DOI: 10.3303/CET2188060 
 

 
 

 
 
 

 

 
 

 
 
 

 
 
 

 
 
 

 

 
 

 
 

 
 
 

 
 
 

 
 
 

 
 
 

 
 
 

 
 
 

 
 

Paper Received: 20 May 2021; Revised: 27 September 2021; Accepted: 2 October 2021 
Please cite this article as: Shukla G., Chaturvedi N.D., 2021, A Goal Programming Approach for Optimizing Natural Gas Transportation 
Network, Chemical Engineering Transactions, 88, 361-366  DOI:10.3303/CET2188060 

CHEMICAL ENGINEERING TRANSACTIONS 

VOL. 88, 2021 

A publication of 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Yee Van Fan, Jiří J. Klemeš

Copyright © 2021, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-86-0; ISSN 2283-9216 

A Goal Programming Approach for Optimizing Natural Gas 
Transportation Network 

Gaurav Shukla, Nitin Dutt Chaturvedi* 
Department of Chemical and Biochemical Engineering, Indian Institute of Technology Patna, Bihta, Patna, 801106, Bihar, 
India 
nitind@iitp.ac.in 

Natural gas (NG) is widely transported through the high-pressure pipeline network. For maintaining the high 
pressure in the gas pipeline network, compressor stations are installed at fixed intervals along with the pipeline. 
The compression procedures involve huge amounts of energy and investment. So designing any pipeline 
network in NG transportation also needs to include conflicting goals of various criteria with the priority like 
maximization of demand satisfaction and minimization of compression energy and investment cost. In this 
paper, a goal programming approach dependent upon lexicographic methodology is proposed for solving the 
multi-criteria problem in NG transportation where each criterion is incorporated on a priority basis to achieve 
targeted goals. The relevance of the proposed model is explained via an example. Maximization in demand 
satisfaction is a prime objective and remaining targeted goals are attained on a priority basis. The deviation 
variable in the proposed model is solved in GAMS by using the CPLEX solver. These deviation variables provide 
flexibility to the decision-maker to change their priorities according to the social, environmental, and economical 
constraints. 

1. Introduction

Natural gas (NG) is a low-carbon, environmentally friendly, and high-quality energy source. Because of its 
convenience and economic feasibility, the pipeline is considered one of the most common ways to supply a 
huge volume of NG from sources station to demand end (Wang et al., 2018). Normally, in pipeline transportation 
the primary aims of decision-makers is to maximize demand satisfaction or maximum delivery, the economic 
benefit, reduce the environmental effect satisfy multiple objectives concurrently (Wu et al., 2018). The regulatory 
body must first define the pipeline project design criteria for designing a transport network for an attractive return 
on investment as well as the needs of current and potential customers. 
The goal of multi-objective optimization is to identify the best operational strategies that achieve two or multiple 
goals simultaneously. Martin et al. (2006) used a mixed-integer linear programming (MILP) framework to 
investigate the optimum supply chain's sensitivity to increases in fuel and installation cost (pipes, loading tanks, 
etc.) as well as changes in energy consumption. Tabkhi et al. (2010) suggested an algorithm for reducing the 
high-pressure gas network's expenditure and operational costs. Further, Chebouba (2015) addresses gas 
pipeline line pack control, which involves making trade-offs between reducing energy consumption in 
compression stations and enhancing gas line pack. Da Silva et al. (2016) suggested a MILP framework in a 
multi-objective optimization system to identify the least transportation cost along with optimizing compressor 
station (CS) gas flow. Recently, Su et al. (2019) suggested a multi-objective optimization model using grid 
concept for the trade-off between consistency and energy demand in pipeline network for the policy-making 
process capacities at altered situations. Sukharev et al. (2019) proposed multiple-criteria decision analysis 
based on the Monte Carlo method for cross-country gas. Schlossera et al (2019) proposed an energy-efficient 
model utilizing pinch analysis for vapor-compression heat pumps for various industries. 
Since multiple objectives need to be achieved, the researcher developed a mathematical model which 
represents those purpose. The goal programming (GP) techniques have become a widely used approach in 
multi-criteria decision-making problems. Each criterion in the GP technique is given a goal or target value to 
reach, and undesirable deviations from this set of target levels are then reduced by the decision variables. 

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mailto:nitind@iitp.ac.in


Carvalho et al. (2012) proposed the MILP model which provides a Pareto frontier set of solutions representing 
optimal trade-offs between economic and environmental goals. Recently, Osiadacz and Isoli (2020) proposed 
a bi-objective methodology for a trade-off between reducing compressor operating costs and increasing gas 
network capability. From the literature review, it can be concluded that the optimization in gas transportation is 
well studied while the multi-objective problems which consider demand satisfaction, investment cost, and 
compression energy simultaneously in pipeline transportation are not focused simultaneously. In this paper, an 
optimization model based on lexicographical GP is proposed to minimize investment cost, compression energy 
requirement in transporting NG, and maximize demand satisfaction. In this view, the problem statement and 
model formulation are discussed in the next sections. 

2. Problem Statement and Mathematical Formulation

A GP approach is introduced in NG transportation which considers three sustainability measures i.e. demand 
satisfaction, energy consumption, investment cost. Consider a gas allocation network consist several existing 
CS which are supplying natural gas flow to demand compressor station. For an allocation network, the following 
parameter is given. 
• Maximum available flow (Fi) of each existing CS ith (i=1, 2, 3, ..., n) and demand CS jth (1, 2, 3, ..., m).
• Pressure (Pi) of each existing CS and demand station (Pj).
• Projected goal Gk to be achieved for kth (k=1, 2, 3, ..., x) criteria along with its priority.
• Cost Factor (Cij) corresponding to each route.

A symbolic representation of the problem statement is shown in Figure 1. Table 1 shows the nomenclature 
section. 

Table 1: Nomenclature 

Set/Parameter/ 
Variable 

Context 

I 
J 
K 
µ(i/j)
λ 
F 
Fo
∗ 

P0 
𝐺𝑘 
Pj, Pi 
Fi 
Cij 
fij 
𝑑𝑘
−/𝑑𝑘

+

i/i Source CS 
j/j Demand CS 
k / k index of criteria 
Energy index of supply CS/ demand CS 
Compression Energy Index 
Volumetric flow rate (m3/s) 
Volumetric flow rate at atmospheric condition (Sm3/s) 
Pressure at atmospheric condition (kPa) 
Projected goal to be achieved for kth criteria 
Demand and supply pressure, (kPa) 
Maximum limit of supply flow from CS, (Sm3/s) 
The investment cost for supplying flow from ith station to jth demand 
Flow supply from compression station (i) to demand station (j), (Sm3/s) 
Positive / Negative deviation variable for over-achievement for kth criteria 

Figure 1: Representation of the problem 

Following constraints comprises the model. It consists of material balance and, thermodynamic related 
constraints. 

Goal objective: 

Maximization of demand satisfaction, Minimization of Energy Utilization, Minimization of Investment 

Source Compression Station 
Demand Station 

Maximum Available flow 

Compressor Station Pressure 

investment cost  

Energy constraint 

Investment constraint 

Flow Requirement 

Maximum Pressure 

Given Data 

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2.1 Material balances: 

The flow from CS (i) to demand (j) is symbolized by 𝑓𝑖𝑗. The flow balance for each source CS is addressed by 
Eq(1). 

∑ 𝑓𝑖𝑗
m
j=1 =  Fi   ∀ 𝑖 ∈  I (1) 

2.2 Thermodynamic related Constraints: 

The compression work or energy requirement is directed by the initial and final states along with the volumetric 
flow. For isothermal compression, the energy requirement (E) can be expressed as Eq(2): (Shukla and 
Chaturvedi, 2020) 

Net Energy,E =

{
 
 

 
 F0

∗ [(P0ln(
Pj

P0
)) − (P0ln(

Pi

P0
))]   For isothermal process

(
n

n−1
)P

0

1
n P

i

𝑛−1
n F0

∗ [(
Pj

Pi
)

𝑛−1

𝑛
− 1]   For polytropic process 

  (2) 

Where F0
∗ represents the volumetric flow rate at atmospheric condition, P0 is the atmospheric pressure, Pj and 

Pi are the demand and supply pressures while 𝑛 is the polytropic index. 

The Energy Index (EI) can be estimated utilizing Eq(3). 

 μ(i j⁄ ) =

{
 

 P0ln(
P(i j)⁄

P0
)   For isothermal process

   (
n

n−1
)P0ln(

P(i j⁄ )

Po
)

(𝑛−1)
𝑛

  For polytropic process  

∀𝑖 ∈ 𝐼,𝑗 ∈ 𝐽 (3) 

Where 𝜇(𝑖/𝑗) shows the energy index for the compressor station. 

The quantity (𝜇𝑗 − 𝜇𝑖) can be stated as the compression energy index, 𝜆𝑖𝑗 (CEI) and expressed as Eq(4); 
𝜆𝑖𝑗 = (𝜇𝑗 − 𝜇𝑖) ∀𝑖 ∈ 𝐼,𝑗 ∈ 𝐽 (4) 

GP introduces an auxiliary deviation variable that acts as a 'facilitator’ to frame the model. Goal constraint is 
defined in Eq(5) and Eq(6). 

∑ ∑ 𝑓𝑖𝑗
𝑚
𝑗=1

𝑛
𝑖=1 + 𝑑𝑘

− − 𝑑𝑘
+ = 𝐺𝑘

1   ∀𝑘 ∈ 𝐾 (5) 

 ∑ ∑ (𝑓𝑖𝑗 ∗
𝑚
𝑗=1

𝑛
𝑖=1 𝜆𝑖𝑗) + 𝑑𝑘

− − 𝑑𝑘
+ =  𝐺𝑘

2   ∀𝑘 ∈ 𝐾 (6) 

There are two types of deviation variables; negative deviation (𝑑𝑗
−) for under performance of the objective and

positive deviation (𝑑𝑗
+) for over performance of the objective or vice versa depending on nature of objective.

These negative and positive deviations depict the gap between the goal and the obtained result. Unlike the 
linear programming model that directly determines the solution via optimizing objectives; the GP tries to reduce 
undesirable deviations between the goal's expectation level and the solution. 

2.3 Objective functions: 

The objective function for achieving multiple goals based on priority order can be expressed by Eq(7). 

𝑍 = 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 { 𝑑𝑘=2
− , 𝑑𝑘=1

− ,𝑑𝑘=3
+ ,𝑑𝑘=4

+  } (7) 

The GP model minimizes unwanted deviations in a lexicographical way. This approach determines the optimal 
solution of the overall problem by sequentially solving several sub-problems which occur on a priority basis for 
each goal. According to priority, a sub-problem is solved for minimizing the unwanted deviation variable of the 
current goal. Then, this deviation variable value becomes a constraint for the next sub-problem which is solved 
for minimizing the unwanted deviation of subsequent goals on a priority basis. 

3. Proposed Algorithm

The algorithm for the proposed methodology in NG transportation concludes the following steps. 
Step 1: Calculate the energy index of supply CS and demand CS using Eq(3). 
Step 2: Calculate the compression energy index using Eq(4). 
Step 3: Formulate the model with multiple goals which minimize unwanted deviations. 
Step 4: To achieve the current goal solve the model for minimizing unwanted deviation. 
Step 5: Repeat Step 4 for satisfying each goal while using the previous unwanted deviation as a constraint. 

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Step 6: Tabulate the result of the deviation variable. 
Flow chart showing a proposed algorithm for obtaining optimal network for NG transportation. 

Figure 2: Flowchart for the goal programming approach 

4. Illustrative Example

The suggested procedure is explained via an illustrative example in this section. In this example, two demands 
CS at 500 kPa and 620 kPa are to be satisfied by four existing CS. Table 2 shows the available flow at a specific 
pressure is given: 

Table 2: Data for Source Compressor Station 

Compressor 
Station 

Pressure 
(kPa) 

Maximum available 
Flow (Sm3/s) 

𝑋1 200 300 
𝑋2 300 240 
𝑋3 130 360 
𝑋4 140 490 

Table 3: Cost factor and Compression Energy Index for demand 1 and demand 2 

Compressor 
Station 

Cost Factor 
for demand 1 

($.s/Sm3) 

Cost Factor 
for demand 2 

($.s/Sm3) 

CEI for 
demand 1 
(kJ/Sm3) 

CEI for 
demand 2 
(kJ/Sm3) 

𝑋1 10 15 92.58 114.65 
𝑋2 8 13 51.76 73.56 
𝑋3 15 20 62.44 84.24 
𝑋4 7 12 22.61 44.41 

Is the problem solved 
for all goals? 

Calculate the compression energy index 

Calculate the energy index of the compressor station 

Formulate the model with multiple goals which 
minimize unwanted deviations 

Start 

Tabulate the result of the deviation variable 

End 

To achieve the current goal solve the model for 
minimizing unwanted deviation 

Yes 

No 

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The compression process is assumed to be isothermal. The compression energy index (CEI) for supplying the 
flow is calculated Using Eq(3) and Eq(4). Table 3 shows the CEI's and cost factors corresponding to the various 
transporting route is given. 
Table 4 shows the value of criteria i.e. each demand, total investment cost, and total compression energy for 
demands. 

Table 4: Values (Goal) of criteria 

Criteria Value 

Demand D1 (Sm3/s) 600 
Demand D2 (Sm3/s) 680 
Investment Cost ($) 90,000 
Total Compression energy (kJ/s) 18,000 

The goal constraint equation for satisfying the demand 1 is given in Eq(8). 

𝑓11 + 𝑓21 + 𝑓31 + 𝑓41 + 𝑑1
− − 𝑑1

+ = 600 (8) 

The equation limits demand satisfaction to the desired goal. Hence (𝑑1
−) should be minimized. Similarly, the

demand balance equation on the priority-based satisfying the demand 2. 

𝑓12 + 𝑓22 + 𝑓32 + 𝑓42 + 𝑑2
− − 𝑑2

+ = 680 (9) 

This equation, limits demand satisfaction to the desired goal. Hence (𝑑2
−) should be minimized.

The equation for achieving the goal to restrict compression energy consumption is given in Eq(10). 

92.85𝑓11 + 1.76𝑓21 + 62.44𝑓31 + 22.61𝑓41 + 114.65𝑓12 + 73.56𝑓22 + 84.24𝑓32 + 44.21𝑓42 + 𝑑3
− − 𝑑3

+ = 90,000 (10)

Their energy consumption is the desired goal, so the unwanted deviation i.e. (𝑑3
+) should be minimized.

The equation for achieving the goal to get a maximum investment of $18000 is given in Eq(11). 

10𝑓11 + 8𝑓21 + 15𝑓31 + 7𝑓41 + 15𝑓12 + 13𝑓22 + 20𝑓32 + 12𝑓42 + 𝑑4
− − 𝑑4

+ = 18,000 (11) 

This equation is for attaining profit of desired goal, so the unwanted deviation i.e. (𝑑4
+) should be minimized.

For the multi-criteria problem, minimizing the deviation of projected goals on a priority basis using the 
Lexicographic method. The GP model is solved in GAMS by using a CPLEX solver. The interpretation of the 
result shown in Table (5) is as follows: 

Table 5: Result for deviation variables of each criterion 

Variable 
1

d


1
d



2
d



2
d



3
d



3
d 

4
d



4
d

Value 50 0 0 0 0 2,199.4 1,490 0 

To design of allocation network in NG transportation needs to combine several conflicting goals simultaneously. 
This multi-criteria decision variable is solved using the GP model and the solution provides the deviation in 
criteria utilized by the decision-maker for designing an optimal allocation network. In this example, the priority is 
satisfying the demand 2. The result shows the zero deviation i.e. demand 2 is satisfied. For demand 1, a negative 
deviation of 50 Sm3/s shows of flow is required i.e. supplied flow from supply stations to demand 1 is lesser than 
the targeted demand. In the next priority for the goal of compression energy requirement, a positive deviation of 
2,199.4 kJ/s of energy is obtained. It means [energy consumed by the CS is greater than the required goal. So 
for satisfying the energy demand, external source CS needs to be installed. For targeting the investment cost, 
a positive deviation of zero shows that the goal is achieved, and a negative deviation of $ 1,490.5 total 
investment is lesser than the required goal. The proposed model provides flexibility to the decision-maker to 
change their priorities according to environmental and economical constraints. 

5. Conclusion

Transportation of NG consists of several conflicting goals that need to be addressed simultaneously. In this 
paper, the multi-criteria problem in NG transportation is solved using GP in a lexicographical manner. GP is 
used as it seems to be most suitable and appropriate tool in multi-objective problems where predefined goal or 
priority is given. In gas pipeline transportation, three objectives are considered which are maximization of 

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demand satisfaction, minimization of investment cost associated with flow transportation, and compression 
energy requirement. The prioritized criteria will govern the result of optimization and also we can draw the 
following conclusion. 
 For demand 2, the goal objective for satisfying the demand is completely attainable with the available source

CS.
 The model suggests that the achievement of the goal set for demand 1 is not attainable without any

additional external CS.
 The positive deviation shows that goal associated to meet the energy consumption in the natural gas

transportation network is not fulfilled which means external CS is needed to fulfill the energy demand.
 For the goal of investment cost, a positive deviation of zero shows that the goal is achieved and a negative

deviation of $ 1,490 shows that the investment is lesser than the required goal. So the decision-maker can
utilize the investment budget for installing the external CS.

The result shows the model can efficiently find an optimum transportation network of NG within a given criterion. 
It can also help monitor the amount of compression energy used, the cost of expenditure, and the fulfillment of 
demand. The developed model allows policymakers to discuss various scenarios with various probable norms 
under various priority rankings. 

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