \cup \overset{´}{L}) P$ )
$M$	A large number
$x c o s t k_{k}^{e k, t}$	Installation cost of new refinery k in level ek during time t ( $t \in T, e k \in E K, k \in \overset{´}{K}) P$ )
$x c o s t l_{l}^{e l, t}$	Installation cost of new distribution center l in level el during time t ( $l \in \overset{´}{L}, t \in T$ )
$u c o s t k_{k}^{u k, t}$	Expansion cost of existing refinery k in level uk during time t ( $t \in T, p \in P, k \in K, u k \in U K) P$ )
$u c o s t l_{l}^{u l, p, t}$	Expansion cost of existing distribution center l in level ul for product p during time t ( $t \in T, p \in P, l \in L, u l \in L) P$ )
$y c o s t k l_{k, l}^{e v, t}$	Expansion cost of pipeline between existing refinery k and existing distribution center l in level lv during time t ( $t \in T, e v \in E V, k \in K, l \in L) P$ )
$y c o s t l p l_{l p, l}^{e v, t}$	Expansion cost of pipeline between existing distribution center lp and existing distribution center l in level lv during time t ( $t \in T, e v \in E V, l p, l \in L, l \neq l p$ )
$h c o s t k_{k}$	Storage cost of crude oil in refinery k during time t ( $t \in T, k \in (K \cup \overset{´}{K})$ )
$h c o s t l_{p, l}$	Storage cost of product p in distribution center l during time t ( $t \in T, p \in P, l \in (L \cup \overset{´}{L})$ )
$q c o s t k l_{k, l}^{t}$	Flow cost of crude oil between refinery k and distribution center l during time t ( $l \in (L \cup \overset{´}{L}), k \in (K \cup \overset{´}{K}), t \in T$ )
$q c o s t l p l_{l p, l}^{t}$	Flow cost of products between existing distribution center lp and existing distribution center l in level lv during time ( $t \in T, l p \in L, l \in (L \cup \overset{´}{L}) O R t \in T, l p \in \overset{´}{L}, l \in \overset{´}{L}$ )

Table 8. A-3: Parameters (Continued)

$r c o s t k l_{k, l, v}^{l v, r v, t}$	Installation cost of new pipeline rv between refinery k and distribution center l in level lv during time t ( $l v \in L V, r v \in R V, l \in (L \cup \overset{´}{L}) O R t \in T, l p \in \overset{´}{L}, l \in \overset{´}{L}$ )
$y c o s t k l_{k, l}^{e v, t}$	Expansion cost of pipeline between existing refinery k and existing distribution center l in level lv during time t ( $t \in T, e v \in E V, k \in K, l \in L$ )
$y c o s t l p l_{l p, l}^{e v, t}$	Expansion cost of pipeline between existing distribution center lp and existing distribution center l in level lv during time t ( $t \in T, e v \in E V, l p, l \in L, l \neq l p$ )
$h c o s t k_{k}$	Storage cost of crude oil in refinery k during time t ( $t \in T, k \in (K \cup \overset{´}{K})$ )
$h c o s t l_{p, l}$	Storage cost of product p in distribution center l during time t ( $t \in T, p \in P, l \in (L \cup \overset{´}{L})$ )
$q c o s t k l_{k, l}^{t}$	Flow cost of crude oil between refinery k and distribution center l during time t ( $t \in T, l \in (L \cup \overset{´}{L}), k \in (K \cup \overset{´}{K})$ )
$q c o s t l p l_{l p, l}^{t}$	Flow cost of products between existing distribution center lp and existing distribution center l in level lv during time ( $t \in T, l p \in L, l \in (L \cup \overset{´}{L}) O R t \in T, l p \in \overset{´}{L}, l \in \overset{´}{L}$ )
$r c o s t k l_{k, l, v}^{l v, r v, t}$	Installation cost of new pipeline rv between refinery k and distribution center l in level lv during time t ( $l v \in L V, r v \in R V, t \in T, l \in (L \cup \overset{´}{L}), k \in (K \cup \overset{´}{K})$ v=pipeline mode)
$r c o s t l p l_{l p, l, p}^{l v, r v, t}$	Installation cost of new pipeline rv between other distribution centers lp and the corresponding distribution center l in level lv during time t ( $l v \in L V, r v \in R V, t \in T, l, l p \in (L \cup \overset{´}{L})$ )
$n c o s t k l_{k, l, v}^{l c v, t}$	Unit cost of transportation mode v between refinery k and distribution center l in capacity level lcv during time t( $l c v \in L C V, v \in V, t \in T, l \in (L \cup \overset{´}{L}), k \in (K \cup \overset{´}{K})$ )
$n c o s t l m_{l, m, v}^{l c v, t}$	customer area m in capacity level lcv during time t ( $l c v \in L C V, v \in V, t \in T, l \in (L \cup \overset{´}{L}), m \in M$ )
$n c o s t l p l_{l p, l, v}^{l c v, t}$	Unit cost of transportation mode v between distribution centers l and lp in capacity level lcv during time t ( $l c v \in L C V, v \in V, t \in T, l, l p \in (L \cup \overset{´}{L})$ )
$i c o s t_{p}^{t}$	Importation cost of product p during time t ( $t \in T, p \in P$ )
$n c o s t l_{l}^{e z, t}$	Installation cost of storage tank with capacity ez in distribution center l during time t ( $t \in T, e z \in E Z, l \in \overset{´}{L}$ )
$p c o s t k_{k}^{t}$	Operation cost of refinery k during time t ( $t \in T, k \in (K \cup \overset{´}{K})$ )
$p c o s t l_{p, l}^{t}$	Operation cost of product p in distribution center l during time t ( $t \in T, l \in (L \cup \overset{´}{L}), p \in P$ )
$F c o s t k_{k}^{t}$	Fixed operation cost of refinery k during time t ( $t \in T, k \in (K \cup \overset{´}{K})$ )
$F c o s t l_{l}^{t}$	Fixed operation cost of distribution center l during time t ( $t \in T, l \in (L \cup \overset{´}{L})$ )
$O P^{t}$	Crude oil price during time t( $t \in T$ )
$E R P P_{p}^{t, e}$	Exportation cost of product p during time t by stakeholder e ( $t \in T, p \in P$ )
$W C o s t_{l e v}^{t}$	Labor force cost with education level lev during time t ( $t \in T$ )
$c l c o s t k_{k}^{t}$	Closing down cost of refinery during time t ( $t \in T, k \in K$ )
$N k_{e n}^{k}$	Zero and one matrix indicating area en in which candidate refinery k has been constructed ( $e n \in E N, k \in \overset{´}{K}$ )
$N e k_{e n}^{k}$	Zero and one matrix indicating area en in which existing refinery k has been expanded ( $e n \in E N, k \in K$ )
$N l_{e n}^{l}$	Zero and one matrix indicating area en where candidate distribution center l has been constructed ( $e n \in E N, l \in \overset{´}{L}$ )
$N e l_{e n}^{l}$	Zero and one matrix indicating area en, in which existing distribution center l has been constructed ( $e n \in E N, l \in \overset{´}{L}$ )

Table 9. A-3: Parameters (Continued)

$P u l k$	Pollution per barrel of crude oil refined
$P u l l$	Pollution in distribution centers
$P u l v_{l c v}^{v}$	Pollution per kilometer of transportation mode v with capacity level lcv ( $l c v \in L C V, v \in V$ )
$P e r$	Pollution coefficients for capacity expansion of refinery and distribution center
$d i s_{m}^{l}$	Distance between distribution center l and customer area m ( $m \in M, l \in (L \cup \overset{´}{L})$ )
$W_{e n}^{\overset{´}{e n}}$	Coefficient of significance of labor force transfer from area en to area ( $e n \in E N, e n \in \overset{´}{E N}$ )
$N l a b_{e n}^{l e v}$	New labor force with education level lev in area en ( $e n \in E N, l e v \in L E V$ )
$W N K_{l e v}$	Required labor force with education level lev for a new refinery( $l e v \in L E V$ )
$W E K_{l e v}$	Required labor force with education level lev for capacity expansion of existing refinery ( $l e v \in L E V$ )
$W N L_{l e v}$	Required labor force with education level lev for a new distribution center ( $l e v \in L E V$ )
$W E L_{l e v}$	Required labor force with education level lev for capacity expansion of existing distribution center ( $l e v \in L E V$ )
$M a x n u m_{e n}$	Maximum construction and expansion of refineries and distribution centers in each area ( $e n \in E N$ )

Table 10. A-5: Positive integer variables

$n k l_{p, k, l, v}^{l c v, t, e}$	Number of fleet of transportation mode v at capacity level lcv between refinery k and DC l and by stakeholder e during t ( $t \in T, l c v \in L C V, e \in E, v \in V, k \in (K \cup \overset{´}{K}), l \in (L \cup \overset{´}{L})$ )
$n l p l_{p, l p, l, v}^{l c v, t, e}$	Number of fleet of transportation mode v at capacity level lcv between DC lp and DC l by stakeholder e during t ( $t \in T, l c v \in L C V, e \in E, v \in V, p \in P, l, l p \in (L \cup \overset{´}{L})$ )
$n l m_{p, l, m, v}^{l c v, t, e}$	Number of fleet of transportation mode v at capacity level lcv between DC l and customer zone m by stakeholder e during t ( $t \in T, l c v \in L C V, e \in E, v \in V, p \in P, l, l p \in (L \cup \overset{´}{L})$ )
$n_{p, l}^{e z, t, e}$	The number of storage tanks in capacity level ez in product p in DC l in by stakeholder e during t ( $t \in T, e \in E, e z \in E Z, p \in P, l \in (L \cup \overset{´}{L})$ )
$H E N K_{e n, k}^{\overset{´}{e n}, l e v, t}$	The number of labor forces in level lev, which are worked in new refinery k in region en from region in capacity level lcv during t ( $l e v \in L E V, t \in T, e n, \overset{´}{e n} \in E N, k \in \overset{´}{K}$ )
$H E E K_{e n, k}^{\overset{´}{e n}, l e v, t}$	TThe number of labor forces in level lev, which are worked in refinery k in region en from region in capacity level lcv during t ( $t \in T, e n, \overset{´}{e n} \in E N, l e v \in L E V, k \in K$ )
$H E N l_{e n, l}^{\overset{´}{e n}, l e v, t}$	The number of labor forces in level lev, which are worked in DC l in region en from region in capacity level lcv during t ( $t \in T, e n, \overset{´}{e n} \in E N, l e v \in L E V, l \in \overset{´}{L}$ )
$H E N l_{e n, l}^{\overset{´}{e n}, l e v, t}$	The number of labor forces in level lev, which are worked in DC l in region en from region in capacity level lcv during t ( $t \in T, e n, \overset{´}{e n} \in E N, l e v \in L E V, l \in L$ )

Table 11. A-4: Positive variables

$q k l_{p, k, l, v}^{t, e}$	Rate of crude oil stream from refinery k to distribution center l via transportation mode v by stakeholder e during time t ( $t \in T, p \in P, e \in E, v \in V, k \in (K \cup \overset{´}{K}), l \in (L \cup \overset{´}{L})$ )
$q l m_{p, l, m, v}^{t, e}$	Rate of product stream from distribution center l to customer m via transportation mode v by stakeholder e during time t ( $t \in T, p \in P, e \in E, v \in V, k \in (K \cup \overset{´}{K}), m \in M$ )

Table 12. A-4: Positive variables

$q l p l_{p, l p, l, v}^{t, e}$	Rate of product stream from other distribution centers lp to distribution center l via transportation mode v by stakeholder e during time t ( $t \in T, e \in E, v \in V, l, l p \in (L \cup \overset{´}{L})$ )
$i_{p, l}^{t, e}$	Imports of product p from distribution center l by stakeholder e during time t ( $t \in T, p \in P, e \in E, l \in (L \cup \overset{´}{L})$ )
$v l_{p, l}^{t, e}$	Inventory of product p in distribution center l by stakeholder e during time t ( $t \in T, p \in P, e \in E, l \in (L \cup \overset{´}{L})$ )
$v k_{k}^{t, e}$	Inventory in refinery k by stakeholder e during time t ( $t \in T, e \in E, k \in (K \cup \overset{´}{K})$ )
$E p_{p, l}^{t, e}$	Exports of product p from distribution center l by stakeholder e during time t ( $t \in T, p \in P, e \in E, l \in (L \cup \overset{´}{L})$ )
$ξ_{l}^{t, e}$	Percentage of the usage of distribution center l by stakeholder e during time t ( $t \in T, e \in E, l \in L$ )
$R L a b_{e n}^{l e v, t}$	Number or required labor forces in level lev during time t( $t \in T, e n \in E N, l e v \in L E V$ )
$A L a b_{e n}^{l e v, t}$	Available labor force with education level lev in area en during time t ( $t \in T, e n \in E N, l e v \in L E V$ )

Table 13. A-6: Binary variables

$x k_{k}^{e k, t, e} = {\begin{matrix} 1 I f t h e r e f i n e r y k i s i n s t a l l e d i n l e v e l e k \\ d u r i n g t i m e t b y e \\ 0 O t h e r w i s e (t \in T, e k \in E K, e \in E, k \in \overset{´}{K}) \end{matrix}$
$τ l_{p, l}^{u l, t, e} = {\begin{matrix} 1 I f t h e d i s t r i b u t i o n c e n t e r l i s e x p a n d e d \\ i n l e v e l u l d u r i n g t i m e t b y e \\ 0 O t h e r w i s e (t \in T, u l \in U L, e \in E, l \in L, p \in P) \end{matrix}$
$τ k_{k}^{u k, t, e} = {\begin{matrix} 1 I f t h e r e f i n e r y k i s e x p a n d e d i n l e v e l u k \\ d u r i n g t i m e t b y e \\ 0 O t h e r w i s e (t \in T, u k \in U K, e \in E, k \in K) \end{matrix}$
$y k l_{k, l}^{e v, t, e} = {\begin{matrix} 1 I f e x i s t i n g r o u t e b e t w e e n e x i s t i n g r e f i n e r y \\ k a n d D C i s e x p a n d e d w i t h l e v e l e v i n \\ p e r i o d t b y e \\ 0 O t h e r w i s e (t \in T, e v \in E V, e \in E, k \in K, l \in L) \end{matrix}$
$y l p l_{l p, l}^{e v, t, e} = {\begin{matrix} 1 I f e x i s t i n g r o u t e b e t w e e n e x i s t i n g D C \\ l p a n d e x i s t i n g D C i s e x p a n d e d w i t h l e v e l \\ e v i n p e r i o d t b y e \\ 0 O t h e r w i s e (t \in T, e v \in E V, e \in E, l, l p \in L) \end{matrix}$
$z_{l}^{e z, t, e} = {\begin{matrix} 1 I f t h e n u m b e r o f s t o r a g e t a n k s i s f i x e d \\ w i t h l e v e l e z i n n e w l e v e l \\ D C l i n p e r i o d t b y e \\ 0 O t h e r w i s e (t \in T, e z \in E Z, e \in E, \overset{´}{l} \in L) \end{matrix}$
$r k l_{k, l, v}^{l v, r v, t, e} = {\begin{matrix} 1 I f r o u t e r v b e t w e e n r e f i n e r y k a n d D C l \\ i s i n s t a l l e d w i t h l e v e l l v i n p e r i o d t b y e \\ 0 O t h e r w i s e \\ (t \in T, e v \in E V, k \in (K \in \overset{´}{K}), l \in \overset{´}{L} \in L) \end{matrix}$
$r l p l_{l p, l, v}^{l v, r v, t, e} = {\begin{matrix} 1 I f r o u t e r v b e t w e e n D C l p a n d D C l \\ i s i n s t a l l e d w i t h l e v e l l v i n p e r i o d t b y e \\ 0 O t h e r w i s e (t \in T, e v \in E V, l, l p \in (\overset{´}{L} \in L) \end{matrix}$
$r l p {\overset{´}{l}}_{l p, l, v}^{t} = {\begin{matrix} 1 I f r o u t e r v b e t w e e n D C l p a n d D C l \\ i s i n s t a l l e d i n p e r i o d t b y e \\ 0 O t h e r w i s e (t \in T, e \in E, l, l p \in (\overset{´}{L} \in L) \end{matrix}$

Table 14. A-6: Binary variables (Continued)

S_{l ​ p, l}^{t} ​ (A ​ u ​ x ​ i ​ l ​ i ​ a ​ r ​ y ​ v ​ a ​ r ​ i ​ a ​ b ​ l ​ e) = {\begin{matrix} 1 I ​ f ​ t ​ h ​ e ​ r ​ e ​ i ​ s ​ a ​ f ​ l ​ o ​ w ​ f ​ r ​ o ​ m ​ D ​ C \\ l ​ p ​ t ​ o ​ D ​ C ​ l ​ i ​ n ​ p ​ e ​ r ​ i ​ o ​ d ​ t \\ 0 O t h e r w i s e (l, l p \in (\overset{´}{L} \in L) \end{matrix}

ψ ​ k_{k}^{t, e} = {\begin{matrix} 1 I ​ f ​ k ​ i ​ s ​ c ​ l ​ o ​ s ​ e ​ d ​ i ​ n ​ p ​ e ​ r ​ i ​ o ​ d ​ t \\ 0 O ​ t ​ h ​ e ​ r ​ w ​ i ​ s ​ e ​ (t \in T, k \in K, e \in E) \end{matrix}

Table 15. Appendix B: Optimal values in the case study

$x k_{k}^{e k, t}, k \in \overset{´}{k}$ (Binary variable)
ek	k	t	e	Value
1	3	1	3	1
1	2	1	2	1
1	1	1	1	1

$n s u$
t	p	value	t	p	value	t	p	value	t	p	value
1	1	127.83	1	2	48.748	2	1	151.88	2	2	60.114
3	1	183.33	3	2	63.028

$ψ_{k}^{t, e}$
t	e	k	value	t	e	k	value	t	e	k	value
3	3	3	1	3	3	2	1	3	3	1	1
2	2	2	1	2	1	3	1	2	1	1	1
1	1	3	1	1	1	2	1	1	1	1	1

$r k l_{k, l, v}^{l v, r v, t}, k \in K, l \in L$ (Binary variable)
k	l	lv	rv	v	t	e	value
3	3	8	1	3	1	3	1
2	3	8	1	3	1	2	1
2	2	8	1	3	1	2	1
1	2	8	1	3	1	1	1
1	1	8	1	3	1	1	1

$v l_{p, l}^{t}, l \in L$ (1000 barrel per day)
p	l	t	e	value	p	l	t	e	value	p	l	t	e	value	p	l	t	e	value
2	2	1	1	35.192	2	2	2	1	28.559	1	1	2	1	21.98	1	2	1	2	19.294
2	3	2	3	4.9

$q k l_{p, k, l, v}^{t}, l \in \overset{´}{L}, k \in \overset{´}{K}$ (1000 barrel per day)
p	k	l	v	t	e	value	p	k	l	v	t	e	value	p	k	l	v	t	e	value
1	3	3	3	3	3	65.407	1	3	3	3	1	3	65.407	1	2	3	3	2	2	49.1
1	2	2	3	3	2	47.997	1	1	1	3	1	1	43.605	1	1	2	3	3	1	34.55
2	3	3	3	2	3	22.363	1	2	3	3	3	2	17.41	2	2	2	3	2	2	5.64

$q k l_{p, k, l, v}^{t}, l \in L, k \in \overset{´}{K}$ (1000 barrel per day)
p	k	l	v	t	e	value	p	k	l	v	t	e	value	p	k	l	v	t	e	value
1	1	1	3	2	1	200	1	1	1	3	3	1	158.0	1	3	3	3	2	3	150.9
1	2	2	3	1	2	146.0	1	2	2	3	3	2	117.4	1	3	3	3	3	3	117.18
1	1	1	3	1	1	102.9	1	2	2	3	2	2	82.68	1	3	3	3	1	3	61.25
2	3	3	3	3	3	60.04	2	2	3	3	2	2	52.84	2	3	3	3	1	3	50.04
2	1	1	3	1	1	49.95	2	2	3	3	1	2	49.60	2	1	1	3	3	1	45.91
2	3	3	3	2	3	44.22	2	2	3	3	3	2	35.86	2	1	2	3	1	1	35.58
2	2	2	3	2	2	31.52	2	1	1	3	2	1	31.38	2	1	2	3	3	1	24.70

$H E N K_{e n, k}^{\overset{´}{e n}, l e v, t}$
K	en	$\overset{´}{e n}$	lev	t	Value	kk	en	$\overset{´}{e n}$	lev	t	Value	kk	en	$\overset{´}{e n}$	lev	t	Value
3	1	7	1	1	6000	2	4	4	1	1	6000	1	2	6	1	1	60000
3	1	7	2	1	400	2	4	4	2	1	400	1	2	1	2	1	400

$q l m_{p, l, m, v}^{t}, l \in L, m \in \overset{´}{M}$ (1000 barrel per day)
p	l	m	v	t	e	value	p	l	m	v	t	e	value	p	l	m	v	t	e	value
1	1	1	2	3	1	180	1	2	3	1	3	2	165.4	1	3	2	2	3	3	162.
1	3	1	2	2	3	150.	1	1	2	2	1	1	146.5	1	1	3	1	2	1	127.
1	2	3	1	1	2	100		1 3	1	1	1	3	73.25	2	3	2	1	2	3	61.9
2	3	1	1	3	3	55.2	1	3	2	2	1	3	53.41	1	2	3	1	2	2	52.8
1	1	2	2	2	1	50.9	2	3	1	2	1	3	50	2	1	3	1	1	1	49.9
1	3	2	2	2	2	49.1	1	2	1	2	2	2	49.1	2	3	1	1	2	2	44.8
2	3	3	2	3	2	35.8	2	2	1	1	2	1	35.19	1	2	2	2	3	1	34.5
2	1	2	2	3	1	31.7	2	1	3	1	2	1	31.38	2	2	2	2	3	2	28.5
1	2	1	1	1	2	26.7	2	2	1	1	3	1	24.70	1	3	1	1	3	3	20
1	3	3	2	3	2	14.5	2	1	3	2	3	1	14.13	2	3	2	1	3	3	9.65
2	2	3	2	2	2	8.61	2	3	2	1	2	2	8.04	1	3	2	2	3	2	2.86
2	2	2	2	1	1	0.35	2	3	3	2	1	3	0.04

$n l m_{p, l, m, v}^{l c v, t, e}, l \in L, m \in M$ (Number)
l	m	t	lcv	v	p	e	value	l	m	t	lcv	v	p	e	value	l	m	t	lcv	v	p	e	value
2	3	3	1	1	1	2	2	3	3	3	1	2	2	2	1	3	3	3	1	2	1	2	1
3	3	1	1	2	2	3	1	3	2	3	1	2	1	3	1	3	2	3	1	2	1	2	1
3	2	3	1	1	2	3	1	3	2	2	1	2	1	2	1	3	2	2	1	1	2	3	1
3	2	2	1	1	2	2	1	3	2	1	1	2	1	3	1	3	2	1	1	1	2	2	1
3	1	3	1	1	2	3	1	3	1	3	1	1	1	3	1	3	1	3	1	1	1	3	1
3	1	2	1	2	1	3	1	3	1	2	1	1	2	2	1	3	1	1	1	2	2	3	1
3	1	1	1	2	2	3	1	3	1	1	1	1	1	3	1	2	3	2	1	2	2	2	1
2	3	2	1	1	1	2	1	2	3	1	1	1	1	2	1	2	2	3	1	2	2	2	1
2	2	3	1	2	1	1	1	2	2	1	1	2	2	1	1	2	1	3	1	1	2	1	1
2	1	2	1	2	1	2	1	2	1	2	1	2	1	2	1	2	1	1	1	1	1	2	1
1	3	3	1	2	2	1	1	1	3	2	1	1	2	1	1	1	3	2	1	1	1	1	1
1	3	1	1	1	2	1	1	1	2	3	1	2	2	1	1	1	2	2	1	2	1	1	1
1	2	1	1	2	1	1	1	1	1	3	1	2	1	1	1

$A l a b_{e n}^{l e v, t}$
en	lev	t	Value	en	lev	t	Value	en	lev	t	Value
3	1	3	1.2e+5	3	1	2	80000	1	1	2	65158
2	2	3	1.06e+5	2	2	2	70546	8	1	3	64800
8	1	2	43200	4	1	3	41091	3	1	1	40000
5	1	3	38532	2	1	3	36564	1	1	1	35579
2	2	1	35473	8	2	3	30600	6	1	3	30333
1	2	3	27032	5	1	2	25688	3	2	3	29712
4	1	2	25394	2	1	2	22376	8	1	1	21600
8	2	2	20400	6	1	2	20222	3	2	2	19808
1	2	2	17888	7	1	3	16002	4	1	1	15697
2	1	1	14188	5	1	1	12844	4	2	3	11381
9	1	3	10701	7	1	2	10668	9	2	3	10599
8	2	1	10200	6	1	1	10111	3	2	1	9904
5	2	3	9633	1	2	1	9144	6	2	3	7581
4	2	2	7454	9	1	2	7134	9	2	2	7066
5	2	2	6422	7	1	1	5334	6	2	2	5054
7	2	3	3990	4	2	1	3927	9	1	1	3567
9	2	1	3533	5	2	1	3211	7	2	2	2660
6	2	1	2527	7	2	1	1330

Equations336

M a x imi z e P_{e} = R F_{e} - C F_{e}

M a x imi z e P_{e} = R F_{e} - C F_{e}

R F_{e}

R F_{e}

+ t \in T \sum p \in P \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum E p_{p, l}^{t, e} E R P P_{p}^{t, e}

C_{e \forall e \in E}

C_{e \forall e \in E}

+ t \in T \sum u l \in U L \sum p \in P \sum l \in L_{e} \sum u cos t l_{l}^{u l, p, t} τ l_{p, l}^{u l, t, e} + t \in T \sum u k \in U K \sum k \in K_{e} \sum c l cos t k_{k}^{t} ψ k_{k}^{t, e} + t \in T \sum l v \in L V \sum k \in (K_{e} \cup \overset{ˊ}{K_{e}}) \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum

v = 3 \sum r cos t k l_{k, l, v}^{l v, r v, t} r_{k, l, v}^{l v, r v, t, e} + t \in T \sum l p \in L_{e} \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum l v \in L V \sum v = 3 \sum r v \in R V \sum r cos tl p l_{l p, l, p}^{l v, r v, t} r l p l_{l p, l, v}^{l v, r v, t, e} + t \in T \sum l p \in \overset{ˊ}{L_{e}} \sum l \in \overset{ˊ}{L_{e}} \sum

l v \in L V \sum v = 3 \sum r v \in R V \sum r cos tl p l_{l p, l, p}^{l v, r v, t} r l p l_{l p, l, v}^{l v, r v, t, e} + t \in T \sum l \in L_{e} \sum k \in K_{e} \sum e v \in E V \sum y cos t k l_{k, l}^{e v, t} y k l_{k, l}^{e v, t, e} + t \in T \sum l \in L_{e} \sum l p \in L_{e} \sum e v \in E V \sum

y cos tl p l_{l p, l}^{e v, t} y l p l_{l p, l}^{e v, t, e} + t \in T \sum l \in \overset{ˊ}{L_{e}} \sum p \in P \sum ez \in E Z \sum n cos t_{l}^{ez, t} n_{p, l}^{ez, t, e} + t \in T \sum l \in L_{e} \sum p \in P \sum F cos t l_{l}^{t} (ξ_{l}^{t, e} i c l_{p, l} + u l \in U L \sum c a p l_{l}^{u l}

τ l_{p, l}^{u l, t, e}) + t \in T \sum k \in K_{e} \sum F cos t k_{k}^{t} (i c k_{k} (1 - \overset{ˊ}{t} ⩽ t \in T \sum ψ k_{k}^{\overset{ˊ}{t}, e}) + u k \in U K \sum c a p k_{k}^{u k} τ k_{k}^{u k, t, e}) + t \in T \sum k \in \overset{ˊ}{K_{e}} \sum e k \in E K \sum F cos t k_{k}^{t}

x k_{k}^{e k, t, e} c_{k}^{e k} + t \in T \sum l \in \overset{ˊ}{L_{e}} \sum ez \in E Z \sum p \in P \sum F cos t l_{l}^{t} N c t_{t}^{ez} n_{p, l}^{ez, l, e} + t \in T \sum l \in \overset{ˊ}{L_{e}} \sum v \in V \sum p \in P \sum (k \in (K_{e} \cup \overset{ˊ}{K_{e}}) \sum q k l_{p, k, l, v}^{t, e} +

l p \in \L_{e} \sum q l p l_{p, l p, l, v}^{t, e} - l p \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum q l p l_{p, l, l p, v}^{t, e}) p cos t l_{p, l}^{t} + t \in T \sum l \in \overset{ˊ}{L_{e}} \sum v \in V \sum p \in P \sum (k \in (K_{e} \cup \overset{ˊ}{K_{e}}) \sum q k l_{p, k, l, v}^{t, e} + l p \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum

q l p l_{p, l p, l, v}^{t, e} - l p \in \overset{ˊ}{L_{e}} \sum q l p l_{p, l, l p, v}^{t, e}) p cos t l_{p, l}^{t} + t \in T \sum k \in (K_{e} \cup \overset{ˊ}{K_{e}}) \sum O P^{t} h cos t k_{k} v k_{k}^{t, e} + p \in P \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}, e \in E_{P S}) \sum p r^{e, p, t}

h cos t l_{p, l} v l_{p, l}^{t, e} + t \in T \sum k \in (K_{e} \cup \overset{ˊ}{K_{e}}) \sum v \in V \sum p \in P \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum q k l_{p, k, l, v}^{t, e} q cos t k l_{k, l}^{t} + t \in T \sum p \in P \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum l p \in \overset{ˊ}{L_{e}} \sum q l p l_{p, l p, l, v}^{t, e}

q cos tl p l_{l p, l}^{t} + t \in T \sum p \in P \sum l \in \overset{ˊ}{L_{e}} \sum l p \in \overset{ˊ}{L_{e}} \sum q l p l_{p, l p, l, v}^{t, e} q cos tl p l_{l p, l}^{t} + t \in T \sum v \in V \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum k \in (K_{e} \cup \overset{ˊ}{K_{e}}) \sum l c v \in L C V \sum n cos t k l_{k, l, v}^{l c v, t}

nk l_{p, k, l, v}^{l c v, t, e} + t \in T \sum v \in V \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum m \in M \sum p \in P \sum l c v \in L C V \sum n cos tl m_{l, m, v}^{l c v, t} n l m_{p, l, m, v}^{l c v, t, e} + t \in T \sum v \in V \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum l p \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum

l c v \in L C V \sum p \in P \sum n cos tl p l_{l, l p, v}^{l c v, t} n l p l_{p, l p, l, v}^{l c v, t, e} + t \in T \sum p \in P \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum i_{p, l}^{t, e} i cos t_{p}^{t} + t \in T \sum l e v \in L E V \sum W C os t_{l e v}^{t} [k \in \overset{ˊ}{K_{e}} \sum \overset{e n}{ˊ} \in E N \sum

e n \in E N \sum H E N K_{e n, k}^{\overset{e n}{ˊ}, l e v, t} + k \in K_{e} \sum \overset{e n}{ˊ} \in E N \sum e n \in E N \sum H E E K_{e n, k}^{\overset{e n}{ˊ}, l e v, t} + l \in \overset{ˊ}{L_{e}} \sum \overset{e n}{ˊ} \in E N \sum e n \in E N \sum H E N L_{e n, l}^{\overset{e n}{ˊ}, l e v, t} + l \in L_{e} \sum

\overset{e n}{ˊ} \in E N \sum e n \in E N \sum H E E L_{e n, l}^{\overset{e n}{ˊ}, l e v, t}]

M inimi z e P u l_{e}

M inimi z e P u l_{e}

x k_{k}^{e k, t, e} P u l k + t \in T \sum l \in \overset{ˊ}{L_{e}} \sum e n \in E N \sum ez \in E Z \sum p \in P \sum

λ E_{e n} N l_{e n}^{l} c_{l}^{ez} n_{p, l}^{ez, t} P u l l + t \in T \sum e n \in E N \sum u k \in U K \sum

k \in K_{e} \sum (λ E_{e n} N e k_{e n}^{k} c_{k}^{u k} τ_{k}^{u k, t, e} P u l k) P er +

t \in T \sum e n \in E N \sum p \in P \sum u l \in U L \sum l \in L_{e} \sum (λ E_{e n} N e l_{e n}^{l} c_{l}^{u l}

τ_{p, l}^{u l, t, e} P u l l) P er + t \in T \sum l \in (L_{e} \cup \overset{ˊ}{L_{e}}) \sum m \in M \sum

l c v \in L C V \sum v \in V \sum P u l v_{l c v}^{v} n l m_{p, l, m, v}^{l c v, t, e} d i s_{m}^{l}

M a x imi z e

M a x imi z e

l e v \in L E V \sum k \in \overset{ˊ}{K_{e}} \sum e n \in E N \sum \overset{e n}{ˊ} \in E N \sum N k_{e n}^{k} W_{e n}^{\overset{e n}{ˊ}}

H E N K_{e n, k}^{\overset{e n}{ˊ}, l e v} + l e v \in L E V \sum k \in K_{e} \sum e n \in E N \sum \overset{e n}{ˊ} \in E N \sum

N e k_{e n}^{k} W_{e n}^{\overset{e n}{ˊ}} H E E K_{e n, k}^{\overset{e n}{ˊ}, l e v} + l e v \in L E V \sum l \in \overset{ˊ}{L_{e}} \sum

e n \in E N \sum \overset{e n}{ˊ} \in E N \sum N l_{e n}^{l} W_{e n}^{\overset{e n}{ˊ}} H E N l_{e n, l}^{\overset{e n}{ˊ}, l e v} +

l e v \in L E V \sum l \in L_{e} \sum e n \in E N \sum \overset{e n}{ˊ} \in E N \sum N e l_{e n}^{l} W_{e n}^{\overset{e n}{ˊ}}

H E E l_{e n, l}^{\overset{e n}{ˊ}, l e v}

t \in T \sum e l \in E L \sum x l_{l}^{e l, t, e} \leq 1 \forall e \in E, l \in \overset{ˊ}{L_{e}}

t \in T \sum e l \in E L \sum x l_{l}^{e l, t, e} \leq 1 \forall e \in E, l \in \overset{ˊ}{L_{e}}

t \in T \sum e k \in E K \sum x k_{k}^{e k, t, e} \leq 1 \forall e \in E, k \in \overset{ˊ}{K_{e}}

t \in T \sum e k \in E K \sum x k_{k}^{e k, t, e} \leq 1 \forall e \in E, k \in \overset{ˊ}{K_{e}}

t \in T \sum z_{l}^{ez, t, e} \leq 1 \forall ez \in E Z, l \in \overset{ˊ}{L_{e}}, p \in P

t \in T \sum z_{l}^{ez, t, e} \leq 1 \forall ez \in E Z, l \in \overset{ˊ}{L_{e}}, p \in P

M \times z_{l}^{ez, t, e} \geq n_{p, l}^{ez, t, e} \forall ez \in E Z, l \in \overset{ˊ}{L_{e}}, p \in P, e \in E

M \times z_{l}^{ez, t, e} \geq n_{p, l}^{ez, t, e} \forall ez \in E Z, l \in \overset{ˊ}{L_{e}}, p \in P, e \in E

t \in T \sum u k \in U K \sum τ k_{k}^{u k, t, e} \leq 1 \forall k \in K_{e}, e \in E

t \in T \sum u k \in U K \sum τ k_{k}^{u k, t, e} \leq 1 \forall k \in K_{e}, e \in E

t \in T \sum u l \in U L \sum e \in E_{l} \sum τ l_{p, l}^{u l, t, e} \leq 1 \forall l \in L_{e}, p \in P, e \in E_{l}

t \in T \sum u l \in U L \sum e \in E_{l} \sum τ l_{p, l}^{u l, t, e} \leq 1 \forall l \in L_{e}, p \in P, e \in E_{l}

t \in T \sum e v \in E V \sum y k l_{k, l}^{e v, t} \leq 1 \forall l \in L_{e}, k \in K_{e}, e \in E

t \in T \sum e v \in E V \sum y k l_{k, l}^{e v, t} \leq 1 \forall l \in L_{e}, k \in K_{e}, e \in E

t \in T \sum e v \in E V \sum y l p l_{l p, l}^{e v, t} \leq 1 \forall l p & l \in L_{e}, e \in E_{l}

t \in T \sum e v \in E V \sum y l p l_{l p, l}^{e v, t} \leq 1 \forall l p & l \in L_{e}, e \in E_{l}

t \in T \sum ψ k_{k}^{t, e} \leq 1 \forall k \in K_{e}, e \in E

t \in T \sum ψ k_{k}^{t, e} \leq 1 \forall k \in K_{e}, e \in E

ξ_{l}^{e} \leq 1 \forall l \in L_{e}, e \in E, t \in T

ξ_{l}^{e} \leq 1 \forall l \in L_{e}, e \in E, t \in T

e \in E \sum ξ_{l}^{e} \leq 1 \forall l \in L_{e}, t \in T

e \in E \sum ξ_{l}^{e} \leq 1 \forall l \in L_{e}, t \in T

t \in T \sum u k \in U K \sum τ k_{k}^{u k, t, e} \leq M (1 - t \in T \sum ψ k_{k}^{t, e}) \forall k \in K_{e}, e \in E

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TopicsProcess Optimization and Integration · Sustainable Supply Chain Management · Global Energy and Sustainability Research

Full text

A Dynamic Sustainable Competitive Petroleum Supply Chain Model for Various Stakeholders with Shared Facilities

††thanks:

Nazanin Moradinasab1, Hassan Jafarzadeh2, M. R. Amin-Naseri3 and Cody H. Fleming 4 1 Ph.D. Industrial and Systems Engineering, Tarbiat Modares University, Tehran, Iran 2 Graduate Research Assistant, Engineering Systems and Environment, University of Virginia, Charlottesville, VA 3 Associate Professor, Industrial and Systems Engineering, Tarbiat Modares University, Tehran, Iran 4Assistant Professor, Engineering Systems and Environment, University of Virginia, Charlottesville, VA

Abstract

Petroleum industry is the world’s biggest energy source, and its associated industries such as production, distribution, refining and retail are considered as the largest ones in the world. Having the increasing price and government’s job creation and international environmental policies, the petroleum companies try to maximize the number of created job, and their profit and minimize the air pollution simultaneously. To meet these objectives, an effective detailed and precise planning is needed. On the other hand, the dynamic environment and the presence of various stakeholders add to the complexity of planning and design of petroleum supply chain. Therefore, the multi- period, multi-objective, multi-level and multi-product dynamic sustainable competitive petroleum supply chain (DSCPSC) model taking into consideration the various stakeholders have been proposed in this paper. The proposed model is an MILP model and GAMS 24.1.2 software has been used to run it for a part of real petroleum supply chain data. Sensitivity analysis was then performed to determine the sensitivity of the results to the variation of the coefficients in objective function. Sensitivity analysis reveals that the highest variations of the objective function were observed with respect to the variable costs, facility installation costs and pipeline transportation costs.

Index Terms:

Competition, petroleum supply chain, sustainability, Mix integer linear problem.

I Introduction

Petroleum supply chain (PSC) is a large network of facilities, infrastructures and complex processes from crude oil extraction to the delivery of the final product to the final customer. This network consists of downstream, midstream and upstream sectors in a wide geographical level in which there are many PSC network design decisions in strategic, tactical and operational levels. Obviously, the management and planning of such network of decisions is very complicated and difficult. Literature survey shows that much work has been done on PSC, in most of which a part of network design decisions shown in Table I have been studied. As observed, some of the PSC has been investigated by applying simplifying assumptions in most works thereby a general and optimal response has not been achieved. Consideration of all aspects of the problem clearly increases the complexity of the problem and the time required to solve it. Therefore, a balance must be kept between simplifying assumptions and problem complexity. As we know, one of the inevitable factors in PSC is the environmental regulations imposed on the network whereas only Ribas (2010) has considered environmental impacts as a constraint in his model [1]. On the other hand, since PSC is one of the industries requiring high investment, it can consequently lead to a high volume of employment. Therefore, consideration of employment planning in PSC is of special significance for governments. Maximization of employment and minimization of pollution are therefore important and inevitable factors together with profit maximization in PSC. Taking these three objectives into account at the same time forms the sustainable supply chain based on Carter and Rogers’ definition.

According to the Carter and Rogers’ sustainability definition, “the strategic, transparent integration and achievement of an organization’s social, environmental, and economic goals is the systematic coordination of key inter-organizational business processes for improving the long term economic performance of the individual company and its supply chain” [2]. However, only Moradinasab et al. (2018) have considered these objects simultaneously[3].

According to Table I, most researches have concentrated on planning models at the tactical level. Only Khosrojerdi et al. (2012), Nasab and Amin-Naseri (2016) and Nasab et al. (2018) have considered both tactical and strategic levels in their studies[4, 5, 6].

Another important factor is the presence of different stakeholders in PSC. As stated, PSC is a large network of different centers managed by various stakeholders, each looking to maximize their profits. As Table I shows, only Fernandes (2013) and Moradinasab et al. (2018) have imagined different stakeholders in their works while others have considered only one stakeholder in PSC. Moradinasab et al. (2018) have developed a sustainable competitive petroleum supply chain (SCPSC) model to minimize pollution while maximizing the profits and job creation. They have considered a government and private sectors as stakeholders which have non-cooperative competition to determine their refined product prices. They have modelled the integrated petroleum supply chain by taking into account two approaches of capacity expansion including new facilities and pipelines installation

and capacity expansion of the present facilities and pipelines. Integrated and simultaneous consideration of these two approaches is important in PSC[3]. According to Table I, Al-Qahtani and Elkamel (2008) have studied only the capacity expansion of the facilities in PSC [14]. In addition, only 7 references have considered location and allocation of the facilities in the midstream sector (Table I). As observed in the Table, the feature of reduced capacity or close down of facilities has not been considered in any of the researches. However, one of the important features in PSC is the possibility of close down or capacity reduction during different time periods. The major and important reason for this consideration is the high fixed and variable costs of existing facilities compared with new ones. This assumption makes the capacity reduction and close down of some of the existing facilities possible by increasing the PSC capacity via constructing new facilities and expansion of existing facilities during different time periods. Simultaneous consideration of all three approaches of new facilities construction, existing facilities expansion and capacity reduction and/or close down of the existing facilities is referred to as the dynamic design of supply chain [26]. Athough the dynamic design of supply chain is important in the petroleum supply chain supply chain because of the high fixed and variable costs for facilities, it is not investigated in any of the sudies which have done so far.

In this paper, multi- period, multi objective, multi-level and multi-product dynamic sustainable competitive petroleum supply chain (DSCPSC) in the presence of various stakeholders with shared facilities in the three levels of refineries, distribution centers and customers is proposed. Being strategic, the petroleum upstram sector including crude oil fields and storage teminals have not been consieded in this study. Since these sectors are critical and strategic for governments, they are kept under governement control and not privatized (i.e. they are ignored in the competitive petroleum supply chain). The decision variables in this model including location, new facilities and pipelines installation, capacity expansion in the existing facilities and pipelines, close down of facilities, inventory, production, imports, exports, routing and selection of transportation modes in strategic and tactical levels. In the model presented at the next sections the routing part has been defined as the shortest path between defined centers, while to have a more detailed model shortest traveling time and also the congestion of the routes can be added to it as well [27].

In this work all the three aspects including installation, closing down and capacity expansion of the facilities and pipelines are studied at the same time which makes the presented model dynamic. Another aspect of the dynamism in the proposed model is associated with the distribution centers. Distribution centers are included in the shared resources of stakeholders, that means different stakeholders can use the same distribution center within the same time period, but the percentage of using a distribution center by stockholders can change within different time periods. Each stakeholder pays the costs of using a distribution center based on the percentage of using the center. In addition to optimal use of the capacity of a distribution center, one of the advantages of this approach is the minimization of each stakeholder’s expenses.

The optimization model generated for this problem is in MILP form, because it has shown its strength in providing a convenient way to capture most aspects of the different problems. That is why it is utilized in different modeling research fields such as path planning [28], scheduling [29], logistics [30], dynamic control problems [31], chemical processing [32], pricing and inventory Control [33] and etc. Another advantage of MILP models is the existence of powerful solvers in the market such as CPLEX. A downside of MILP is the complexity of the model. Using integer variables makes the model NP-hard and increases the size of solution space exponentially. A technique to overcome this weakness is applying heuristic algorithms along with traditional MILP solving methods. For instance, rapid path planning algorithms such as [34, 35] can be used when a part of the modeling are dealing with finding a shortest path between different facilities.

In this paper, the problem definition is given in the next section. Following the statement of problem assumptions, modeling is dealt with in section 4. Fuzzy planning for solving multi-objective models is presented in section 5. Finally, section 6 gives the numerical results obtained by executing the model and conclusions and suggestions for future researches are provided in section 7.

II Problem Definition

In this work, petroleum supply chain is studied in three levels including refineries, distribution centers, and customer centers in presence of different stakeholders. Having various stakeholders in a network leads to competition between them. In this network, the demand of each customer domain can be met by different stakeholders which in this study are different supply chains. For optimal use of distribution centers, each of them is assumed to be owned by several stakeholders and each stakeholder paying the fixed and variable costs of the distribution center according to the portion of center capacity usage.

Profit maximization is one of the most common objective functions according to the literature survey. The environmental regulations and the significance of the increasing job opportunities make the minimization of the pollution, maximization of profit and job creation important for governments. These have not been paid enough attention in the literature, therefore this study emphasizes more on these objectives. The supply chain in which the objectives of minimizing the pollution, maximizing the job opportunities and maximizing the overall profit are modeled simultaneously, is called sustainable supply chain. Having different stakeholders, the DSCPSC model is used to study the petroleum supply chain in this work. The DSCPSC model optimizes the supply chain decisions for each stakeholder simultaneously, i.e. optimal locations for installing facilities (refineries, distribution centers and pipelines), capacities expansions, the selection between the two installation and capacity expansion policies (refineries, distribution centers and pipelines), close down of the existing facilities (refineries, distribution centers and pipelines), inventory, pipeline route, selection of transportation modes, the amounts of crude oil and refined product and the amounts of import and exports.

In this work, different mode of transportation is available for stakeholders. They transport refined products such as gasoline, gasoil and kerosene from refineries to distribution centers by one of the transportation modes and ultimately sells it to the final customers.

III Problem Assumptions

This section explains the problem assumption in DSCPSC mathematical model. The following assumptions help to convert a real problem to the proposed mathematical model:

The supply chain consists of the three levels of refineries, distribution centers, and final customers. 2. 2.

Customer demands are definite and clear. 3. 3.

The supply chain includes several levels and products and different transportation modes. 4. 4.

The location of the existing refineries, distribution centers and existing pipelines is fixed and may not be changed. 5. 5.

Capacity expansion may be performed in existing refineries, distribution centers and pipelines and not in new facilities and pipelines. 6. 6.

Closing down the facilities may occur in existing refineries, distribution centers and existing pipelines and not in new facilities and pipelines. 7. 7.

The stream of refined products is categorized into three parts in this supply chain. These categories and the corresponding modes of transportation are as follows:

•

Transportation between refineries and distribution centers, which are carried out by roads, rail and pipelines.

•

Transportation between distribution centers, which are carried out by roads, rail and pipelines.

•

Transportation between distribution centers and final customers, which are carried out by roads and rail.

• 8. 8.

New pipelines may be constructed in the first and second transportation categories. 9. 9.

The number and type of storage tanks for each refined product in each distribution center are determined according to the input stream into the distribution center. 10. 10.

The distribution centers may either export or import the refined products. 11. The costs of capacity expansion, construction and closing down an existing facility and the fixed and variable costs for each type of facility are known given its capacity. 11. 11.

The inventory holding costs for each of the facilities in each time period have been considered in this supply chain. 12. 12.

A part of the inventory is kept as the confidence inventory in each facility to prevent shortages. 13. 13.

All possible levels of capacity for new refineries, distribution centers and pipelines are known. 14. 14.

All possible levels of capacity expansion for existing refineries, distribution centers and crude oil pipelines are known. 15. 15.

In addition to economical profit function, two other functions including minimization of pollution and maximization of job creation have also been considered.

IV Problem Modeling

The formulation and mathematical DSCPSC model, an MILP model, is presented in this part. The definitions of the parameters, sets and variables used in the model are given in Appendix A.

IV-A Objective function

DSCPSC model is modeled as a multi-objective linear programming model with three objective functions. The first objective function is profit maximization and the second objective function is the minimizing the amount of pollutions resulting from facilities and transportation modes. The third objective function is maximizing the number of created job. The objective functions are formulated as follows:

IV-A1 Total profit maximization

According to Equation 1, each stakeholder’s profit is obtained from the difference between incomes and expenses. The incomes for each of the stakeholders, obtained by Equation IV-A1, includes those from the sale and export of the products.

[TABLE]

According to Equation IV-A1, which shows each of the stakeholder’s expenses, the first and second relations represent the installation cost of new refineries and distribution centers, respectively. Capacity expansion costs of existing refineries and distribution centers are given by relations 3 and 4. Closing down cost of existing refineries are shown by relation 5 and relations 6-8 show installation costs of new pipelines. Capacity expansion cost of existing pipelines are calculated using relations 9-10. Installation cost of new storage tanks in distribution centers are obtained by relation 11. The facilities constant and variable costs are given by relations 12-17. Relations 18-19 are used to obtain inventory costs of new and existing refineries and distribution centers. The costs of refined products transported between refineries and distribution centers to the final customers (i.e. is done by one of the transportation modes) are determined using relations 20-25. Product importation cost are calculated using relation 26. Finally, employment costs for refineries and distribution centers are obtained using relation 27.

{strip}

[TABLE]

IV-A2 Minimizing the pollution created by facilities and transportation modes

In Equation IV-A2, the first and second terms show the pollution by new refineries and distribution centers, respectively. Parameter $\lambda E_{en}$ is an indication of the sensitivity of an area to the pollution. The amount of pollution by capacity expansion of existing refineries and distribution centers are calculated by the fourth and fifth terms in Equation 4, respectively. Finally, the pollution by transportation modes are obtained by the last two terms in Equation IV-A2.

[TABLE]

IV-A3 Maximization of job creation

Based on Equation IV-A3, created job opportunities by installing new refineries and distribution centers are calculated by the first two terms and those by capacity expansion of the existing refineries and distribution centers are obtained using the second two terms.

[TABLE]

IV-B Model constraints

Several constraints have been considered in this model, as follows:

IV-B1 Network design constraints

[TABLE]

According to inequalities 6 and 7 , at most one facility (refinery or distribution center) at a given capacity level is selected by a stakeholder in each candidate site. The number of storage tanks for each refined product (p) in each distribution center is determined only once in the model, as shown by relations 8-9. Constraints 10 and 11 show that only one capacity expansion with a known capacity level and only by one stakeholder takes place in each refinery or distribution center, respectively. Similarly, considering constraints 12 and 13, for each pipeline from refineries to distribution centers and between distribution centers, respectively, only one capacity expansion with a known capacity level and only by one stakeholder takes place. Constraint 14 shows that a refinery closes down only once during modeling periods. It must be pointed out that, once the refinery was closed down it may not be opened again. As stated in the problem definition section, distribution centers are shared sources and are simultaneously used by several stakeholders. $\xi_{l}^{t,e}$ indicates the share of stakeholder e in the distribution center l during time period t and is less than one for each stakeholder according to constraint 15. Constraint 16 shows that the sum of stakeholders’ shares in a distribution center must be less than one. Constraint 17 shows if a distribution center is decided to be closed down within time periods, it should not be expanded. Input streams to existing refineries and distribution centers are limited to their current capacity and the capacity expansions is performed on them based on equations IV-B1 and IV-B1. According to inequalities IV-B1 and IV-B1, the stream between refineries and distribution centers must fall within a certain range (between the defined minimum and maximum amounts of stream). Therefore, a refinery or distribution center is constructed if the stream from the refineries to the distribution centers is greater than minimum defined stream to each refinery and distribution center. In addition, if and only if a distribution center is constructed, the refined products are stored in it, which is shown by Equation IV-B1. The transportation of refined products between refineries and distribution centers takes place by one of the transportation modes of pipeline, road or rail. Constraints IV-B1-IV-B1 show the streams of refined products and selection of transportation modes between existing refineries and distribution centers. Constraints IV-B1-IV-B1 show the streams of refined products and selection of transportation modes between new refineries and distribution centers. Constraint IV-B1 shows that the sum of the product streams passing through a pipeline must be smaller than its capacity. Relations IV-B1-IV-B1 are similar to constraints IV-B1-IV-B1 and are used to determine the streams of refined products and select the transportation mode between two distribution centers. Similar to constraint IV-B1, constraint IV-B1 shows that the sum of the product streams passing through a pipeline must be smaller than its capacity. The refined products are transported from the distribution centers to customers via road and rail modes, but not pipelines. The selection of transportation mode between distribution centers and customers is determined by relation IV-B1.

Constraint IV-B1 shows transportation mode capacity. The maximum number of available facilities from each transportation mode is shown by equation IV-B1. Constraints IV-B1-IV-B1 show that the stream between a refinery and distribution center exits if and only if new refineries and distribution centers are constructed. According to inequality IV-B1, the sum of storage tank capacities in each distribution center must be lower than its capacity. Constraints IV-B1-IV-B1 show that a new pipeline to or from a new facility is constructed if and only if those new facilities are constructed. Constraint IV-B1 shows the limitation of oil stream input to refineries to the crude oil extracted from oil fields. Constraints IV-B1-IV-B1 have been defined to prevent loop formation. Considering constraint IV-B1, the demand of each area customer for each product must be supplied by all the stakeholders. The amount of refined products supplied to the final customers by each stakeholder must be equal to the demand planned for each stakeholder, which is shown by inequality IV-B1. Equations IV-B1 and IV-B1 show that the amount of input stream to each facility must be equal to the corresponding output. The inventory stored in each facility must be less than the storage capacity of the facility, which is shown by Equations IV-B1-IV-B1. In addition, the inventory stored in each facility must be greater than the defined minimum inventory, which is shown by Equations IV-B1-IV-B1.

IV-B2 Job creation constraints

[TABLE]

According to Equation IV-B2, the number of required labor force in each time period must be smaller than the number of available people. The number of available labor force in a time period is obtained by subtracting the number of required labor force from the number of available labor force in the previous period and adding the number of new standby labor force, which is shown in Equation IV-B2. The number of required labor force in each time period, which is the number of labor force employed in the refineries and distribution centers, is obtained from Equation IV-B2. According to Equations IV-B2-IV-B2, the number of required labor force in each refinery and distribution center must be equal to the number of employees. Equation IV-B3 is referred to as coverage constraint, based on which the sum of constructions, capacity expansions and existing facilities in each area must be less than the minimum defined for that area.

IV-B3 Coverage constraint

[TABLE]

V Fuzzy programming technique for a multi-objective, linear problem

Bellman and Zadeh proposed fuzzy theory for uncertainty management for the first time [36]. The concept of fuzzy theory with appropriate objective functions was used by Zimmermann for solving multi objective, linear programming. This method can consider minimization of some objective functions along with maximization of other objective functions to establish a balance between them [37] . In other words, fuzzy approach can be used to obtain unique responses for multi objective optimization problems. Consider a mathematical programming problem with two objective functions such as Equation V:

[TABLE]

The steps corresponding to fuzzy theory for solving a multi objective problem are as follows:

Step 1: Only one objective function is maximized with all constraints at each time, as follows:

[TABLE]

The results of solving models are called $f^{*}$ and $g^{*}$ .

Step 2: Only one objective function is minimized with all constraint at each time, as follows:

[TABLE]

The results of solving models are called $f^{-}$ and $g^{-}$ .

Step 3: Finally, the objective functions are obtained using Equations 92 and Appendix A: parameters, sets and variables used in the mode based on f(x) and g(x) functions, which are maximization and minimization objective functions, respectively.

[TABLE]

These objective functions must be greater than $\lambda$ . Therefore, the multi objective, linear programming model is modeled using fuzzy programming as follows:

[TABLE]

Multi- period, multi objective, multi-stakeholder, multi-level and multi-product DSCPSC model is converted into a single-objective problem via fuzzy approach using the above linear programming model.

VI Numerical results

As stated, the multi objective, multi-stakeholder, multi-level and multi-product DSCPSC model is an MILP model in which the outputs are flow rates, installation of new facilities, expansion of existing facilities, installation of new pipelines, expansion of existing pipelines, exports, imports, determination of each of the stakeholder’s share of distribution centers, inventory and selection of transportation modes for each of the stakeholders. Three stakeholders have been considered in the model.

Since the proposed model is a multi-objective model, fuzzy programming technique is used to solve and convert it to a single-objective model. Therefore, three objective functions are converted into one and three constraints are added to the model constraints. The model is thus solved using GAMS 24.1.2 software. In order to use fuzzy planning technique, the maximum and minimum values of each function are calculated. The maximum and minimum values are then placed in constraint 94. Therefore, optimal responses are obtained by considering one objective function. Model parameter values are shown in Appendix A.. Model objective function values are shown in Table II. The corresponding costs for each of the stakeholders including pipeline construction and expansion, facility construction and expansion, transportation of products, inventory and fixed and variable costs in each of the equilibria are shown in Figure 1.

As observed, the corresponding pipeline and facility construction and expansion, inventory and fixed and variable costs for the first stakeholder are greater compared with those of the second and third stakeholders. It is also observed that fixed and variable, transportation and facility construction and expansion costs are remarkable in comparison with other costs.

According to Figure 2, the amounts of products transported by the first stakeholder are greater than those by the second and third stakeholders. This verifies the differences between transportation costs in different stakeholders in Figure 1.

Sensitivity analysis is performed for further study. The objective of performing sensitivity analysis is the investigation of analysis results and the analysis of their variations to parameter changes. The variations in results with objective function coefficient changes (cost parameters) have been considered in sensitivity analysis.

Given that parameter changes individually lead to small changes in the objective function in this problem, the parameters are divided into 8 groups and the variations in the objective function with changes in each group of parameters have been studied (Table III). Each of these examples studies the variations of a group of parameters. The comparison of the values of the objective function in each of these examples with the corresponding optimal values is shown in Table III.

The percentage changes of the objective function for each of the first, second and third stakeholders are shown in Figures 3, 4 and 5, respectively.

As seen, with a 30% change in parameters, for all three stakeholders, the highest change in the objective function value is in the 2.5-4% range, which corresponds to the variable cost group. In addition to variable costs, the highest changes in the objective function correspond to the variations in facility construction and transportation costs. The reason for the higher change in the objective function with changes in variable costs is the associated construction and transportation costs, which make up the highest share of costs compared with other costs, according to Figure 1.

VII Conclusions

Multi- period, multi objective, multi-level, multi-stakeholder and multi-product DSCPSC model has been investigated considering the competition between different stakeholders using job creation maximization, profit maximization and pollution minimization objective functions. The proposed model is an MILP model. In addition to different stakeholders, the dynamic design of the supply chain has also been considered. Dynamic design of the supply chain simultaneously considers all three approaches of new facility construction, expansion of existing facilities and capacity reduction and/or facility close down. The other aspect of dynamism is related to the distribution centers in this model. Distribution centers are one of the sources shared between stakeholders. Different stakeholders can use a distribution center within a period of time and pay its expenses according to their use of the distribution center. In addition to the optimal use of the capacity of a distribution center, another advantage of this approach is the minimization of each stakeholder’s expenses. Considering that the model is a multi-objective one, fuzzy approach has been used to convert it to a single-objective model. The proposed model was executed using GAMS 24.1.2 software for a part of the real PSC data. Sensitivity analysis was then performed to determine the sensitivity of the results to the variation of the coefficients of objective function (cost parameters). According to the sensitivity analysis, the highest variations of the objective function were observed with respect to the changes in coefficients of variable, facility construction and transportation costs. The reason for this observation is that these coefficients make up a great part of the whole costs for each of the stakeholders.

Appendix A: parameters, sets and variables used in the mode

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