CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 52, 2016 

A publication of 

 

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

Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-42-6; ISSN 2283-9216 
 

Modelling of Multi-Scale Uncertainties in  

Biofuel Supply Chain Optimization 

Dajun Yue, Fengqi You* 
Cornell University, 318 Olin Hall, Ithaca, New York 14853, USA  

fengqi.you@cornell.edu 

Although strategic and operational uncertainties differ in significance of impact, a “one-size-fits-all” approach 

has been typically used to tackle all types of uncertainty in the optimal design and operations of supply chains. 

In this work, we propose a stochastic robust optimization model that handles multi-scale uncertainties in a 

holistic framework, aiming to optimize the expected economic performance while ensuring the robustness of 

operations. Stochastic programming and robust optimization approaches are integrated in a nested manner to 

reflect the decision maker’s different levels of conservatism towards strategic and operational uncertainties. 

The resulting multi-level mixed-integer linear programming model is solved by a decomposition-based column-

and-constraint generation algorithm. To illustrate the application, a county-level case study on optimal design 

and operations of a spatially-explicit biofuel supply chain in Illinois is presented, which demonstrates the 

advantages and flexibility of the proposed modeling framework and efficiency of the solution algorithm. 

1. Introduction 

When hedging against uncertainties in the optimal design and operations of supply chains, only one uniform 

approach has been typically used to tackle all types of uncertainty (Shah, 2004). However, uncertainties at 

strategic and operational scales may differ in their significance of impact (Yue et al., 2014). Strategic 

uncertainties have impacts over a significant duration of the project’s lifetime. Once realized, they would 

remain unchanged for a considerable period of time. Examples of strategic uncertainties include climate and 

weather, technology evolution, incentives and policies, and network stability (Gao et al., 2015). In contrast, 

operational uncertainties change more frequently and often lead to immediate adjustment in operational 

decisions. Examples of operational uncertainties include variations in raw material quality and composition, 

supply and demand, cost and price, as well as in lead times of production, transportation, and material 

handling activities (Gebreslassie et al, 2012). Moreover, the realizations of operational uncertainties may be 

associated with that of strategic uncertainties. For example, the yields of agricultural products are expected to 

be dependent on the climate and weather. In addition, a decision maker may hold different levels of 

conservatism towards strategic and operational uncertainties. For instance, one might be less conservative 

towards strategic uncertainties and willing to explore all possibilities, but be more conservative towards 

operational uncertainties considering factors such as demand fulfill rate (Garcia and You, 2015a). 

Due to all the concerns above, a “one-size-fits-all” approach may not be sufficient for efficiently handling multi-

scale uncertainties in optimal design and operations of supply chains. Therefore, a heterogeneous modeling 

framework is required. Consequently, the goal of this work is to propose a holistic modeling framework and 

solution algorithm to handle both strategic and operational uncertainties for supply chain optimization 

problems. Major novelties of this work are summarized below: 

 A novel framework that handles multi-scale uncertainties in optimal design and operations of supply chains; 

 A nested stochastic robust optimization framework that models the different characteristics of and the 

decision maker’s different levels of conservatism towards strategic and operational uncertainties via 

integration of stochastic programming and robust optimization approaches; 

 A decomposition-based C&CG algorithm that efficiently solves the resulting multi-level MILP problem. 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1652035 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Yue D., You F., 2016, Modelling of multi-scale uncertainties in biofuel supply chain optimization, Chemical 
Engineering Transactions, 52, 205-210  DOI:10.3303/CET1652035   

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2. General modeling framework 

2.1 Deterministic model 
Before introducing the proposed stochastic robust optimization model for supply chain design and operations 

under multi-scale uncertainties, we first present the corresponding deterministic model. A general form of the 

deterministic model is given below, based on which an uncertainty counterpart will be developed. 

              min , , , : ,  , 0,1 , ,
nstr opr m r

C x y C x y z Ax By d Ex Fy Gz h x y z                           (PD)  

where x denotes the binary variables for strategic decisions, which model the selection of facility locations; 

choice of manufacturing technologies, storage types, and transportation modes, etc.; y denotes the continuous 

variables for strategic decisions, which include the capacities of manufacturing facilities and warehouses; z 

represents the continuous operational decisions, which involve the quantities of raw materials to acquire from 

suppliers and products to sell to customers, target of production to reach, as well as amounts of materials to 

transfer through the transportation links and to hold in the inventory.  

The deterministic model (PD) can be divided into two parts. The first part corresponds to supply chain design. 

 ,strC x y  stands for the amortized capital cost associated only with strategic decisions. The inequality 

 Ax By d  represents the strategic-level constraints, involving network design, capacity limits, as well as 

economic models for capital investment and fixed operating cost. The second part corresponds to supply 

chain operations.  , ,oprC x y z  stands for the operational cost that is influenced by both strategic and 

operational decisions. The inequality   Ex Fy Gz h  represents the operational level constraints, covering 

material balances, bounding constraints, and cost calculations of various supply chain activities. Problem (PD) 

is often formulated as a mixed-integer linear programming (MILP) model. 

2.2 Stochastic robust optimization model 

When supply chain design decisions must be made before the realization of uncertain parameters, two-stage 

optimization models are often employed (Birge and Louveaux, 2011). In such models, the first-stage 

(strategic) decisions are made “here-and-now” before realization of any uncertainty, and the second-stage 

(operational) decisions are made in a “wait-and-see” manner after all uncertainties are revealed. Instead of 

handling all types of uncertainty with a uniform approach, we handle strategic and operational uncertainties 

differently and reflect the decision maker’s different levels of conservatism with the following stochastic robust 

optimization model. 

              min , , : 0,1 ,
opr nstr m

C x y E C x y Ax By dx y                (P0)  

where     
 

          



       , maxmin , , , : , ,

opr
opr r

zu U
C x y C x y z u Gz h E x F y K u z .   denotes a 

particular realization of strategic uncertainties, and u  denotes a particular realization of operational 

uncertainties. As can be seen, the proposed model (P0) explicitly reflects the decision maker’s different levels 

of conservatism. An expectation over all strategic uncertainty realizations is considered in the objective 

function, showing that the decision maker is considering all possibilities. On the other hand, the value of 

  ,
opr

C x y  represents the worst-case operating cost under strategic uncertainty realization   at given first-

stage decisions according to the max-min problem (Shi and You, 2016). Therefore, the decision maker 

hedges against the worst-case realization of operational uncertainties defined by the uncertainty set U  in 

problem (P0) (Gong et al., 2016). Please also note that the uncertainty set U  for operational uncertainties is 

indexed by realization   of the strategic uncertainty. In this way, we allow the range of variation in operational 

uncertainties to be dependent on the realization of strategic uncertainties. This is critical as strategic 

uncertainties may have significant impacts on operational uncertainties. As there can be infinite realizations of 

strategic uncertainties, the expectation function in (P0) is usually transformed approximately into a tractable 

optimization problem by introducing a finite number of scenarios. We name the resulting model a nested 

stochastic robust optimization model, because a robust optimization problem is nested in a two-stage 

stochastic program. 

 

     


 
     

 
min , , : , 0,1 ,

opr nstr m
ss

s S

C x y p C x y Ax By d x y                 (P1) 

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where     


       , maxmin , , , : ,  ,
s s

opr
opr r

s s s s s s s s s s s
zu U

C x y C x y z u Gz h E x F y K u z s S . The index   in 

(P0) is replaced by scenario s. The expectation function is replaced by the summation over all scenarios, 

where sp  denotes the probability of scenario s of strategic uncertainty realization. This is relevant when we 

have a number of long-term projections at hand (e.g., good, normal, and bad scenarios for the future climate) 

or we have a good estimate of the distribution of the strategic uncertainties (e.g., process conversion 

efficiency from historical data). The proposed stochastic robust optimization model (P1) is a multi-level MILP, 

which cannot be solved directly by any off-the-shelf solver. 

3. Solution strategy 

To solve problem (P1), we take advantage of the C&CG algorithm (Zeng and Zhao, 2013). It can be shown 

that the value of uncertain parameters at the optimal solution will be at one of the extreme points of the 

uncertainty set. Therefore, if we enumerate all the extreme points, then problem (P1) can be reduced to an 

equivalent (probably large-scale) MILP by removing the max-min function. Specifically, we can replace the 

content in the curly bracket in problem (P1) with the following. 

           , , , ,, , , , , ,
opr

opr r
s s l s l s l s s s s s l s

C x y C x y z Gz h E x F y K u z l L                (EP) 

where ,s lu  denotes an extreme point of uncertainty set sU .  

Next, we decompose the problem using partial enumeration, and obtain a master problem that has the same 

structure as the equivalent MILP but contains only a subset of the extreme points. Thus, the master problem 

constitutes a valid relaxation and provides a lower bound. In order to obtain upper bounds and feasible 

solutions, we solve a series of max-min operational-level subproblems at given first-stage decisions and 

strategic uncertainty realization s. It can be proved that a set of new extreme points would be generated in 

each iteration if the termination criterion is not met. Based on these extreme points, we can add an optimality 

cut to the master problem, which is in the form of (EP) but corresponds only to one new extreme point. Explicit 

form of the master and subproblems are omitted here. Implementation of the improved C&CG algorithm is 

shown in Figure 1. 

 

Figure 1: Flowchart of the improved C&CG algorithm  

4. Case study 

4.1 Case description 
To demonstrate the applicability of the proposed modeling framework and solution algorithm, we consider an 

application on the optimal design and operations of a potential biofuel supply chain, which shows great 

promise for sustainable energy supply (Gong et al., 2015). We modify the deterministic spatially explicit and 

multi-period MILP model in (You et al., 2012) to handle strategic and operational uncertainties in a biofuel 

supply chain. The underlying supply chain superstructure is shown in Figure 2a. In this case study, the 

biomass feedstock is corn stover, and the biofuel product is fuel ethanol (Yue et al., 2015). The biomass-to-

biofuel conversion is via a bio-chemical pathway. The supply chain optimization is performed at the county 

level in Illinois (Yue et al., 2013). We consider 25 biomass suppliers, 40 biofuel customers, and 10 candidate 

207



sites for building biorefineries. Three capacity levels of ethanol production at biorefineries are considered, 

namely 10 – 50, 50 – 100, and 100 – 150 MM gallons/year. We assume that corn stover can only be 

harvested in October and November (Yue and You, 2015). The first-stage decisions include the selection of 

location and determination of capacity for biorefineries, which must be made before the realization of both 

strategic and operational uncertainties. The second-stage decisions involve the various continuous variables 

on biomass acquisition, biofuel sales, inter-site transportation, and inventory management. These second-

stage decisions are made after the realization of both strategic and operational uncertainties. Complete 

recourse is assumed by allowing the use of imported biofuel from an external source to satisfy the unmet 

demand with a penalty cost. 

 

Figure 2: (a) Superstructure of the three-echelon biofuel supply chain; and optimal supply chain layout of (b) 

stochastic robust solution, (c) deterministic solution, and (d) standard stochastic programming solution. 

Backgrounds represent the nominal biomass availability in Illinois  

The objective is to minimize the expected total cost over different scenarios of strategic uncertainty realization 

while ensuring the robustness against operational uncertainties. As shown in Table 1, the strategic 

uncertainties considered include technology evolution and climate and weather. Uncertainty in technology 

evolution is reflected as the deviation of actual process conversion efficiency from the designated value, which 

may originate from design or implementation errors. Uncertainty of climate and weather is reflected in the 

variations of precipitation and insolation, which greatly impact the yield of biomass feedstock. The operational 

uncertainties include variations in biomass supply and biofuel demand. 

Table 1: Scenarios and uncertainty sets 

  Strategic uncertainties Operational uncertainties 

Scenario Probability Conversion Climate Biomass availability Biofuel demand 

Nominal    Normal a d 

S1 1/12 0.9  

Bad [0.7a, 0.9a] 

[0.9d, 1.1d] 

S2 1/6   

S3 1/12 1.1  

S4 1/12 0.9  

Normal [0.9a, 1.1a] S5 1/6   

S6 1/12 1.1  

S7 1/12 0.9  
Good 

 
[1.1a, 1.3a] S8 1/6   

S9 1/12 1.1  

 

4.2 Results and discussions 
The stochastic robust optimal solution under both strategic and operational uncertainties are shown in Figure 

2(b). The optimal first-stage decisions indicate that a total of six biorefineries should be built. For comparison, 

we also solve a deterministic MILP by setting the process conversion efficiency, biomass availability, and 

biofuel demand at the nominal values. The resulting supply chain layout is given in Figure 2(c). To compare 

with the solutions from a uniform modeling approach, we also solve the problem using a standard two-stage 

stochastic programming approach with 90 scenarios. The resulting supply chain layout is shown in Figure 

(a) (b) (c) (d) 

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2(d). As can be observed, the stochastic programming solution is like a compromise solution between the 

stochastic robust solution and the deterministic solution.  

To compare all three supply chain designs in an uncertain environment, we perform a simulation in all the 

aforementioned 90 scenarios for each supply chain design. The results are plotted in Figure 3. The triangles 

denote the perfect information solutions, which are obtained by solving a deterministic model for each 

scenario that optimizes both strategic and operational decisions. The circles denote the stochastic robust 

solutions corresponding to Figure 2(b). The squares denote the deterministic solutions corresponding to 

Figure 2(c), among which the crossed squares indicate that ethanol import from the external source are 

needed in 55 out of the 90 scenarios. The diamonds denote the standard stochastic programming solutions 

corresponding to Figure 2(d), among which the crossed diamonds indicate that ethanol import from the 

external source are needed in 14 out of the 90 scenarios. From the horizontal lines, we can see that the 

perfect information solutions set the baseline for expected cost. The expected cost of the stochastic robust 

solution is higher than that of the stochastic programming solution. The deterministic solution has an expected 

cost that is even slightly higher than the stochastic robust solution. This indicates that deterministic 

optimization at nominal values without considering uncertainties can lead to lower demand fulfill rate and 

higher expected total cost. 

 

Figure 3: Simulation results for different supply chain designs 

To take a close look at the stochastic robust solution, we present its cost breakdown and inventory profile in 

the scenario where all uncertain parameters are set to their nominal values, as shown in Figure 4. 

 

Figure 4: (a) Cost breakdown and (b) inventory profile of the stochastic robust solution under nominal scenario 

4.3 Computational performances 
We solve the MILP master problem and subproblem with CPLEX 12. All models and solution procedures are 

coded in GAMS 24.5. The optimality tolerance for CPLEX 12 is set to 0. As aforementioned, the C&CG 

(a) (b) 

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algorithm involves iterative solution of one master problem and multiple subproblems. The subproblem in all 

iterations has 65 binary variables, 2,161 continuous variables, and 11,103 constraints. The master problem in 

the first iteration has 30 binary variables, 63,421 continuous variables, and 19,551 constraints. As the 

algorithm iterates, the size of the master problem increases because an optimality cut is added at the end of 

each iteration. Among the 36 instances shown in Figure 5, the minimum number of iterations is 1, and the 

maximum number of iterations is 8. The master problem at the 8th iteration has 30 binary variables, 506,941 

continuous variables, and 155,694 constraints. The shortest solution time is 117 CPU s, while the longest 

solution time is about 14.5 CPU h. Overall, the C&CG is shown to be capable of achieving a small optimality 

gap in its early iterations and converge within a reasonable number of iterations and solution time. 

5. Conclusions 

A nested stochastic robust optimization model was proposed to simultaneously handle strategic and 

operational supply chain uncertainties. Considering the cases that a decision maker is less conservative 

towards strategic uncertainties and more conservative towards operational uncertainties, the authors modeled 

the multi-scale uncertainties via integration of stochastic programming and robust optimization approaches. 

The resulting formulation was a multi-level MILP involving expectation over multiple scenarios as well as max-

min problems that guarantee the robustness of operations. The authors employed a decomposition-based 

C&CG algorithm for the solution, which was based on partial enumeration. The application was demonstrated 

by a county-level case study on optimal design of a spatially-explicit biofuel supply chain in Illinois. 

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