CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 76, 2019 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar S. Varbanov, Timothy G. Walmsley, Jiří J. Klemeš, Panos Seferlis 
Copyright © 2019, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-73-0; ISSN 2283-9216 

Chance Constrained Optimization of Biodiesel Supply Chain 

Kan Rungphanich, Kitipat Siemanond* 

The Petroleum and Petrochemical College, Chulalongkorn University, Bangkok, Thailand 

kitipat.s@chula.ac.th 

Biodiesel is an alternative and renewable biofuel for blending with fossil-based diesel. One of the challenges is 

to design the supply-chain network for biohydrodeoxygenated diesel (BHD) under uncertainty from both raw 

materials availability and the demand. The supply chain with four echelons; suppliers, factories, inventories and 

customers has been studied. The biodiesel factory has been simulated by commercial simulation program Pro/II 

to find the utility cost and biodiesel production capacity. In this study, the stochastic optimization is used to solve 

mathematical model of the BHD supply chain network. The proposed model is based on stochastic mixed-

integer programming with chance constraints. The chance constrained optimization has been applied to design 

supply chain network under the uncertainty at different levels of confidence. Then optimized supply chain 

network has been investigated on stability of the supply chain network under the uncertainties, accuracy of the 

profit and the effect of penalty. This study shows profit and trade-off between profit and penalty cost with different 

levels of confidence of BHD supply chain under uncertainties in raw-material supply and biodiesel demand. The 

results show that the chance constrained optimization can be used to design optimum supply chain network 

under uncertainties within levels of confidence. 

1. Introduction 

Energy is one of the necessities in everyday life and its demand is getting higher every year. The U.S. Energy 

Information Administration (2019) reported that the total energy usage in 2018 was 105.72x109 GJ and they 

predict that it will grow to 112.50x109 GJ in 2050. The transportation segment is the second largest. Therefore, 

the sustainable energy has been concerned. The biodiesel is one of the solutions to sustain the usage of diesel 

in transportation segment. The first generation of biodiesel is Fatty Acid Methyl Esters (FAME) via 

transesterification process. However due to high oxygenated contents in this type of biodiesel, the fuel cannot 

be used practically as transportation fuel. The next generation of biodiesel is Bio Hydrogenated Diesel (BHD) 

via dehydrogenation process. This method improves fuel properties by removing oxygenated group from product 

(Chen et al., 2019). When production scale of biodiesel increases, the supply chain problems of uncertainty 

occurs (Gao and You, 2017) especially in availability of feedstock and demand for biodiesel. For the availability 

of feedstock, currently the main raw material for biodiesel is edible plants (Avhad and Marchetti, 2015) that 

share demand with food production. For the demand, biodiesel and diesel are needed for transportation fuel 

leading to uncertain demand of biodiesel. Therefore, the stochastic optimization is used to handle this problem. 

The chance constrained optimization is one of the stochastic optimization introduced by Charnes and Cooper 

(1959) and Miller and Wagner (1965). This method is used to find the optimum solutions under uncertainties at 

certain probability, in the other hand, called level of confidence. 

In this work, the supply chain of biodiesel under uncertainties has been developed. The uncertainties in this 

model occur in suppliers and customers. The BHD plants have been simulated by process simulation software 

Pro/II to find correlation between feed and operation cost. Finally, the optimized supply chain has been 

investigated on the stability of supply chain, validation of profit and sensitivity analysis on penalty. 

2. Methodology 

The chance constrained mixed integer nonlinear programming (MINLP) has been used to optimize the supply 

chain of biodiesel produced from steric acid in vegetable oil. This supply chain consists of four echelons: 

vegetable-oil suppliers (i), BHD plants (j), BHD inventories (k) and BHD customers (l) as shown in Figure 1(a). 

 
 
 
 
 
 
 
 
 
 
                                                                                                                                                                 DOI: 10.3303/CET1976096 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 16/03/2019; Revised: 07/06/2019; Accepted: 24/06/2019 
Please cite this article as: Rungphanich K., Siemanond K., 2019, Chance Constrained Optimization of Biodiesel Supply Chain, Chemical 
Engineering Transactions, 76, 571-576  DOI:10.3303/CET1976096 
  

571



The objective of optimization is maximization of profit under uncertainties (at suppliers and customers) with 

different levels of confidence. 

2.1 Mathematical model 

The mathematical model for this supply chain is expressed as shown below. 

maximize(𝑧𝑧) = 𝑝𝐵𝐷 ∙ ∑ ∑ (𝑥𝑘𝑙 − 𝑃𝑙
𝑝𝑜

)𝑙𝑘  − [ ∑ ∑ 𝑐𝑡,𝑖𝑗 ∙ 𝑥𝑖𝑗𝑗 + ∑ ∑ 𝑐𝑡,𝑗𝑘 ∙ 𝑥𝑗𝑘𝑘𝑗𝑖 + ∑ ∑ 𝑐𝑡,𝑘𝑙 ∙ 𝑥𝑘𝑙𝑙𝑘  ]  

           −[𝑝𝑆 ∙ ∑ 𝑥𝑖𝑗𝑗 + 𝑝𝐻2 ∙  ∑ 𝑥𝐻2,𝑗𝑗 + 𝑝𝐻𝑈 ∙ ∑ 𝐻𝑈𝑗𝑗 + 𝑝𝐶𝑈 ∙ ∑ 𝐶𝑈𝑗𝑗 + 𝑝𝐸𝑈 ∙ ∑ 𝐸𝑈𝑗𝑗 ] − 𝑐𝑝 ∙ ∑ 𝑃𝑙
𝑛𝑒

𝑙   
(1) 

s.t. 

∑ 𝐶𝑛,𝐵𝐷 ∙ (∑ 𝑥𝑖𝑗𝑖 )
𝑛

𝑛 = ∑ 𝑥𝑗𝑘𝑘   (2) 

∑ 𝑥𝑗𝑘𝑗 = ∑ 𝑥𝑘𝑙𝑙   (3) 

∑ 𝑥𝑗𝑘𝑘 ≥ 𝑆𝑃𝑗,𝑎𝑣,𝐿𝐵  (4) 

∑ 𝑥𝑗𝑘𝑘 ≤ 𝑆𝑃𝑗,𝑎𝑣,𝑈𝐵  (5) 

∑ 𝑥𝑗𝑘𝑗 ≤ 𝑆𝑃𝑘,𝑎𝑣  (6) 

Pr(∑ 𝑥𝑖𝑗𝑗 ≤ 𝑆𝑃𝑖,𝑟𝑎𝑛𝑑 ) ≥ 𝛼𝑖   (7) 

Pr(∑ 𝑥𝑘𝑙𝑘 ≥ 𝑆𝑃𝑙,𝑟𝑎𝑛𝑑 ) ≥ 𝛽𝑙   (8) 

𝑥𝐻2,𝑗 = ∑ 𝐶𝑛,𝐻2 ∙ (∑ 𝑥𝑖𝑗𝑖 )
𝑛

𝑛   (9) 

𝐻𝑈𝑗 = ∑ 𝐶𝑛,𝐻𝑈 ∙ (∑ 𝑥𝑖𝑗𝑖 )
𝑛

𝑛   (10) 

𝐶𝑈𝑗 = ∑ 𝐶𝑛,𝐶𝑈 ∙ (∑ 𝑥𝑖𝑗𝑖 )
𝑛

𝑛   (11) 

𝐸𝑈𝑗 = ∑ 𝐶𝑛,𝐸𝑈 ∙ (∑ 𝑥𝑖𝑗𝑖 )
𝑛

𝑛   (12) 

The objective function is to maximize profit that consists of four parts: revenue from selling BHD, transportation 

cost, operation cost and penalty cost as shown in Eq(1). The penalty in this model is only under demand in 

selling BHD or opportunity loss. However, over-demand products are not sold to customers. Eq(2) deals with 

mass balance and conversion of vegetable oil to biodiesel at plants. Eq(3) deals with mass balance at 

inventories. Eqs(4,5) deal with the minimum and maximum capacity of plants, respectively. Eq(6) deals with the 

maximum capacity of inventories. Eqs(7,8) deal with suppliers and customers uncertainties in terms of chance 

constraints at different levels of confidence (αi and βl, respectively). Eqs(9-12) deal with operating utilities of 

plants. The chance constraints are transformed to deterministic equivalent form (Charnes and Cooper, 1959). 

Eq(7,8) can be rewritten to Eq(13,14) where 𝜙−1 is quantile function. 

∑ 𝑥𝑖𝑗𝑗 ≤ 𝑆𝑃𝑖,𝑎𝑣 + 𝜙
−1(1 − 𝛼𝑖 ) ∙ 𝑆𝑃𝑖,𝑠𝑑   (13)  ∑ 𝑥𝑘𝑙𝑘 ≥ 𝑆𝑃𝑙,𝑎𝑣 − 𝜙

−1(1 − 𝛽𝑙 ) ∙ 𝑆𝑃𝑙,𝑠𝑑   (14) 

The Eq(14) has been modified to Eq(15-17) to calculate penalty in the model. 

∑ 𝑥𝑘𝑙𝑘 + 𝑃𝑙
𝑛𝑒 − 𝑃𝑙

𝑝𝑜
≥ 𝑆𝑃𝑙,𝑎𝑣 − 𝜙

−1(1 − 𝛽𝑙 ) ∙ 𝑆𝑃𝑙,𝑠𝑑  (15) 

∑ 𝑥𝑘𝑙𝑘 − 𝑃𝑙
𝑝𝑜

≤ 𝑆𝑃𝑙,𝑎𝑣 − 𝜙
−1(1 − 𝛽𝑙 ) ∙ 𝑆𝑃𝑙,𝑠𝑑   (16)  ∑ 𝑥𝑘𝑙𝑘 + 𝑃𝑙

𝑛𝑒 ≥ 𝑆𝑃𝑙,𝑎𝑣 − 𝜙
−1(1 − 𝛽𝑙 ) ∙ 𝑆𝑃𝑙,𝑠𝑑  (17) 

2.2 Biodiesel plant simulation 

The kinetic models are used to calculate biodiesel produce, hydrogen, hot utility, cold utility and electric energy 

usage in biodiesel plant. The kinetic models of BHD process can be expressed by Arrhenius equation with 

constant parameters, shown in Table 1. The biodiesel plant has been simulated using kinetic models. The 

product specification of biodiesel is 90 % w/w purity. The simulation data of biodiesel produce, hydrogen, hot 

utility, cold utility and electric energy usage has been collected at different feed flow rates of steric acid then 

found the correlation with steric acid feed condition as shown in Figure 1(b). 

Table 1 Activation energy and pre-exponential factor for kinetic model (Kumar et al., 2014). 

Reaction Ea (kJ/mol) A0 (s-1) 

C17H35 COOH + 2H2 → C18H37OH + H2O 175.4 5.57x10
12 

C18H37 OH → C17H36 + H2 + CO 250.0 1.34x10
21 

C18H37 OH + H2 → C18H38 + H2O 190.9 4.77x10
13 

C18H37 OH + 2H2 → C15 H32 + C3H8 + H2O 387.7 5.08x10
32 

C18H37 OH + 2H2 → C16 H34 + C2H6 + H2O 377.2 1.08x10
32 

Steric acid is C17H35 COOH          BHD is C15H32, C16H34, C17H36 and C18H38 

572



 

Figure 1: (a) The supply chain diagram for this work. (b) Correlation of utility data from simulation program Pro/II 

2.3 The investigation on optimized supply chain network 

The optimized supply chain network has been investigated on validation of profit, stability of supply chain and 

sensitivity analysis on penalty lost. The stability of network can be obtained from non-violation condition in 

chance constraints; Eqs(7,8) where SPI,rand and SPl,rand are randomly generated based on normal distribution 

data of steric acid availability and BHD demand, respectively. Then the levels of confidence of suppliers and 

customers are validated from total number of feasible data points over total number of random date points.  

The validation of the optimized supply chain network can be hard to solve because the optimized supply chain 

network can be infeasible for example the random capacity of supplier can be less than the required value in 

optimized network. This leads to error in the calculation. Therefore, the network needs to recalculate from the 

initial state. The validation model can be expressed as shown below. 

max(𝑧𝑧) = 𝑝𝐵𝐷 ∙ ∑ ∑ (𝑥𝑟,𝑘𝑙 − 𝑃𝑙
𝑝𝑜

)𝑙𝑘  − [ ∑ ∑ 𝑐𝑡,𝑖𝑗 ∙ 𝑥𝑟,𝑖𝑗𝑗 + ∑ ∑ 𝑐𝑡,𝑗𝑘 ∙ 𝑥𝑟,𝑗𝑘𝑘𝑗𝑖 + ∑ ∑ 𝑐𝑡,𝑘𝑙 ∙ 𝑥𝑟,𝑘𝑙𝑙𝑘  ]  

   −[𝑝𝑆 ∙ ∑ 𝑥𝑟,𝑖𝑗𝑗 + 𝑝𝐻2 ∙  ∑ 𝑥𝐻2,𝑗𝑗 + 𝑝𝐻𝑈 ∙ ∑ 𝐻𝑈𝑗𝑗 + 𝑝𝐶𝑈 ∙ ∑ 𝐶𝑈𝑗𝑗 + 𝑝𝐸𝑈 ∙ ∑ 𝐸𝑈𝑗𝑗 ] − 𝑐𝑝 ∙ ∑ 𝑃𝑙
𝑛𝑒

𝑙  
(18) 

s.t.

∑ 𝐶𝑛,𝐵𝐷 ∙ (∑ 𝑥𝑟,𝑖𝑗𝑖 )
𝑛

𝑛 = ∑ 𝑥𝑟,𝑗𝑘𝑘   (19) 

∑ 𝑥𝑟,𝑗𝑘𝑗 = ∑ 𝑥𝑟,𝑘𝑙𝑙   (20) 

∑ 𝑥𝑟,𝑗𝑘𝑘 ≥ 𝑆𝑃𝑗,𝑎𝑣,𝐿𝐵  (21) 

∑ 𝑥𝑟,𝑗𝑘𝑘 ≤ 𝑆𝑃𝑗,𝑎𝑣,𝑈𝐵   (22) 

∑ 𝑥𝑟,𝑗𝑘𝑗 ≤ 𝑆𝑃𝑘,𝑎𝑣  (23) 

∑ 𝑥𝑟,𝑖𝑗𝑗 ≤ 𝑆𝑃𝑖,𝑟𝑎𝑛𝑑  (24) 

∑ 𝑥𝑟,𝑘𝑙𝑘 + 𝑃𝑙
𝑛𝑒 − 𝑃𝑙

𝑝𝑜
 ≥ 𝑆𝑃𝑙,𝑟𝑎𝑛𝑑  (25) 

∑ 𝑥𝑟,𝑘𝑙𝑘 − 𝑃𝑙
𝑝

≤ 𝑆𝑃𝑙,𝑟𝑎𝑛𝑑  (26) 

∑ 𝑥𝑟,𝑘𝑙𝑘 + 𝑃𝑙
𝑛 ≥ 𝑆𝑃𝑙,𝑟𝑎𝑛𝑑   (27) 

𝑥𝐻2,𝑗 = ∑ 𝐶𝑛,𝐻2 ∙ (∑ 𝑥𝑟,𝑖𝑗𝑖 )
𝑛

𝑛   (28) 

𝐻𝑈𝑗 = ∑ 𝐶𝑛,𝐻𝑈, ∙ (∑ 𝑥𝑟,𝑖𝑗𝑖 )
𝑛

𝑛   (29) 

𝐶𝑈𝑗 = ∑ 𝐶𝑛,𝐶𝑈 ∙ (∑ 𝑥𝑟,𝑖𝑗𝑖 )
𝑛

𝑛   (30) 

𝐸𝑈𝑗 = ∑ 𝐶𝑛,𝐸𝑈 ∙ (∑ 𝑥𝑟,𝑖𝑗𝑖 )
𝑛

𝑛   (31) 

𝑥𝑟,𝑖𝑗 + 𝑅𝑖𝑗 = 𝑥𝑠,𝑖𝑗 (32) 

𝑥𝑟,𝑗𝑘 + 𝑅𝑗𝑘 = 𝑥𝑠,𝑗𝑘  (33) 

𝑥𝑟,𝑘𝑙 + 𝑅𝑘𝑙 = 𝑥𝑠,𝑘𝑙 (34) 

𝑀 ∙ (𝑦𝑖 − 1) ≤ 𝑆𝑃𝑖,𝑟𝑎𝑛𝑑 − ∑ 𝑥𝑠,𝑖𝑗𝑗  ≤ 𝑀 ∙ 𝑦𝑖 (35) 

𝑦𝑖 (∑ (𝑥𝑟,𝑖𝑗 − 𝑥𝑠,𝑖𝑗 ))𝑗 + (1 − 𝑦𝑖 )(∑ (𝑥𝑟,𝑖𝑗 − 𝑆𝑃𝑖,𝑟𝑎𝑛𝑑 ))𝑗  = 0 (36) 

Eqs(19-31) show the modification from the chance constrained model. Eqs(32-36) deal with recalculating the 

initial optimized supply chain network (subscript s) to valid supply chain network (subscript r). Therefore, the 

valid values can be used to calculate the validated profit. Finally, the sensitivity analysis is used to see penalty 

cost of opportunity loss affecting on profit at different levels of confidence. This data can be obtained by varying 

573



penalty cost of opportunity loss with the same prices and costs then compare the validate profit at different 

levels of confidence. 

3. Result and discussion 

3.1 An illustrative example 

The hypothetical case is provided to show the effectiveness of model. Table 2 shows statistical data related to 

capacity of suppliers, plants, inventory and demand of customers. Table 3 shows steric acid price, operation 

cost, penalty cost and biodiesel price. Table 4 shows transportation cost in dollar per litter. 

Table 2 Data related to capacity and demand. 

  Supplier Plant Inventory Customer 

  i1 i2 i3 j1 j2 k1 k2 l1 l2 l3 

Average Capacity or 

Demand (L/d) 

UB: 10,000 12,500 9,500 8,000 15,000 15,000 10,000 - - - 

LB: - - - 5,000 6,000 - - 3,000 7,500 5,000 

Standard Deviation (L/d)  1,000 1,250 950 - - - - 300 750 500 

Table 3 Fluid price, utility cost and penalty cost 

Fluid price  Utility cost  Penalty cost 

Steric acid 20  $/L  Hot utility 5  $/kWh  Opportunity loss 62.375  $/L 

Hydrogen 5  $/L  Cold utility 2.5  $/kWh  Over-demand loss 0  $/L 

Biodiesel 49.9  $/L  Electric energy 1.25  $/kWh     

Table 4 Transportation cost in dollar per litter 

Transportation cost from 
supplier to plant ($/L) 

 Transportation cost from 
plant to inventory ($/L) 

 Transportation cost from inventory 
to customer ($/L) 

 j1 j2   k1 k2   l1 l2 l3 

i1 2 4  j1 3 1  k1 2 3 2 

i2 3 5  j2 2 4  k2 1 4 2 

i3 1 3          

3.2 Optimization of biodiesel supply chain network results 

Figure 2(a) shows result of optimized network by chance constrained optimization at different levels of 

confidence. The total value of each bar represents the revenue for each case. The level of confidence of 0.50 

is representative of deterministic optimization. The results show that the higher profit can be obtained at higher 

levels of confidence for each suppliers and customers as shown in Figure 2(a). This happen because the 

capacity for each supplier decreases but the BHD demand increases above the average value when using the 

deterministic equivalent form from Eqs(13,14). Therefore, the required quantity of biodiesel in supply chain 

increases, resulting in increases of BHD revenue and total cost increase. 

For the validation part of optimal supply chain, the average validated profit is lower than the optimized profit as 

shown in Figure 2(b). This comes from the deterministic equivalent chance constrained model which does not 

calculate the penalty quantity (both of opportunity loss and over-demand loss) in the system. When deterministic 

equivalent form is used, the uncertain values are converted to certain value. Therefore, the model trends to 

satisfy the constraint at the new certain value and does not have penalty quantity in the network. The validation 

model is used to handle this problem. 

Next, the trend of validated profit becomes lower at higher level of confidence. This comes from the over-demand 

loss is affected the revenue even though the penalty cost for over-demand loss is 0 $/L. At higher levels of 

confidence for each suppliers and customers, the total quantity of biodiesel in network increases. Therefore, 

more biodiesel is flowing through the network but the revenue stays the same due to the over-demand products. 

The transportation and operation costs are higher in contrast of the penalty cost due to more biodiesel in the 

network. Although, the opportunity loss is lower but not trade-off with other costs. The trend of profit is going 

down at higher level of confidence. This case study shows that chance constrained programming only gives 

lower opportunity loss than deterministic programming. To improve the profit of chance constrained supply 

chain, the sensitivity analysis of penalty cost for opportunity loss will be done in next part. 

574



 

Figure 2: The results from (a) chance constrained model (b) validation model.  

3.3 Feasibility of network and sensitivity analysis on penalty cost 

The next investigation is to study the feasibility of constraints; Eqs(7,8) as shown in Figure 3(a). The values of 

levels of confidence for each supplier and customers from the stability test are away or align on the diagonal 

line of the graph in Figure 3(a). This means that the calculated level of confidence is greater than or equal to 

the level of confidence used in Eqs(7,8). This proves that the chance constrained optimization can find the 

optimized supply chain satisfying the condition of level of confidence. 

Figure 3(b) shows the result of sensitivity analysis on penalty cost for opportunity loss. The result shows that 

deterministic supply chain has higher slope than chance constrained ones and their slope decreases when level 

of confidence increases and when the penalty cost for opportunity loss becomes larger value, the deterministic 

supply chain will become less profitable than chance constrained ones. For this study, at 62.375 $/L penalty 

cost for opportunity loss, the penalty cost is smaller than the other costs meaning that the decrease of 

opportunity loss from chance constrained programming is meaningless. However, when the penalty cost for 

opportunity loss keeps increasing, the penalty cost and the decrease of the quantity of opportunity becomes 

more relevant. At 83.17 $/L penalty cost for opportunity loss, deterministic supply chain gives profit of 196,323.27 

$/d down from 210,327.65 $/d which is lower than chance constrained supply chain at 0.80 level of confidence 

which gives 210,159.77 $/d. Furthermore, at 124.75 $/L penalty cost for opportunity loss, deterministic supply 

chain gives profit of 168,314.53 $/d which is much lower than chance constrained supply chain at 0.80 level of 

confidence that gives 202,311.33 $/d. 
 

 

Figure 3:The result on (a) stability of supply chain (b) sensitivity analysis on penalty cost of opportunity loss. 

The supply chain system designed at lower level of confidence gives more opportunity loss. Therefore, the 

supply chain by deterministic optimization has more opportunity loss than one by the chance constrained 

optimization which has higher level of confidence. The opportunity loss is the main factor that affects the 

sensitivity of penalty cost in this study. Therefore, higher opportunity loss in the system means that more 

sensitive to penalty cost. For this case study, at penalty cost for opportunity loss of 83.17 $/L, the chance 

constrained optimization with level of confidence of 0.80 can improve the profit from deterministic optimization 

as shown in Figure 3(b). 

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4. Conclusions 

The basic concept of chance constrained programming is used to design the optimum BHD supply chain network 

under the uncertainties within certain levels of confidence. However, the validation step is needed to reflect the 

real profit value by using the result from the deterministic equivalent solving method and transform into the 

validated result under the uncertainties. The validated result shows that more stable network with less profit is 

obtained when the level of confidence increases. For this case, the deterministic optimization gives more profit 

than chance constrained optimization but less stability due to more opportunity loss occurring in the system. 

Finally, the sensitivity analysis shows that the chance constrained optimization is less sensitive to penalty cost 

due to the less opportunity loss occurring in the system. Therefore, the chance constrained optimization helps 

increase the profit of supply chain compared to one obtained from deterministic optimization when the penalty 

cost for opportunity loss has the significant value on the network. Further research can be conducted by 

improving on the accuracy on the result of the chance constrained programming when compared with validation 

step or using the joint probability chance constraint instead. 

Nomenclatures 

Set 

𝑖 Set of suppliers  𝑘 Set of inventories 

𝑗 Set of plants  𝑙 Set of customers 

Subscript 

𝑟𝑎𝑛𝑑 The random value  𝐻2 For hydrogen 

𝑎𝑣 The average value  𝐵𝐷 For biodiesel 

𝑠𝑑 The standard deviation value  𝐻𝑈 For hot utility 

𝑈𝐵 Upper bound  𝐶𝑈 For cold utility 

𝐿𝐵 Lower bound  𝐸𝑈 For electric energy 

𝑆 For steric acid    

Parameter 

𝑝 Price per litter  𝐶𝑛 Correlation coefficient 

𝑐𝑡  Transportation cost per litter  𝑛 Correlation order 

𝑐𝑝  Opportunity lost per litter  𝑥𝑠 Optimized supply chain network 

𝑆𝑃 Capacity or demand value  𝑀 Large number 

𝛼/𝛽 Levels of confidence    

Variable 

𝑥 Quantity in supply chain network  𝑥𝑟  Validated supply chain network 

𝑃𝑙
𝑝𝑜

 Over-demand penalty quantity  𝑅 Remainder in network 

𝑃𝑙
𝑛𝑒 Opportunity lost quantity  𝑦 Decision variable for validation model 

Acknowledgments 

Authors would like to thank Thai Government budget fund and the Petroleum and Petrochemical College, 

Chulalongkorn university for financial support throughout the duration of this research. 

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