#  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

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] |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

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:

  1. 1.

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.

$\displaystyle Maximize\;P_{e}=RF_{e}-CF_{e}$ (1)

$\displaystyle RF_{e}$ $\displaystyle=\sum_{t\in T}\sum_{p\in P}\sum_{v\in
V}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{m\in
M}qlm_{p,l,m,v}^{t,e}pr^{e,p,t}$ $\displaystyle+\sum_{t\in T}\sum_{p\in
P}\sum_{l\in(L_{e}\cup\acute{L_{e}})}Ep_{p,l}^{t,e}ERPP_{p}^{t,e}$ (2)

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.

$\displaystyle C_{e\;\forall e\,\in E}$ $\displaystyle=\sum_{t\in
T}\sum_{k\in\acute{K_{e}}}\sum_{ek\in
EK}xcostk_{k}^{ek,t}xk_{k}^{ek,t,e}+\sum_{t\in T}\sum_{el\in
EL}\sum_{l\in\acute{L_{e}}}xcostl_{l}^{el,t}xl_{l}^{el,t,e}+\sum_{t\in
T}\sum_{uk\in UK}\sum_{k\in K_{e}}ucostk_{k}^{uk,t}\tau k_{k}^{uk,t,e}$
$\displaystyle+\sum_{t\in T}\sum_{ul\in UL}\sum_{p\in P}\sum_{l\in
L_{e}}ucostl_{l}^{ul,p,t}\tau l_{p,l}^{ul,t,e}+\sum_{t\in T}\sum_{uk\in
UK}\sum_{k\in K_{e}}clcostk_{k}^{t}\psi k_{k}^{t,e}+\sum_{t\in T}\sum_{lv\in
LV}\sum_{k\in(K_{e}\cup\acute{K_{e}})}\sum_{l\in(L_{e}\cup\acute{L_{e}})}$
$\displaystyle\sum_{v=3}rcostkl_{k,l,v}^{lv,rv,t}r_{k,l,v}^{lv,rv,t,e}+\sum_{t\in
T}\sum_{lp\in L_{e}}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{lv\in
LV}\sum_{v=3}\sum_{rv\in
RV}rcostlpl_{lp,l,p}^{lv,rv,t}rlpl_{lp,l,v}^{lv,rv,t,e}+\sum_{t\in
T}\sum_{lp\in\acute{L_{e}}}\sum_{l\in\acute{L_{e}}}$ $\displaystyle\sum_{lv\in
LV}\sum_{v=3}\sum_{rv\in
RV}rcostlpl_{lp,l,p}^{lv,rv,t}rlpl_{lp,l,v}^{lv,rv,t,e}+\sum_{t\in
T}\sum_{l\in L_{e}}\sum_{k\in K_{e}}\sum_{ev\in
EV}ycostkl_{k,l}^{ev,t}ykl_{k,l}^{ev,t,e}+\sum_{t\in T}\sum_{l\in
L_{e}}\sum_{lp\in L_{e}}\sum_{ev\in EV}$ $\displaystyle
ycostlpl_{lp,l}^{ev,t}ylpl_{lp,l}^{ev,t,e}+\sum_{t\in
T}\sum_{l\in\acute{L_{e}}}\sum_{p\in P}\sum_{ez\in
EZ}ncost_{l}^{ez,t}n_{p,l}^{ez,t,e}+\sum_{t\in T}\sum_{l\in L_{e}}\sum_{p\in
P}Fcostl_{l}^{t}(\xi_{l}^{t,e}icl_{p,l}+\sum_{ul\in UL}capl_{l}^{ul}$
$\displaystyle\tau l_{p,l}^{ul,t,e})+\sum_{t\in T}\sum_{k\in
K_{e}}Fcostk_{k}^{t}(ick_{k}(1-\sum_{\acute{t}\leqslant t\in T}\psi
k_{k}^{\acute{t},e})+\sum_{uk\in UK}capk_{k}^{uk}\tau
k_{k}^{uk,t,e})+\sum_{t\in T}\sum_{k\in\acute{K_{e}}}\sum_{ek\in
EK}Fcostk_{k}^{t}$ $\displaystyle xk_{k}^{ek,t,e}c_{k}^{ek}+\sum_{t\in
T}\sum_{l\in\acute{L_{e}}}\sum_{ez\in EZ}\sum_{p\in
P}Fcostl_{l}^{t}Nct_{t}^{ez}n_{p,l}^{ez,l,e}+\sum_{t\in
T}\sum_{l\in\acute{L_{e}}}\sum_{v\in V}\sum_{p\in
P}(\sum_{k\in(K_{e}\cup\acute{K_{e}})}qkl_{p,k,l,v}^{t,e}+$
$\displaystyle\sum_{lp\in\L_{e}}qlpl_{p,lp,l,v}^{t,e}-\sum_{lp\in(L_{e}\cup\acute{L_{e}})}qlpl_{p,l,lp,v}^{t,e})pcostl_{p,l}^{t}+\sum_{t\in
T}\sum_{l\in\acute{L_{e}}}\sum_{v\in V}\sum_{p\in
P}(\sum_{k\in(K_{e}\cup\acute{K_{e}})}qkl_{p,k,l,v}^{t,e}+\sum_{lp\in(L_{e}\cup\acute{L_{e}})}$
$\displaystyle
qlpl_{p,lp,l,v}^{t,e}-\sum_{lp\in\acute{L_{e}}}qlpl_{p,l,lp,v}^{t,e})pcostl_{p,l}^{t}+\sum_{t\in
T}\sum_{k\in(K_{e}\cup\acute{K_{e}})}OP^{t}hcostk_{k}vk_{k}^{t,e}+\sum_{p\in
P}\sum_{l\in(L_{e}\cup\acute{L_{e}},e\in E_{PS})}pr^{e,p,t}$ $\displaystyle
hcostl_{p,l}vl_{p,l}^{t,e}+\sum_{t\in
T}\sum_{k\in(K_{e}\cup\acute{K_{e}})}\sum_{v\in V}\sum_{p\in
P}\sum_{l\in(L_{e}\cup\acute{L_{e}})}qkl_{p,k,l,v}^{t,e}qcostkl_{k,l}^{t}+\sum_{t\in
T}\sum_{p\in
P}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{lp\in\acute{L_{e}}}qlpl_{p,lp,l,v}^{t,e}$
$\displaystyle qcostlpl_{lp,l}^{t}+\sum_{t\in T}\sum_{p\in
P}\sum_{l\in\acute{L_{e}}}\sum_{lp\in\acute{L_{e}}}qlpl_{p,lp,l,v}^{t,e}qcostlpl_{lp,l}^{t}+\sum_{t\in
T}\sum_{v\in
V}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{k\in(K_{e}\cup\acute{K_{e}})}\sum_{lcv\in
LCV}ncostkl_{k,l,v}^{lcv,t}$ $\displaystyle nkl_{p,k,l,v}^{lcv,t,e}+\sum_{t\in
T}\sum_{v\in V}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{m\in M}\sum_{p\in
P}\sum_{lcv\in LCV}ncostlm_{l,m,v}^{lcv,t}nlm_{p,l,m,v}^{lcv,t,e}+\sum_{t\in
T}\sum_{v\in
V}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{lp\in(L_{e}\cup\acute{L_{e}})}$
$\displaystyle\sum_{lcv\in LCV}\sum_{p\in
P}ncostlpl_{l,lp,v}^{lcv,t}nlpl_{p,lp,l,v}^{lcv,t,e}+\sum_{t\in T}\sum_{p\in
P}\sum_{l\in(L_{e}\cup\acute{L_{e}})}i_{p,l}^{t,e}icost_{p}^{t}+\sum_{t\in
T}\sum_{lev\in LEV}WCost_{lev}^{t}[\sum_{k\in\acute{K_{e}}}\sum_{\acute{en}\in
EN}$ $\displaystyle\sum_{en\in EN}HENK_{en,k}^{\acute{en},lev,t}+\sum_{k\in
K_{e}}\sum_{\acute{en}\in EN}\sum_{en\in
EN}HEEK_{en,k}^{\acute{en},lev,t}+\sum_{l\in\acute{L_{e}}}\sum_{\acute{en}\in
EN}\sum_{en\in EN}HENL_{en,l}^{\acute{en},lev,t}+\sum_{l\in L_{e}}$
$\displaystyle\sum_{\acute{en}\in EN}\sum_{en\in
EN}HEEL_{en,l}^{\acute{en},lev,t}]$

####  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.

$\displaystyle Minimize\;Pul_{e}$ $\displaystyle=\sum_{t\in
T}\sum_{k\in\acute{K_{e}}}\sum_{en\in EN}\sum_{ek\in EK}\lambda
E_{en}Nk_{en}^{k}c_{k}^{ek}$ $\displaystyle xk_{k}^{ek,t,e}Pulk+\sum_{t\in
T}\sum_{l\in\acute{L_{e}}}\sum_{en\in EN}\sum_{ez\in EZ}\sum_{p\in P}$
$\displaystyle\lambda E_{en}Nl_{en}^{l}c_{l}^{ez}n_{p,l}^{ez,t}Pull+\sum_{t\in
T}\sum_{en\in EN}\sum_{uk\in UK}$ $\displaystyle\sum_{k\in K_{e}}(\lambda
E_{en}Nek_{en}^{k}c_{k}^{uk}\tau_{k}^{uk,t,e}Pulk)Per+$
$\displaystyle\sum_{t\in T}\sum_{en\in EN}\sum_{p\in P}\sum_{ul\in
UL}\sum_{l\in L_{e}}(\lambda E_{en}Nel_{en}^{l}c_{l}^{ul}$
$\displaystyle\tau_{p,l}^{ul,t,e}Pull)Per+\sum_{t\in
T}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{m\in M}$ $\displaystyle\sum_{lcv\in
LCV}\sum_{v\in V}Pulv_{lcv}^{v}nlm_{p,l,m,v}^{lcv,t,e}dis_{m}^{l}$

####  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.

$\displaystyle Maximize$ $\displaystyle\;S_{e}=$ $\displaystyle\sum_{lev\in
LEV}\sum_{k\in\acute{K_{e}}}\sum_{en\in EN}\sum_{\acute{en}\in
EN}Nk_{en}^{k}W_{en}^{\acute{en}}$ $\displaystyle
HENK_{en,k}^{\acute{en},lev}+\sum_{lev\in LEV}\sum_{k\in{K_{e}}}\sum_{en\in
EN}\sum_{\acute{en}\in EN}$ $\displaystyle
Nek_{en}^{k}W_{en}^{\acute{en}}HEEK_{en,k}^{\acute{en},lev}+\sum_{lev\in
LEV}\sum_{l\in\acute{L_{e}}}$ $\displaystyle\sum_{en\in EN}\sum_{\acute{en}\in
EN}Nl_{en}^{l}W_{en}^{\acute{en}}HENl_{en,l}^{\acute{en},lev}+$
$\displaystyle\sum_{lev\in LEV}\sum_{l\in{L_{e}}}\sum_{en\in
EN}\sum_{\acute{en}\in EN}Nel_{en}^{l}W_{en}^{\acute{en}}$ $\displaystyle
HEEl_{en,l}^{\acute{en},lev}$

###  IV-B Model constraints

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

####  IV-B1 Network design constraints

$\displaystyle\sum_{t\in T}\sum_{el\in EL}xl_{l}^{el,t,e}\leq
1\;\;\;\;\;\forall e\in E,l\in\acute{L_{e}}$ (6)

$\displaystyle\sum_{t\in T}\sum_{ek\in EK}xk_{k}^{ek,t,e}\leq
1\;\;\;\;\;\forall e\in E,k\in\acute{K_{e}}$ (7)

$\displaystyle\sum_{t\in T}z_{l}^{ez,t,e}\leq 1\;\;\;\;\;\;\;\forall ez\in
EZ,l\in\acute{L_{e}},p\in P$ (8)

$\displaystyle M\times z_{l}^{ez,t,e}\geq n_{p,l}^{ez,t,e}\;\;\;\forall ez\in
EZ,l\in\acute{L_{e}},p\in P,e\in E$ (9)

$\displaystyle\sum_{t\in T}\sum_{uk\in UK}\tau k_{k}^{uk,t,e}\leq
1\;\;\;\;\;\;\;\forall k\in K_{e},e\in E$ (10)

$\displaystyle\sum_{t\in T}\sum_{ul\in UL}\sum_{e\in E_{l}}\tau
l_{p,l}^{ul,t,e}\leq 1\;\;\;\;\;\;\;\forall l\in L_{e},p\in P,e\in E_{l}$ (11)

$\displaystyle\sum_{t\in T}\sum_{ev\in EV}ykl_{k,l}^{ev,t}\leq
1\;\;\;\;\;\;\;\forall l\in L_{e},k\in K_{e},e\in E$ (12)

$\displaystyle\sum_{t\in T}\sum_{ev\in EV}ylpl_{lp,l}^{ev,t}\leq
1\;\;\;\;\;\;\;\forall lp\&l\in L_{e},e\in E_{l}$ (13)

$\displaystyle\sum_{t\in T}\psi k_{k}^{t,e}\leq
1\;\;\;\;\;\;\;\;\;\;\;\;\forall k\in K_{e},e\in E$ (14)

$\displaystyle\xi_{l}^{e}\leq 1\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in L_{e},e\in
E,t\in T$ (15)

$\displaystyle\sum_{e\in E}\xi_{l}^{e}\leq 1\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in L_{e},t\in T$ (16)

$\displaystyle\sum_{t\in T}\sum_{uk\in UK}\tau k_{k}^{uk,t,e}\leq
M(1-\sum_{t\in T}\psi k_{k}^{t,e})\;\;\;\forall k\in K_{e},e\in E$ (17)

$\displaystyle\sum_{p\in P}\sum_{v\in
V}\sum_{l\in(L_{e}\cup\acute{L_{e}})}qkl_{p,k,l,v}^{t,e}(1-\mu_{p})+\nu
k_{k}^{t-1,e}\leq ick_{k}$ $\displaystyle(1-\sum_{\acute{t}\leq t\in T}\psi
k_{k}^{t,e})+\sum_{\acute{t}\leq t\in T}\sum_{uk\in UK}capk_{k}^{uk}\tau
k_{k}^{uk,\acute{t},e}\;\;\;\;\;\;\;\;$ $\displaystyle\;\;\;\forall k\in
K_{e},e\in E_{k},t\in T$ (18)

$\displaystyle\sum_{e\in E_{l}}\sum_{v\in V}\sum_{lp\in
L_{e}}qlpl_{p,lp,l,v}^{t,e}+\sum_{e\in E_{l}}\sum_{v\in
V}\sum_{k\in(K_{e}\cup\acute{K_{e}})}qkl_{p,k,l,v}^{t,e}$
$\displaystyle-\sum_{e\in E_{l}}\sum_{v\in
V}\sum_{l\in(L_{e}\cup\acute{L_{e}})}qlpl_{p,l,lp,v}^{t,e}+\sum_{e\in
E_{l}}i_{p,l}^{t,e}+\sum_{e\in E_{l}}vl_{p,l}^{t-1,e}$
$\displaystyle\leq\sum_{e\in E_{l}}icl_{p,l}\xi_{l}^{t,e}+\sum_{e\in
E_{l}}\sum_{\acute{t}\leq t\in T}\sum_{ul\in UL}capl_{l}^{ul}\tau
l_{p,l}^{ul,\acute{t},e}\;\;\;\;\;\;\;\;\;\;$ $\displaystyle\;\;\;\forall l\in
L_{e},t\in T$ (19)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{ek\in
EK}Mk_{k}^{ek}Nck_{k}^{ek}xk_{k}^{ek,\acute{t},e}\leq\sum_{v\in V}\sum_{p\in
P}\sum_{l\in(L_{e}\cup\acute{L_{e}})}$ $\displaystyle
qkl_{p,k,l,v}^{t,e}(1\setminus\mu_{p})\leq\sum_{\acute{t}\leq t\in
T}\sum_{ek\in EK}Nck_{k}^{ek}xk_{k}^{ek,\acute{t},e}\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\forall k\in\acute{K_{e}{}},t\in T,e\in E_{k}$ (20)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{ez\in
EZ}Ml_{p,l}^{ez}Nct_{l}^{ez}n_{p,l}^{ez,\acute{t},e}\leq\sum_{v\in
V}\sum_{k\in(K_{e}\cup\acute{K_{e}})}qkl_{p,k,l,v}^{t,e}$
$\displaystyle+i_{l}^{p,t,e}+\sum_{v\in
V}\sum_{lp\in(L_{e}\cup\acute{L_{e}})}qlpl_{p,lp,l,v}^{t,e}-\sum_{v\in
V}\sum_{lp\in\acute{L_{e}}}qlpl_{p,l,lp,v}^{t,e}$
$\displaystyle\leq\sum_{\acute{t}\leq t\in T}\sum_{ez\in
EZ}Nct_{l}^{ez}n_{p,l}^{ez,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\forall l\in\acute{L_{e}{}},e\in E_{l},t\in T,p\in P$ (21)

$\displaystyle\sum_{ez\in EZ}n_{p,l}^{ez,t,e}\leq M\sum_{el\in
EL}x_{l}^{el,t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in\acute{L_{e}{}},e\in E_{l},t\in T,p\in P$ (22)

$\displaystyle qkl_{p,k,l,v}^{t,e}\leq\sum_{lcv\in
LCV}trc_{v}^{lcv}nkl_{p,k,l,v}^{lcv,t,e}TPP+\sum_{\acute{t}\leq t\in
T}\sum_{lv\in LV}$ $\displaystyle\sum_{rv\in
RV}clv_{lv}rkl_{k,l,v}^{lv,rv,\acute{t},e}+Rkl_{k,l}(icapkl_{k,l}+\sum_{\acute{t}\leq
t\in T}\;\;\;\;\;\;\;\;\;\;\;\;$ $\displaystyle\sum_{ev\in
EV}capkl_{k,l}^{ev}ykl_{k,l}^{ev,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\in L_{e}{},k\in K_{e}{},e\in E_{l},t\in T,p\in P,v\in
V$ (23)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}\sum_{v\in V}rkl_{k,l,v}^{lv,rv,\acute{t},e}+\sum_{v\in
V}rk\acute{l}_{k,l,v}^{t,e}\leq 1\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in L_{e}{},k\in K_{e}{},e\in E,t\in T$ (24)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}\sum_{v\in V}rkl_{k,l,v}^{lv,rv,\acute{t},e}+Rkl_{k,l}\leq
1\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in L_{e}{},k\in K_{e}{},e\in E,t\in T$ (25)

$\displaystyle nkl_{p,k,l,v}^{lcv,t,e}\leq M\sum_{v\in
V}rk\acute{l}_{k,l,v}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in L_{e}{},k\in
K_{e}{},e\in E,t\in T,p\in P,lcv\in LCV$ (26)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{ev\in
EV}ykl_{k,l}^{ev,\acute{t},e}\leq
Rkl_{k,l}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in L_{e}{},k\in K_{e}{},e\in E,t\in T$ (27)

$\displaystyle
rkl_{k,l,v}^{lv,rv,t,e}=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in L_{e}{},k\in K_{e}{},e\in
E,t\in T,p\in P,v\in V$ $\displaystyle v\neq
pipeline\;\;route\;\;transportation\;\;mode$ (28)

$\displaystyle
rk\acute{l}_{k,l,v}^{t,e}=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in L_{e}{},k\in K_{e}{},e\in
E,t\in T,p\in P$ $\displaystyle v=pipeline\;\;route\;\;transportation\;\;mode$
(29)

$\displaystyle qkl_{p,k,l,v}^{t,e}\leq\sum_{lcv\in
LCV}trc_{v}^{lcv}nkl_{p,k,l,v}^{lcv,t,e}TPP+\sum_{\acute{t}\leq t\in
T}\sum_{lv\in LV}$ $\displaystyle\sum_{rv\in
RV}clv_{lv}rkl_{k,l,v}^{lv,rv,\acute{t},e}\;\;\;\;\forall
l\in(\acute{L_{e}{}\cup L_{e}{})},\forall k\in\acute{K_{e}{}},e\in E,$
$\displaystyle t\in T,p\in P,v\in V\;\;\;OR\;\;\;\forall
l\in\acute{L_{e}{}},\forall k\in\acute{K_{e}{}},e\in E$ $\displaystyle,t\in
T,p\in P,v\in V\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (30)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}\sum_{v\in V}rkl_{k,l,v}^{lv,rv,\acute{t},e}+\sum_{v\in
V}rk\acute{l}_{k,l,v}^{t,e}\leq 1\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\in(\acute{L_{e}{}\cup L_{e}{})},\forall
k\in\acute{K_{e}{}},e\in E,t\in T\;\;\;\;\;$ $\displaystyle OR\;\;\forall
l\in\acute{L_{e}{}},\forall k\in K_{e}{},e\in E,t\in T\;\;\;\;\;$ (31)

$\displaystyle nkl_{p,k,l,v}^{lcv,t,e}\leq M\sum_{v\in
V}rk\acute{l}_{k,l,v}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\in(\acute{L_{e}{}\cup L_{e}{})},\forall
k\in\acute{K_{e}{}},e\in E,t\in T,p\in P,lcv\in LCV$ $\displaystyle
OR\;\;\forall l\in\acute{L_{e}{}},\forall k\in K_{e}{},e\in E,t\in T,p\in
P,lcv\in LCV$ (32)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}\sum_{v\in V}rkl_{k,l,v}^{lv,rv,\acute{t},e}\leq
1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\forall l\in(\acute{L_{e}{}\cup
L_{e}{})},\forall k\in\acute{K_{e}{}},e\in E,t\in T$ $\displaystyle
OR\;\;\forall l\in\acute{L_{e}{}},\forall k\in K_{e}{},e\in E,t\in T$ (33)

$\displaystyle
rkl_{k,l,v}^{lv,rv,t,e}=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\in(\acute{L_{e}{}\cup L_{e}{})},\forall
k\in\acute{K_{e}{}},e\in E,t\in T,lv\in LV,rv\in RV,$ $\displaystyle v\in
V,v\neq pipeline\;\;route\;\;transportation\;\;mode$ $\displaystyle
OR\;\;\forall l\in\acute{L_{e}{}},\forall k\in K_{e}{},e\in E,t\in T,lv\in
LV,rv\in RV,$ $\displaystyle v\in V,v\neq
pipeline\;\;route\;\;transportation\;\;mode$ (34)

$\displaystyle
rk\acute{l}_{k,l,v}^{t,e}=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in(\acute{L_{e}{}\cup L_{e}{})},\forall k\in\acute{K_{e}{}},e\in E,t\in T,$
$\displaystyle v=pipeline\;\;route\;\;transportation\;\;mode$ $\displaystyle
OR\;\;\forall l\in\acute{L_{e}{}},\forall k\in K_{e}{},e\in E,t\in T,$
$\displaystyle v=pipeline\;\;route\;\;transportation\;\;mode$ (35)

$\displaystyle\sum_{p\in P}qkl_{p,k,l,v}^{t,e}\leq\sum_{\acute{t}\leq t\in
T}\sum_{lv\in LV}\sum_{rv\in
RV}clv_{lv}rkl_{k,l,v}^{lv,rv,\acute{t},e}+Rkl_{k,l}$
$\displaystyle(icapkl_{k,l}+\sum_{\acute{t}\leq t\in T}\sum_{ev\in
EV}capkl_{k,l}^{ev}ykl_{k,l}^{ev,\acute{t},e})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\in(\acute{L_{e}{}\cup
L_{e}{})},k\in(\acute{K_{e}{}\cup K_{e}{})},e\in E,t\in T,p\in P,v\in V\;\;\;$
(36)

$\displaystyle qlpl_{p,lp,l,v}^{t,e}\leq\sum_{lcv\in
LCV}trc_{v}^{lcv}nlpl_{p,lp,l,v}^{lcv,t,e}TPP+Rlpl_{lp,l}\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle(icaplpl_{lp,l}+\sum_{\acute{t}\leq t\in T}\sum_{ev\in
EV}caplpl_{lp,l}^{ev}ylpl_{lp,l}^{ev,\acute{t},e})+\sum_{\acute{t}\leq t\in
T}\sum_{lv\in LV}$ $\displaystyle\sum_{rv\in
RV}clv_{lv}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}\;\;\;\forall l\in
L_{e}\;\;and\;\;lp\in L_{e},e\in E,t\in T$ (37)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}\sum_{v\in V}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}+\sum_{v\in
V}rlp\acute{l}_{lp,l,v}^{t,e}\leq 1\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\in L_{e},lp\in L_{e}{},e\in E,t\in T\;\;\;\;\;$ (38)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}\sum_{v\in V}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}+Rlpl_{lp,l}\leq
1\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in L_{e}{},lp\in L_{e}{},e\in E,t\in T$ (39)

$\displaystyle nlpl_{p,lp,l,v}^{lcv,t,e}\leq M\sum_{v\in
V}rlp\acute{l}_{lp,l,v}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in L_{e}{},lp\in
L_{e}{},e\in E,t\in T,p\in P,lcv\in LCV$ (40)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{ev\in
EV}ylpl_{lp,l}^{ev,\acute{t},e}\leq
Rlpl_{lp,l}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in L_{e}{},lp\in L_{e}{},e\in E,t\in T$ (41)

$\displaystyle qlpl_{p,lp,l,v}^{t,e}\leq\sum_{lcv\in
LCV}trc_{v}^{lcv}nlpl_{p,lp,l,v}^{lcv,t,e}TPP+\sum_{\acute{t}\leq t\in
T}\sum_{lv\in LV}$ $\displaystyle\sum_{rv\in
RV}clv_{lv}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\;and\;lp\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in
T,p\in P\;\;\;$ (42)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}\sum_{v\in V}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}+\sum_{v\in
V}rlp\acute{l}_{lp,l,v}^{t,e}\leq 1\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\;and\;lp\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in
T\;\;\;\;\;\;\;\;$ (43)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}\sum_{v\in V}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}\leq
1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\;and\;lp\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in T$ (44)

$\displaystyle nlpl_{p,lp,l,v}^{lcv,t,e}\leq M\sum_{v\in
V}rlp\acute{l}_{lp,l,v}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\;and\;lp\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in
T,p\in P,lcv\in LCV\;\;\;\;$ (45)

$\displaystyle
rlpl_{lp,l,v}^{lv,rv,t,e}=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\;and\;lp\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in T,p\in P,v\in V$
$\displaystyle v\neq pipeline\;\;route\;\;transportation\;\;mode$ (46)

$\displaystyle
rlp\acute{l}_{lp,l,v}^{t,e}=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\;and\;lp\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in T,p\in P$
$\displaystyle v=pipeline\;\;route\;\;transportation\;\;mode$ (47)

$\displaystyle\;\;\;qlpl_{p,lp,l,v}^{t,e}\leq
Rlpl_{lp,l}(icaplpl_{lp,l}+\sum_{\acute{t}\leq t\in T}\sum_{ev\in
EV}caplpl_{lp,l}^{ev}$ $\displaystyle
ylpl_{lp,l}^{ev,\acute{t},e})+\sum_{\acute{t}\leq t\in T}\sum_{lv\in
LV}\sum_{rv\in
RV}clv_{lv}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\forall l\;and\;lp\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in
T$ $\displaystyle v=pipeline\;\;route\;\;transportation$ (48)

$\displaystyle\;\;\;qlm_{p,l,m,v}^{t,e}\leq\sum_{lcv\in
LCV}trc_{v}^{lcv}nlm_{p,l,m,v}^{lcv,t,e}TPP\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\forall l\;and\;lp\in(\acute{L_{e}{}\cup L_{e}{})},e\in
E,t\in T,v\in V,m\in M,p\in P$ (49)

$\displaystyle\;\;\;\sum_{k\in(K_{e}\cup\acute{K_{e}})}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{p\in
P}nkl_{p,k,l,v}^{lcv,t,e}+\sum_{lp\in(L_{e}\cup\acute{L_{e}})}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\;\;\;\;\;\;$
$\displaystyle\sum_{p\in
P}nlpl_{p,lp,l,v}^{lcv,t,e}+\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{p\in
P}\sum_{m\in M}nlm_{p,l,m,v}^{lcv,t,e}\leq nmax^{lcv}$
$\displaystyle\;\;\;\forall e\in E,t\in T,v\in V,lcv\in LCV\;\;\;\;\;\;$ (50)

$\displaystyle qkl_{p,k,l,v}^{t,e}\leq M\sum_{\acute{t}\leq t\in T}\sum_{el\in
EL}xl_{l}^{el,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in\acute{L_{e}{}},k\in(\acute{K_{e}{}\cup K_{e}{})},e\in E,t\in T\;\;\;$
(51)

$\displaystyle\;\;qkl_{p,k,l,v}^{t,e}\leq M\sum_{\acute{t}\leq t\in
K}\sum_{ek\in
EK}xk_{k}^{ek,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
k\in\acute{K_{e}{}},l\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in T\;\;\;$
(52)

$\displaystyle qlm_{p,l,m,v}^{t,e}\leq M\sum_{\acute{t}\leq t\in T}\sum_{el\in
EL}xl_{l}^{el,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in\acute{L_{e}{}},m\in M,e\in E,t\in T\;\;\;$ (53)

$\displaystyle qlpl_{p,lp,l,v}^{t,e}\leq M\sum_{\acute{t}\leq t\in
T}\sum_{el\in
EL}xl_{l}^{el,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in\acute{L_{e}{}},lp\in\acute{L_{e}{}},e\in E,t\in T\;\;\;$ (54)

$\displaystyle qlpl_{p,lp,l,v}^{t,e}\leq M\sum_{\acute{t}\leq t\in
T}\sum_{el\in
EL}xl_{l}^{el,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
l\in\acute{L_{e}{}},lp\in L_{e}{},e\in E,t\in T\;\;\;$ (55)

$\displaystyle\sum_{t\in T}\sum_{ez\in EZ}\sum_{p\in
P}Nct_{l}^{ez}n_{p,l}^{ez,t,e}\leq\sum_{t\in T}\sum_{el\in
EL}Ncl_{l}^{el}xl_{l}^{el,t,e}$ $\displaystyle\forall l\in\acute{L_{e}{}},e\in
E,t\in T\;\;\;$ (56)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}rkl_{k,l,v}^{lv,rv,\acute{t},e}\leq\sum_{\acute{t}\leq t\in K}\sum_{ek\in
EK}xk_{k}^{ek,\acute{t},e}\;\;\;\;\;\;\;\;\;\;\;\;$ $\displaystyle\forall
k\in\acute{K_{e}{}},l\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in T\;\;\;\;\;$
$\displaystyle v=pipeline\;\;route\;\;transportation\;\;mode$ (57)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}rkl_{k,l,v}^{lv,rv,\acute{t},e}\leq\sum_{t\in T}\sum_{el\in
EL}xl_{l}^{el,t,e}\;\;\;\;\;\;\;\;\;\;\;\;$ $\displaystyle\forall
k\in\acute{K_{e}{}},l\in(\acute{L_{e}{}\cup L_{e}{})},e\in E,t\in T\;\;\;\;\;$
$\displaystyle v=pipeline\;\;route\;\;transportation\;\;mode$ (58)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}\leq\sum_{t\in T}\sum_{el\in
EL}xl_{l}^{el,t,e}\;\;\;\;\;\;\;\;\;\;\;\;$ $\displaystyle\forall
l\in\acute{L_{e}{}},lp\in L_{e}{},e\in E,t\in T\;\;\;\;\;$ $\displaystyle
v=pipeline\;\;route\;\;transportation\;\;mode$ (59)

$\displaystyle\sum_{\acute{t}\leq t\in T}\sum_{lv\in LV}\sum_{rv\in
RV}rlpl_{lp,l,v}^{lv,rv,\acute{t},e}\leq\sum_{t\in T}\sum_{el\in
EL}xl_{l}^{el,t,e}\;\;\;\;\;\;\;\;\;\;\;\;$ $\displaystyle\forall
l\in\acute{L_{e}{}},lp\in\acute{L_{e}{}},e\in E,t\in T\;\;\;\;\;$
$\displaystyle v=pipeline\;\;route\;\;transportation\;\;mode$ (60)

$\displaystyle\sum_{k\in(K_{e}\cup\acute{K_{e}})}\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{p\in
P}\sum_{v\in V}\sum_{e\in E}qkl_{p,k,l,v}^{t,e}(1\setminus\mu_{p})\leq
w_{i}\;\;\;\;\;\;$ $\displaystyle\;\;\;\forall i\in I,t\in T\;\;\;\;\;\;$ (61)

$\displaystyle qlpl_{p,lp,l,v}^{t,e}\leq
Ms_{lp,l}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in
L_{e}{},lp\in L_{e}{},e\in E,p\in P,v\in V,t\in T\;\;\;$ (62)

$\displaystyle qlpl_{p,l,lp,v}^{t,e}\leq
Ms_{l,lp}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in
L_{e}{},lp\in L_{e}{},e\in E,p\in P,v\in V,t\in T\;\;\;$ (63)

$\displaystyle s_{l,lp}^{t,e}+s_{lp,l}^{t,e}\leq
1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in
L_{e}{},lp\in L_{e}{},t\in T,e\in E\;\;\;$ (64)

$\displaystyle qlpl_{p,l,lp,v}^{t,e}\leq
Ms_{l,lp}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in
L_{e}{},lp\in L_{e}{},e\in E,p\in P,v\in V,t\in T\;\;\;$ (65)

$\displaystyle qlpl_{p,lp,l,v}^{t,e}\leq
Ms_{lp,lk}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall lk\in
L_{e}{},lp\in L_{e}{},e\in E,p\in P,v\in V,t\in T\;\;\;$ (66)

$\displaystyle qlpl_{p,lk,l,v}^{t,e}\leq
Ms_{lk,l}^{t,e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall lk\in
L_{e}{},l\in L_{e}{},e\in E,p\in P,v\in V,t\in T\;\;\;$ (67)

$\displaystyle s_{l,lp}^{t,e}+s_{lp,lk}^{t,e}+s_{lk,l}^{t,e}\leq
1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall l\in L_{e}{},lp\in
L_{e}{},lk\in L_{e}{},t\in T,e\in E\;\;\;\;\;\;$ (68)

$\displaystyle\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{v\in V}\sum_{e\in
E}qlm_{p,l,m,v}^{t,e}\geq d_{p,m}^{t}\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\forall p\in P,t\in T,m\in M$ (69)

$\displaystyle\sum_{l\in(L_{e}\cup\acute{L_{e}})}\sum_{v\in V}\sum_{m\in
M}qlm_{p,l,m,v}^{t,e}\geq D_{p}^{t,e}\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\forall p\in P,t\in T,e\in E$ (70)

$\displaystyle\sum_{lp\in(L_{e}\cup\acute{L_{e}})}\sum_{v\in
V}qlpl_{p,lp,l,v}^{t,e}-\sum_{l\in\acute{L_{e}}}\sum_{v\in
V}qlpl_{p,l,lp,v}^{t,e}+\sum_{k\in(K_{e}\cup\acute{K_{e}})}$
$\displaystyle\sum_{v\in
V}qkl_{p,k,l,v}^{t,e}+vl_{p,l}^{t-1,e}+i_{p,l}^{t,e}=\sum_{v\in V}\sum_{m\in
M}qlm_{p,l,m,v}^{t,e}\;\;\;\;\;\;\;\;\;$
$\displaystyle+Ep_{p,l}^{t,e}+vl_{p,l}^{t,e}\;\;\;\;\;\;\;\;\;\forall p\in
P,t\in T,e\in E,l\in\acute{L_{e}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (71)

$\displaystyle\sum_{lp\in L_{e}}\sum_{v\in
V}qlpl_{p,lp,l,v}^{t,e}-\sum_{lp\in(L_{e}\cup\acute{L_{e}})}\sum_{v\in
V}qlpl_{p,l,lp,v}^{t,e}+\sum_{k\in(K_{e}\cup\acute{K_{e}})}$
$\displaystyle\sum_{v\in
V}qkl_{p,k,l,v}^{t,e}+vl_{p,l}^{t-1,e}+i_{p,l}^{t,e}=\sum_{v\in V}\sum_{m\in
M}qlm_{p,l,m,v}^{t,e}\;\;\;\;\;\;\;\;\;$
$\displaystyle+Ep_{p,l}^{t,e}+vl_{p,l}^{t,e}\;\;\;\;\;\;\;\;\;\forall p\in
P,t\in T,e\in E,l\in L_{e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (72)

$\displaystyle vk_{k}^{t,e}\leq ick_{k}(1-\sum_{\acute{t}\leq t\in T}\psi
k_{k}^{t,e})+\sum_{\acute{t}\leq t\in T}\sum_{uk\in UK}capk_{k}^{uk}$
$\displaystyle+\tau k_{k}^{uk,\acute{t},e}\;\;\;\;\;\;\;\;\;\forall t\in
T,e\in E,k\in K_{e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (73)

$\displaystyle\;\;\;\sum_{e\in E}vl_{p,l}^{t,e}\leq\sum_{e\in
E}ic_{l}^{p}\xi_{l}^{t,e}+\sum_{\acute{t}\leq t\in T}\sum_{ul\in UL}\sum_{e\in
E}capl_{l}^{ul}$ $\displaystyle+\tau
l_{p,l}^{ul,\acute{t},e}\;\;\;\;\;\;\;\;\;\forall t\in T,p\in P,l\in
L_{e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (74)

$\displaystyle\;\;\;vk_{k}^{t,e}\leq\sum_{\acute{t}\leq t\in T}\sum_{ek\in
EK}Nck_{k}^{ek}xk_{k}^{ek,\acute{t},e}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
t\in T,k\in\acute{K_{e}}$ (75)

$\displaystyle\;\;\;vl_{p,l}^{t,e}\leq\sum_{\acute{t}\leq t\in T}\sum_{ez\in
EZ}Nct_{l}^{ez}n_{p,l}^{ez,\acute{t},e}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\forall t\in T,p\in P,e\in
E,l\in\acute{L_{e}}$ (76)

$\displaystyle vk_{k}^{t,e}\geq lk_{k}(ick_{k}(1-\sum_{\acute{t}\leq t\in
T}\psi k_{k}^{t,e})+\sum_{\acute{t}\leq t\in T}\sum_{uk\in UK}capk_{k}^{uk}$
$\displaystyle+\tau k_{k}^{uk,\acute{t},e})\;\;\;\;\;\;\;\;\;\forall t\in
T,e\in E,k\in K_{e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (77)

$\displaystyle\;\;\;\sum_{e\in E}vl_{p,l}^{t,e}\geq l_{l}(\sum_{e\in
E}ic_{l}^{p}\xi_{l}^{t,e}+\sum_{\acute{t}\leq t\in T}\sum_{ul\in UL}\sum_{e\in
E}capl_{l}^{ul}$ $\displaystyle+\tau
l_{p,l}^{ul,\acute{t},e})\;\;\;\;\;\;\;\;\;\forall t\in T,p\in P,l\in
L_{e}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (78)

$\displaystyle\;\;\;vk_{k}^{t,e}\geq lk_{k}(\sum_{\acute{t}\leq t\in
T}\sum_{ek\in EK}Nck_{k}^{ek}xk_{k}^{ek,\acute{t},e})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
t\in T,e\in E,k\in\acute{K_{e}}$ (79)

$\displaystyle\;\;\;vl_{p,l}^{t,e}\geq l_{l}(\sum_{\acute{t}\leq t\in
T}\sum_{ez\in EZ}Nct_{l}^{ez}n_{p,l}^{ez,\acute{t},e})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\forall t\in T,p\in P,e\in
E,l\in\acute{L_{e}}$ (80)

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

$\displaystyle RLab_{en}^{lev,t}\leq
ALab_{en}^{lev,t}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
t\in T,en\in EN,lev\in LEV$ (81)

$\displaystyle
ALab_{en}^{lev,t}=ALab_{en}^{lev,t-1}-RLab_{en}^{lev,t-1}+NLab_{en}^{lev,t}\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
t\in T,en\in EN,lev\in LEV$ (82)

$\displaystyle\sum_{\acute{en}\in
EN}\sum_{k\in\acute{K_{e}}}HENK_{en,k}^{\acute{en},lev,t}+\sum_{\acute{en}\in
EN}\sum_{k\in\acute{K_{e}}}HEEK_{en,k}^{\acute{en},lev,t}$
$\displaystyle+\sum_{\acute{en}\in
EN}\sum_{l\in\acute{L_{e}}}HENL_{en,l}^{\acute{en},lev,t}+\sum_{\acute{en}\in
EN}\sum_{l\in\acute{L_{e}}}HEEL_{en,l}^{\acute{en},lev,t}$
$\displaystyle=RLab_{en}^{lev,t}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall
t\in T,en\in EN,lev\in LEV$ (83)

$\displaystyle\sum_{ek\in
EK}Nck_{k}^{ek}xk_{k}^{ek,t,e}WNK_{lev}=\sum_{\acute{en}\in EN}\sum_{en\in
EN}Nk_{en}^{k}\;\;\;$ $\displaystyle
HENK_{en,k}^{\acute{en},lev,t}\;\;\;\;\;\;\;\;\forall t\in T,e\in E,lev\in
LEV,k\in\acute{K_{e}}$ (84)

$\displaystyle\sum_{uk\in UK}capk_{k}^{uk}\tau
k_{k}^{uk,t,e}WEK_{lev}=\sum_{\acute{en}\in EN}\sum_{en\in
EN}Nek_{en}^{k}\;\;\;$ $\displaystyle
HEEK_{en,k}^{\acute{en},lev,t}\;\;\;\;\;\;\;\;\forall t\in T,e\in E,lev\in
LEV,k\in K_{e}$ (85)

$\displaystyle\sum_{el\in
EL}Ncl_{l}^{el}xl_{l}^{el,t,e}WNL_{lev}=\sum_{\acute{en}\in EN}\sum_{en\in
EN}Nl_{en}^{l}\;\;\;\;\;\;\;\;\;$ $\displaystyle
W_{en}^{\acute{en}}HENl_{en,l}^{\acute{en},lev,t}\;\;\;\;\;\;\;\;\forall t\in
T,e\in E,lev\in LEV,l\in\acute{L_{e}}$ (86)

$\displaystyle\sum_{ul\in UL}\sum_{p\in P}capl_{l}^{ul}\tau
l_{l}^{ul,t,e}WEL_{lev}=\sum_{\acute{en}\in EN}\sum_{en\in
EN}Nel_{en}^{l}\;\;\;$ $\displaystyle
HEEL_{en,l}^{\acute{en},lev,t}\;\;\;\;\;\;\;\;\forall t\in T,e\in E,lev\in
LEV,k\in K_{e}$ (87)

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

$\displaystyle\sum_{t\in T}\sum_{ek\in EK}\sum_{e\in
E}\sum_{k\in\acute{K_{e}}}xk_{k}^{ek,t,e}Nk_{en}^{k}+\sum_{t\in T}\sum_{uk\in
UK}\sum_{e\in E}\sum_{k\in K_{e}}$ $\displaystyle\tau
k_{k}^{uk,t,e}Nek_{en}^{k}+\sum_{t\in T}\sum_{el\in EL}\sum_{e\in
E}\sum_{l\in\acute{L_{e}}}xl_{l}^{el,t,e}Nl_{en}^{l}+\sum_{t\in T}$
$\displaystyle\sum_{ul\in UL}\sum_{e\in E}\sum_{l\in L_{e}}\tau
l_{l}^{ul,t,e}Nel_{en}^{l}NR_{en}+ND_{en}\leq Maxnum_{en}$
$\displaystyle\;\;\;\;\forall t\in T,en\in EN$ (88)

##  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:

$\displaystyle
max\;\;f(x)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle
min\;\;g(x)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;subject\;\;to:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;h(x)\leq
0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(89)

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:

$\displaystyle
max\;\;f(x)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;max\;\;g(x)\;\;\;\;\;\;\;$
$\displaystyle\;\;subject\;\;to:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;subject\;\;to:\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;h(x)\leq
0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;h(x)\leq 0$
(90)

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:

$\displaystyle
min\;\;f(x)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;min\;\;g(x)\;\;\;\;\;\;\;$
$\displaystyle\;\;subject\;\;to:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;subject\;\;to:\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;h(x)\leq
0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;h(x)\leq 0$
(91)

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.

$\mu(f)=\left\\{\begin{array}[]{ll}0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;f(x)\leq
f^{-}\\\ \frac{f(x)-f^{-}}{f^{*}-f^{-}}\;\;\;\;\;\;\;\;\;\;f^{-}\leq f(x)\leq
f^{*}\\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;f(x)\geq
f^{*}\end{array}\right.$ (92)

$\mu(g)=\left\\{\begin{array}[]{ll}0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;g(x)\leq
g^{-}\\\ \frac{g^{*}-g(x)}{g^{*}-g^{-}}\;\;\;\;\;\;\;\;\;\;g^{-}\leq g(x)\leq
g^{*}\\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;g(x)\geq
g^{*}\end{array}\right.$ (93)

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

$\displaystyle
max\;\;\lambda\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;\;\;subject\;\;to:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;h(x)\leq
0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;f(x)\geq
f^{-}+\lambda(f^{*}-f^{-})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle\;\;\;\;g(x)\leq
g^{*}+\lambda(g^{*}-g^{-})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(94)

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.

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
| $Pul_{e}$ | 5.19926E+12
| $S_{e}$ | 21570
Stakeholder 2 | $P_{e}$ | 7.23E+14
| $Pul_{e}$ | 3.48061E+12
| $S_{e}$ | 17989
Stakeholder 3 | $P_{e}$ | 8.82E+14
| $Pul_{e}$ | 4.24606E+12
| $S_{e}$ | 19173
$\lambda$ | 0.60105
Figure 1: Stakeholders’ costs

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.

Figure 2: Flow of products transported by stakeholders 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

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.

Figure 3: Rate of changes in objective function values of the first
stakeholder

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.

Figure 4: Rate of change in objective function values of the second
stakeholder Figure 5: Rate of change in objective function values of the third
stakeholder

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.

## References

  * [1] G. Ribas, S. Hamacher, and A. Street, “Optimization of the integrated petroleum supply chain considering uncertainties using stochastic, robust and max-min models,” _International Transactions in Operational Research_ , vol. 17, pp. 777–796, 2010.
  * [2] C. R. Carter and D. S. Rogers, “A framework of sustainable supply chain management: moving toward new theory,” _International journal of physical distribution & logistics management_, vol. 38, no. 5, pp. 360–387, 2008\.
  * [3] N. Moradinasab, M. Amin-Naseri, T. J. Behbahani, and H. Jafarzadeh, “Competition and cooperation between supply chains in multi-objective petroleum green supply chain: A game theoretic approach,” _Journal of cleaner production_ , vol. 170, pp. 818–841, 2018.
  * [4] A. Khosrojerdi, A. Hadizadeh, and J. K. Allen, “Designing a dynamic bi-objective network model for a petroleum supply chain,” in _IIE Annual Conference. Proceedings_. Institute of Industrial and Systems Engineers (IISE), 2012, p. 1.
  * [5] N. M. Nasab and M. Amin-Naseri, “Designing an integrated model for a multi-period, multi-echelon and multi-product petroleum supply chain,” _Energy_ , vol. 114, pp. 708–733, 2016.
  * [6] N. M. Nasab, M. Amin-Naseri, and H. Jafarzadeh, “A benders’ decomposition method to solve a multi-period, multi-echelon, and multi-product integrated petroleum supply chain,” _Process Integration and Optimization for Sustainability_ , vol. 2, no. 3, pp. 281–300, 2018.
  * [7] T. Sear, “Logistics planning in the downstream oil industry,” _Journal of the Operational Research Society_ , vol. 44, no. 1, pp. 9–17, 1993.
  * [8] L. F. Escudero, F. J. Quintana, and J. Salmerón, “Coro, a modeling and an algorithmic framework for oil supply, transformation and distribution optimization under uncertainty,” _European Journal of Operational Research_ , vol. 114, no. 3, pp. 638–656, 1999.
  * [9] M. Dempster, N. H. Pedron, E. Medova, J. Scott, and A. Sembos, “Planning logistics operations in the oil industry,” _Journal of the Operational Research Society_ , vol. 51, no. 11, pp. 1271–1288, 2000.
  * [10] J. Pinto, M. Joly, and L. Moro, “Planning and scheduling models for refinery operations,” _Computers & Chemical Engineering_, vol. 24, no. 9-10, pp. 2259–2276, 2000.
  * [11] S. M. Neiro and J. M. Pinto, “Lagrangean decomposition applied to multiperiod planning of petroleum refineries under uncertainty,” _Latin American applied research_ , vol. 36, no. 4, pp. 213–220, 2006.
  * [12] C. A. Mendez, I. E. Grossmann, I. Harjunkoski, and P. Kaboré, “A simultaneous optimization approach for off-line blending and scheduling of oil-refinery operations,” _Computers & chemical engineering_, vol. 30, no. 4, pp. 614–634, 2006.
  * [13] A. Pongsakdi, P. Rangsunvigit, K. Siemanond, and M. J. Bagajewicz, “Financial risk management in the planning of refinery operations,” _International Journal of Production Economics_ , vol. 103, no. 1, pp. 64–86, 2006.
  * [14] W. B. Al-Othman, H. M. Lababidi, I. M. Alatiqi, and K. Al-Shayji, “Supply chain optimization of petroleum organization under uncertainty in market demands and prices,” _European Journal of Operational Research_ , vol. 189, no. 3, pp. 822–840, 2008.
  * [15] T.-H. Kuo and C.-T. Chang, “Application of a mathematic programming model for integrated planning and scheduling of petroleum supply networks,” _Industrial & Engineering Chemistry Research_, vol. 47, no. 6, pp. 1935–1954, 2008.
  * [16] S. S. Pitty, W. Li, A. Adhitya, R. Srinivasan, and I. A. Karimi, “Decision support for integrated refinery supply chains: Part 1. dynamic simulation,” _Computers & Chemical Engineering_, vol. 32, no. 11, pp. 2767–2786, 2008\.
  * [17] Y. Kim, C. Yun, S. B. Park, S. Park, and L. Fan, “An integrated model of supply network and production planning for multiple fuel products of multi-site refineries,” _Computers & Chemical Engineering_, vol. 32, no. 11, pp. 2529–2535, 2008.
  * [18] K. Al-Qahtani and A. Elkamel, “Multisite facility network integration design and coordination: An application to the refining industry,” _Computers & Chemical Engineering_, vol. 32, no. 10, pp. 2189–2202, 2008.
  * [19] S. MirHassani, “An operational planning model for petroleum products logistics under uncertainty,” _Applied Mathematics and Computation_ , vol. 196, no. 2, pp. 744–751, 2008.
  * [20] P. Guyonnet, F. H. Grant, and M. J. Bagajewicz, “Integrated model for refinery planning, oil procuring, and product distribution,” _Industrial & Engineering Chemistry Research_, vol. 48, no. 1, pp. 463–482, 2008.
  * [21] R. Rocha, I. E. Grossmann, and M. V. P. de Aragão, “Petroleum allocation at petrobras: Mathematical model and a solution algorithm,” _Computers & Chemical Engineering_, vol. 33, no. 12, pp. 2123–2133, 2009.
  * [22] K. Y. Al-Qahtani and A. Elkamel, _Planning and integration of refinery and petrochemical operations_. John Wiley & Sons, 2011.
  * [23] A. Gill, “A supply chain design approach to petroleum distribution,” _Int J Bus Res Manage_ , vol. 2, no. 1, pp. 33–44, 2011.
  * [24] X. Chen, I. Grossmann, and L. Zheng, “A comparative study of continuous-time models for scheduling of crude oil operations in inland refineries,” _Computers & Chemical Engineering_, vol. 44, pp. 141–167, 2012.
  * [25] L. J. Fernandes, S. Relvas, and A. P. Barbosa-Póvoa, “Strategic network design of downstream petroleum supply chains: single versus multi-entity participation,” _Chemical engineering research and design_ , vol. 91, no. 8, pp. 1557–1587, 2013.
  * [26] A. B. Arabani and R. Z. Farahani, “Facility location dynamics: An overview of classifications and applications,” _Computers & Industrial Engineering_, vol. 62, no. 1, pp. 408–420, 2012.
  * [27] B. Benzaman and E. Pakdamanian, “Discrete event simulation of driver’s routing behavior rule at a road intersection,” _arXiv preprint arXiv:1907.00450_ , 2019.
  * [28] H. Jafarzadeh, N. Moradinasab, and M. Elyasi, “An enhanced genetic algorithm for the generalized traveling salesman problem,” _Engineering, Technology & Applied Science Research_, vol. 7, no. 6, pp. 2260–2265, 2017.
  * [29] N. Moradinasab, R. Shafaei, M. Rabiee, and P. Ramezani, “No-wait two stage hybrid flow shop scheduling with genetic and adaptive imperialist competitive algorithms,” _Journal of Experimental & Theoretical Artificial Intelligence_, vol. 25, no. 2, pp. 207–225, 2013.
  * [30] H. Jafarzadeh, N. Moradinasab, H. Eskandari, and S. Gholami, “Genetic algorithm for a generic model of reverse logistics network,” _International Journal of Engineering Innovation & Research_, vol. 6, no. 4, pp. 174–178, 2017.
  * [31] M. G. Earl and R. D’andrea, “Iterative milp methods for vehicle-control problems,” _IEEE Transactions on Robotics_ , vol. 21, no. 6, pp. 1158–1167, 2005.
  * [32] T. Huang and W. Zheng, “Water pollution prevention and control of chemical enterprises based on cooperative game,” _Chemical Engineering Transactions_ , vol. 67, pp. 421–426, 2018.
  * [33] N. Rasouli and I. N. Kamalabadi, “Joint pricing and inventory control for seasonal and substitutable goods mentioning the symmetrical and asymmetrical substitution,” _International Journal of Engineering-Transactions C: Aspects_ , vol. 27, no. 9, pp. 1385–1394, 2014.
  * [34] H. Jafarzadeh, S. Gholami, and R. Bashirzadeh, “A new effective algorithm for on-line robot motion planning,” _Decision Science Letters_ , vol. 3, no. 1, pp. 121–130, 2014.
  * [35] H. Jafarzadeh and C. H. Fleming, “An exact geometry–based algorithm for path planning,” _International Journal of Applied Mathematics and Computer Science_ , vol. 28, no. 3, pp. 493–504, 2018.
  * [36] R. E. Bellman and L. A. Zadeh, “Decision-making in a fuzzy environment,” _Management science_ , vol. 17, no. 4, pp. B–141, 1970.
  * [37] H.-J. Zimmermann, “Fuzzy programming and linear programming with several objective functions,” _Fuzzy sets and systems_ , vol. 1, no. 1, pp. 45–55, 1978.

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

A-1: Sets $K$ | Set of existing refineries
---|---
$\acute{K}$ | Set of new refineries
$L$ | Set of existing DCs
$\acute{L}$ | Set of the new DCs
$M$ | Set of costumer zone
$V$ | Set of transportation modes
$LCV$ | Set of capacity of transportation modes
$P$ | Set of products
$EK$ | Set of capacity of new refineries
$EL$ | Set of capacity of new DCs
$UK$ | Set of capacity expansion of existing refineries
$UL$ | Set of capacity expansion of existing DCs
$EV$ | Set of capacity expansion of existing pipelines
$EZ$ | Set of capacity of storage tanks in DC
$LV$ | Set of capacity of new pipeline
$RV$ | Set of new pipeline routes
$E$ | Set of stakeholders including government and private sectors
$EN$ | Set of regions EN
$LEV$ | Set of education levels LEV
$T$ | Set of Time of periods
A-2: Indices $k$ | Refinery
---|---
$l$ | DC
$m$ | Costumer zone
$v$ | Transportation mode
$lcv$ | Capacity of transportation mode
$p$ | Product
$ek$ | Capacity of new refinery
$el$ | Capacity of new DC
$uk$ | Capacity expansion of existing refinery
$ul$ | Capacity expansion of existing DC
$ev$ | Capacity expansion of existing pipeline
$ez$ | Capacity of the storage tank in DC
$lv$ | Capacity of new pipeline
$rv$ | New pipeline route
$e$ | Beneficiaries including government and private sectors
$en$ | regions
$lev$ | Education levels
$PS$ | Private sectors
$G$ | Government
$t$ | Time of periods
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 ($m\in M,t\in T,p\in P$)
$capl_{l}^{ul}$ |  Capacity expansion for existing distribution center 1 in level ul ($l\in L,ul\in L$)
$capk_{k}^{uk}$ |  Capacity expansion for existing refinery k in level uk ($k\in K,uk\in UK$)
$ick_{k}$ |  Initial capacity of existing refinery k ($k\in K$)
$icl_{p,l}$ |  Initial capacity of existing distribution center 1 ($l\in L,p\in P$)
A-3: Parameters (Continued) $Mk_{\acute{k}}^{ek}$ |  Minimum rate of capacity utilization coefficient of new refinery k in level ek ($ek\in EK,k\in\acute{K}$)
---|---
$Ml_{p,\acute{l}}^{ez}$ |  Minimum rate of capacity utilization coefficient of new distribution center l for product p in level el ($ez\in EZ,p\in P,l\in\acute{L}$)
$ivk_{k}^{0}$ |  Initial inventory level in refinery k ($k\in(\acute{K}\cup K)$)
$ivl_{l}^{0}$ |  Initial inventory level in distribution center l ($l\in(\acute{L}\cup L)$)
$\lambda E_{en}$ |  Coefficient of pollution significance in area en ($en\in EN$)
$Nct_{\acute{l}}^{ez}$ |  Capacity of storage tank in level ez in new distribution center l ($l\in\acute{L},ez\in EZ$)
$Ncl_{l}^{el}$ |  Capacity of new distribution center l in level el ($l\in\acute{L},el\in EL$)
$Nck_{\acute{k}}^{ekl}$ |  Capacity of new refinery k in level ek ($k\in\acute{K},ek\in EK$)
$icapkl_{k,l}$ |  Initial capacity of the pipeline between refinery k and distribution center ($k\in K,l\in L$)
$icaplpl_{lp,l}$ |  Initial capacity of the pipeline between distribution center l and distribution center ($lp,l\in L,lp\neq l$)
$trc_{v}^{lcv}$ |  Capacity of transportation mode v in level lcv ($v\in V,lcv\neq LCV$)
$capkl_{k,l}^{ev}$ |  Capacity expansion of the route between existing refinery k and distribution center l in level ev ($k\in K,l\in L,ev\in EV$)
$caplpl_{lp,l}^{ev}$ |  Capacity expansion of the route between existing distribution center lp and existing distribution center l in level ev ($lp,l\in L,lp\neq l,ev\in EV$)
$TPP$ |  Time period
$clv_{lv}$ |  Capacity of pipeline transportation mode in level lv ($lv\in LV$)
$Rkl_{k,l}$ |  Zero and one matrix indicating the existing routes between existing refinery k and existing distribution center l ($l\in L,k\in K$)
$Rlpl_{lp,l}$ |  Zero and one matrix indicating the existing routes between existing distribution center lp and existing distribution center l ($l,lp\in L,lp\neq l$)
$nmax_{v}^{lcv,t}$ |  Maximum number of available transportation mode v in level lcv in time t ($lcv\in LCV,v\in V,t\in T$)
$w_{i}$ |  Maximum crude oil production from oil field i ($i\in I$)
$\mu_{p}$ |  Production rate of product p from one crude oil barrel ($p\in P$)
$lk_{k}$ |  Minimum inventory level of refinery k ($k\in(K\cup\acute{K})P$)
$ll_{l}$ |  Minimum inventory level of distribution center l ($l\in(L\cup\acute{L})P$)
$M$ |  A large number
$xcostk_{k}^{ek,t}$ |  Installation cost of new refinery k in level ek during time t ($t\in T,ek\in EK,k\in\acute{K})P$)
$xcostl_{l}^{el,t}$ |  Installation cost of new distribution center l in level el during time t ($l\in\acute{L},t\in T$)
$ucostk_{k}^{uk,t}$ |  Expansion cost of existing refinery k in level uk during time t ($t\in T,p\in P,k\in K,uk\in UK)P$)
$ucostl_{l}^{ul,p,t}$ |  Expansion cost of existing distribution center l in level ul for product p during time t ($t\in T,p\in P,l\in L,ul\in L)P$)
$ycostkl_{k,l}^{ev,t}$ |  Expansion cost of pipeline between existing refinery k and existing distribution center l in level lv during time t ($t\in T,ev\in EV,k\in K,l\in L)P$)
$ycostlpl_{lp,l}^{ev,t}$ |  Expansion cost of pipeline between existing distribution center lp and existing distribution center l in level lv during time t ($t\in T,ev\in EV,lp,l\in L,l\neq lp$)
$hcostk_{k}$ |  Storage cost of crude oil in refinery k during time t ($t\in T,k\in(K\cup\acute{K})$)
$hcostl_{p,l}$ |  Storage cost of product p in distribution center l during time t ($t\in T,p\in P,l\in(L\cup\acute{L})$)
$qcostkl_{k,l}^{t}$ |  Flow cost of crude oil between refinery k and distribution center l during time t ($l\in(L\cup\acute{L}),k\in(K\cup\acute{K}),t\in T$)
$qcostlpl_{lp,l}^{t}$ |  Flow cost of products between existing distribution center lp and existing distribution center l in level lv during time ($t\in T,lp\in L,l\in(L\cup\acute{L})ORt\in T,lp\in\acute{L},l\in\acute{L}$)
A-3: Parameters (Continued) $rcostkl_{k,l,v}^{lv,rv,t}$ |  Installation cost of new pipeline rv between refinery k and distribution center l in level lv during time t ($lv\in LV,rv\in RV,l\in(L\cup\acute{L})ORt\in T,lp\in\acute{L},l\in\acute{L}$)
---|---
$ycostkl_{k,l}^{ev,t}$ |  Expansion cost of pipeline between existing refinery k and existing distribution center l in level lv during time t ($t\in T,ev\in EV,k\in K,l\in L$)
$ycostlpl_{lp,l}^{ev,t}$ |  Expansion cost of pipeline between existing distribution center lp and existing distribution center l in level lv during time t ($t\in T,ev\in EV,lp,l\in L,l\neq lp$)
$hcostk_{k}$ |  Storage cost of crude oil in refinery k during time t ($t\in T,k\in(K\cup\acute{K})$)
$hcostl_{p,l}$ |  Storage cost of product p in distribution center l during time t ($t\in T,p\in P,l\in(L\cup\acute{L})$)
$qcostkl_{k,l}^{t}$ |  Flow cost of crude oil between refinery k and distribution center l during time t ($t\in T,l\in(L\cup\acute{L}),k\in(K\cup\acute{K})$)
$qcostlpl_{lp,l}^{t}$ |  Flow cost of products between existing distribution center lp and existing distribution center l in level lv during time ($t\in T,lp\in L,l\in(L\cup\acute{L})OR\;t\in T,lp\in\acute{L},l\in\acute{L}$)
$rcostkl_{k,l,v}^{lv,rv,t}$ |  Installation cost of new pipeline rv between refinery k and distribution center l in level lv during time t ($lv\in LV,rv\in RV,t\in T,l\in(L\cup\acute{L}),k\in(K\cup\acute{K})$v=pipeline mode)
$rcostlpl_{lp,l,p}^{lv,rv,t}$ |  Installation cost of new pipeline rv between other distribution centers lp and the corresponding distribution center l in level lv during time t ($lv\in LV,rv\in RV,t\in T,l,lp\in(L\cup\acute{L})$)
$ncostkl_{k,l,v}^{lcv,t}$ |  Unit cost of transportation mode v between refinery k and distribution center l in capacity level lcv during time t($lcv\in LCV,v\in V,t\in T,l\in(L\cup\acute{L}),k\in(K\cup\acute{K})$)
$ncostlm_{l,m,v}^{lcv,t}$ |  customer area m in capacity level lcv during time t ($lcv\in LCV,v\in V,t\in T,l\in(L\cup\acute{L}),m\in M$)
$ncostlpl_{lp,l,v}^{lcv,t}$ |  Unit cost of transportation mode v between distribution centers l and lp in capacity level lcv during time t ($lcv\in LCV,v\in V,t\in T,l,lp\in(L\cup\acute{L})$)
$icost_{p}^{t}$ |  Importation cost of product p during time t ($t\in T,p\in P$)
$ncostl_{l}^{ez,t}$ |  Installation cost of storage tank with capacity ez in distribution center l during time t ($t\in T,ez\in EZ,l\in\acute{L}$)
$pcostk_{k}^{t}$ |  Operation cost of refinery k during time t ($t\in T,k\in(K\cup\acute{K})$)
$pcostl_{p,l}^{t}$ |  Operation cost of product p in distribution center l during time t ($t\in T,l\in(L\cup\acute{L}),p\in P$)
$Fcostk_{k}^{t}$ |  Fixed operation cost of refinery k during time t ($t\in T,k\in(K\cup\acute{K})$)
$Fcostl_{l}^{t}$ |  Fixed operation cost of distribution center l during time t ($t\in T,l\in(L\cup\acute{L})$)
$OP^{t}$ |  Crude oil price during time t($t\in T$)
$ERPP_{p}^{t,e}$ |  Exportation cost of product p during time t by stakeholder e ($t\in T,p\in P$)
$WCost_{lev}^{t}$ |  Labor force cost with education level lev during time t ($t\in T$)
$clcostk_{k}^{t}$ |  Closing down cost of refinery during time t ($t\in T,k\in K$)
$Nk_{en}^{k}$ |  Zero and one matrix indicating area en in which candidate refinery k has been constructed ($en\in EN,k\in\acute{K}$)
$Nek_{en}^{k}$ |  Zero and one matrix indicating area en in which existing refinery k has been expanded ($en\in EN,k\in K$)
$Nl_{en}^{l}$ |  Zero and one matrix indicating area en where candidate distribution center l has been constructed ($en\in EN,l\in\acute{L}$)
$Nel_{en}^{l}$ |  Zero and one matrix indicating area en, in which existing distribution center l has been constructed ($en\in EN,l\in\acute{L}$)
A-3: Parameters (Continued) $Pulk$ |  Pollution per barrel of crude oil refined
---|---
$Pull$ |  Pollution in distribution centers
$Pulv_{lcv}^{v}$ |  Pollution per kilometer of transportation mode v with capacity level lcv ($lcv\in LCV,v\in V$)
$Per$ |  Pollution coefficients for capacity expansion of refinery and distribution center
$dis_{m}^{l}$ |  Distance between distribution center l and customer area m ($m\in M,l\in(L\cup\acute{L})$)
$W_{en}^{\acute{en}}$ |  Coefficient of significance of labor force transfer from area en to area ($en\in EN,en\in\acute{EN}$)
$Nlab_{en}^{lev}$ |  New labor force with education level lev in area en ($en\in EN,lev\in LEV$)
$WNK_{lev}$ |  Required labor force with education level lev for a new refinery($lev\in LEV$)
$WEK_{lev}$ |  Required labor force with education level lev for capacity expansion of existing refinery ($lev\in LEV$)
$WNL_{lev}$ |  Required labor force with education level lev for a new distribution center ($lev\in LEV$)
$WEL_{lev}$ |  Required labor force with education level lev for capacity expansion of existing distribution center ($lev\in LEV$)
$Maxnum_{en}$ |  Maximum construction and expansion of refineries and distribution centers in each area ($en\in EN$)
A-5: Positive integer variables $nkl_{p,k,l,v}^{lcv,t,e}$ |  Number of fleet of transportation mode v at capacity level lcv between refinery k and DC l and by stakeholder e during t ($t\in T,lcv\in LCV,e\in E,v\in V,k\in(K\cup\acute{K}),l\in(L\cup\acute{L})$)
---|---
$nlpl_{p,lp,l,v}^{lcv,t,e}$ |  Number of fleet of transportation mode v at capacity level lcv between DC lp and DC l by stakeholder e during t ($t\in T,lcv\in LCV,e\in E,v\in V,p\in P,l,lp\in(L\cup\acute{L})$)
$nlm_{p,l,m,v}^{lcv,t,e}$ |  Number of fleet of transportation mode v at capacity level lcv between DC l and customer zone m by stakeholder e during t ($t\in T,lcv\in LCV,e\in E,v\in V,p\in P,l,lp\in(L\cup\acute{L})$)
$n_{p,l}^{ez,t,e}$ |  The number of storage tanks in capacity level ez in product p in DC l in by stakeholder e during t ($t\in T,e\in E,ez\in EZ,p\in P,l\in(L\cup\acute{L})$)
$HENK_{en,k}^{\acute{en},lev,t}$ |  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 ($lev\in LEV,t\in T,en,\acute{en}\in EN,k\in\acute{K}$)
$HEEK_{en,k}^{\acute{en},lev,t}$ |  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 ($t\in T,en,\acute{en}\in EN,lev\in LEV,k\in K$)
$HENl_{en,l}^{\acute{en},lev,t}$ |  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 ($t\in T,en,\acute{en}\in EN,lev\in LEV,l\in\acute{L}$)
$HENl_{en,l}^{\acute{en},lev,t}$ |  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 ($t\in T,en,\acute{en}\in EN,lev\in LEV,l\in L$)
A-4: Positive variables $qkl_{p,k,l,v}^{t,e}$ |  Rate of crude oil stream from refinery k to distribution center l via transportation mode v by stakeholder e during time t ($t\in T,p\in P,e\in E,v\in V,k\in(K\cup\acute{K}),l\in(L\cup\acute{L})$)
---|---
$qlm_{p,l,m,v}^{t,e}$ |  Rate of product stream from distribution center l to customer m via transportation mode v by stakeholder e during time t ($t\in T,p\in P,e\in E,v\in V,k\in(K\cup\acute{K}),m\in M$)
A-4: Positive variables $qlpl_{p,lp,l,v}^{t,e}$ |  Rate of product stream from other distribution centers lp to distribution center l via transportation mode v by stakeholder e during time t ($t\in T,e\in E,v\in V,l,lp\in(L\cup\acute{L})$)
---|---
$i_{p,l}^{t,e}$ |  Imports of product p from distribution center l by stakeholder e during time t ($t\in T,p\in P,e\in E,l\in(L\cup\acute{L})$)
$vl_{p,l}^{t,e}$ |  Inventory of product p in distribution center l by stakeholder e during time t ($t\in T,p\in P,e\in E,l\in(L\cup\acute{L})$)
$vk_{k}^{t,e}$ |  Inventory in refinery k by stakeholder e during time t ($t\in T,e\in E,k\in(K\cup\acute{K})$)
$Ep_{p,l}^{t,e}$ |  Exports of product p from distribution center l by stakeholder e during time t ($t\in T,p\in P,e\in E,l\in(L\cup\acute{L})$)
$\xi_{l}^{t,e}$ |  Percentage of the usage of distribution center l by stakeholder e during time t ($t\in T,e\in E,l\in L$)
$RLab_{en}^{lev,t}$ |  Number or required labor forces in level lev during time t($t\in T,en\in EN,lev\in LEV$)
$ALab_{en}^{lev,t}$ |  Available labor force with education level lev in area en during time t ($t\in T,en\in EN,lev\in LEV$)
A-6: Binary variables
$xk_{k}^{ek,t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;the\;refinery\;k\;is\;installed\;in\;level\;ek\\\
\;\;\;\;\;\;\;during\;time\;t\;by\;e\\\ 0\;\;\;\;\;Otherwise\;\;\;\;(t\in
T,ek\in EK,e\in E,k\in\acute{K})\end{array}\right.$
---
$\tau
l_{p,l}^{ul,t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;the\;distribution\;center\;l\;is\;expanded\\\
\;\;\;\;\;\;in\;level\;ul\;during\;time\;t\;by\;e\\\ 0\;\;\;\;\;Otherwise(t\in
T,ul\in UL,e\in E,l\in L,p\in P)\end{array}\right.$
$\tau
k_{k}^{uk,t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;the\;refinery\;k\;is\;expanded\;in\;level\;uk\\\
\;\;\;\;\;\;\;during\;time\;t\;by\;e\\\ 0\;\;\;\;\;Otherwise\;\;\;\;(t\in
T,uk\in UK,e\in E,k\in K)\end{array}\right.$
$ykl_{k,l}^{ev,t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;existing\;route\;between\;existing\;refinery\\\
\;\;\;\;\;\;k\;and\;DC\;is\;expanded\;with\;level\;ev\;in\\\
\;\;\;\;\;\;period\;t\;by\;e\\\ 0\;\;\;\;\;Otherwise(t\in T,ev\in EV,e\in
E,k\in K,l\in L)\end{array}\right.$
$ylpl_{lp,l}^{ev,t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;existing\;route\;between\;existing\;DC\\\
\;\;\;\;\;\;lp\;and\;existing\;DC\;is\;expanded\;with\;level\\\
\;\;\;\;\;\;ev\;in\;period\;t\;by\;e\\\ 0\;\;\;\;\;Otherwise(t\in T,ev\in
EV,e\in E,l,lp\in L)\end{array}\right.$
$z_{l}^{ez,t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;the\;number\;of\;storage\;tanks\;is\;fixed\\\
\;\;\;\;\;\;with\;level\;ez\;in\;new\;level\\\
\;\;\;\;\;\;DC\;l\;in\;period\;t\;by\;e\\\ 0\;\;\;\;\;Otherwise(t\in T,ez\in
EZ,e\in E,\acute{l}\in L)\end{array}\right.$
$rkl_{k,l,v}^{lv,rv,t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;route\;rv\;between\;refinery\;k\;and\;DC\;l\\\
\;\;\;\;\;\;is\;installed\;with\;level\;lv\;\;in\;period\;t\;by\;e\\\
0\;\;\;\;\;Otherwise\\\ \;\;\;\;(t\in T,ev\in
EV,k\in(K\in\acute{K}),l\in\acute{L}\in L)\end{array}\right.$
$rlpl_{lp,l,v}^{lv,rv,t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;route\;rv\;between\;DC\;lp\;and\;DC\;l\\\
\;\;\;\;\;\;is\;installed\;with\;level\;lv\;\;in\;period\;t\;by\;e\\\
0\;\;\;\;\;Otherwise(t\in T,ev\in EV,l,lp\in(\acute{L}\in
L)\end{array}\right.$
$rlp\acute{l}_{lp,l,v}^{t}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;route\;rv\;between\;DC\;lp\;and\;DC\;l\\\
\;\;\;\;\;\;is\;installed\;in\;period\;t\;by\;e\\\ 0\;\;\;\;\;Otherwise(t\in
T,e\in E,l,lp\in(\acute{L}\in L)\end{array}\right.$
A-6: Binary variables (Continued)
$S_{lp,l}^{t}(Auxiliary\;variable)=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;there\;is\;a\;flow\;from\;DC\\\
\;\;\;\;\;\;\;lp\;to\;DC\;l\;in\;period\;t\\\
0\;\;\;\;\;Otherwise(l,lp\in(\acute{L}\in L)\end{array}\right.$
---
$\psi
k_{k}^{t,e}=\left\\{\begin{array}[]{ll}1\;\;\;\;\;If\;k\;is\;closed\;in\;period\;t\\\
0\;\;\;\;\;Otherwise(t\in T,k\in K,e\in E)\end{array}\right.$
Appendix B: Optimal values in the case study $xk_{k}^{ek,t},k\in\acute{k}$
(Binary variable)
---
ek | k | t | e | Value
1 | 3 | 1 | 3 | 1
1 | 2 | 1 | 2 | 1
1 | 1 | 1 | 1 | 1
$nsu$
---
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 |  |  |  |  |  |
$\psi_{k}^{t,e}$
---
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
$rkl_{k,l,v}^{lv,rv,t},k\in K,l\in L$ (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
$vl_{p,l}^{t},l\in L$ (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 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$qkl_{p,k,l,v}^{t},l\in\acute{L},k\in\acute{K}$ (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
$qkl_{p,k,l,v}^{t},l\in L,k\in\acute{K}$ (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
$HENK_{en,k}^{\acute{en},lev,t}$
---
K | en | $\acute{en}$ | lev | t | Value | kk | en | $\acute{en}$ | lev | t | Value | kk | en | $\acute{en}$ | 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
$qlm_{p,l,m,v}^{t},l\in L,m\in\acute{M}$ (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 |  |  |  |  |  |  |
$nlm_{p,l,m,v}^{lcv,t,e},l\in L,m\in M$ (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 |  |  |  |  |  |  |  |
$Alab_{en}^{lev,t}$
---
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 |  |  |  |