Addressing Uncertainty in Formulated Products and Process Design

Article pubs.acs.org/IECR

Addressing Uncertainty in Formulated Products and Process Design Mariano Martín*,† and Alberto Martínez‡ †

Department of Chemical Engineering, University of Salamanca, Plaza de los Caídos 1-5, 37008 Salamanca, Spain Brussels Innovation Center, Procter and Gamble, Temselaan 100, 1853 Strombeek-Bever, Belgium

‡

S Supporting Information *

ABSTRACT: In this paper, we address the simultaneous design of sustainable formulated products and their production process under uncertainty due to the variability in ingredient prices and lack of knowledge of product processability. The problem is formulated as a multiobjective mixed integer nonlinear programming problem where both endogenous and exogenous uncertainties are considered. A two-step strategy is proposed to address uncertainty in formulated products where endogenous is solved at lab scale. Next, the exogenous uncertainties are considered. For the exogenous uncertainty, we compare two strategies. First, we use a scenario reduction technique to obtain the optimal solution when a probability distribution can be computed from market data. The second one considers a sample approximation approach when we only know the range of values of the uncertain variables. The study has been applied to the production of a family of formulated products with different performances and final prices aiming at product efficiency, profitability, and sustainability. The method allows determining the optimal sustainable detergent composition under uncertainty analyzing only a reduced number of scenarios or small samples with good agreement.

1. INTRODUCTION Current market environment aims to satisfy the consumers’ needs by offering a portfolio of products that matches their demands. This poses a challenge to industry and to the field of process systems engineering to simultaneously design the production process and the final product. In recent years, a number of papers have introduced the concept of product design.1−4 However, product design is typically a complex problem that involves the evaluation of the relationship between the product, its performance, and the customer acceptance, which also involves socio-cultural aspects. Thus, a number of studies have focused on evaluating that relationship to find the physicochemical principles of odor, taste, or efficiency toward treating a certain illness.5−9 The effect of the final product on the processability or technical feasibility of the production process is another aspect that also has been ́ 11 inteaddressed separately.10 Lately, Martiń and Martinez grated product and process design by formulating it as a mathematical optimization problem. That work proposes first a number of strategies to model the effect of the ingredients on different properties of the intermediate or final product as well as its performance and involves pooling constraints to define the product formulation for a family of products and a sustainability metrics. However, the study is deterministic. To come closer to the real operation, we need to address the effect of uncertainty on the product formulation and on the design of the process. Uncertainty has traditionally been a variable that affects process design and control, as it has been reported in many papers over the last decades.12−19 Furthermore, the environment where the production process sits is also affected by uncertainty such as the marketplace and the supply chain.20−24 The design of consumer goods is also affected by the uncertainty but in this particular case, at various levels. For instance, coal blending optimization under uncertainty has been © 2015 American Chemical Society

studied to reduce sulfur emissions due to the variability in coal properties.25 Gasoline blending to achieve the desired fuel properties is another typical example.26 Thus, the sources of uncertainty can be either related to the process or the final product such as the process operating conditions or the properties of the intermediates being processed (Endogenous uncertainty) or externally given by the environment, such as product demand or raw material and finished product prices and availabilities (Exogenous).27 Uncertainty can negatively or positively impact the proper operation and market success in any new or modified product. It can also have a significant impact on how easy or difficult it is to incorporate modifications in future generations of existing products. Figure 1 shows a scheme of the most important uncertainties. After these examples, we briefly describe both types of uncertainty. 1.1. Endogenous. There are two types of endogenous uncertainties, either where the decision maker can alter the probability distribution by increasing the probability of one possibility over the other or where the decision maker can only obtain better information by partially resolving the uncertainty. This last one is the more interesting for formulated product design.28−31 New products are always technically risky because of the lack of knowledge in the transformation from the raw materials to the desired final products or the lack of experience at the time of scaling up to full-size production. These uncertainties are minimized either at an early stage such as in lab or pilot plant scale trials or as the production advances. Experience may not help if the behavior or the response is different to traditional products. For instance, an ingredient that works well in one product might not do so in another because Received: Revised: Accepted: Published: 5990

June 2, 2014 April 30, 2015 May 12, 2015 May 12, 2015 DOI: 10.1021/acs.iecr.5b00792 Ind. Eng. Chem. Res. 2015, 54, 5990−6001

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Industrial & Engineering Chemistry Research

Figure 1. Sources of uncertainties by type.31

this point, product design may have an important contribution to put the product into the market sooner.20−24,36 1.2.3. Political and Cultural Context. The socio-economic situation, including the changes in the governments, may affect the regulations (environmental constraints, health security limits), which may force the companies to modify their products to meet the new constraints.28 In this paper, we deal with a particular example, proposed in Martı ́n and Martı ́nez,11 related to the detergent industry. Needless to say, the strategy can be extended to any other formulated product such as drugs, food, cosmetics, just to mention a few representative cases. For this particular case, the uncertainties determine the composition of the final products and thus their performance, environmental impact, and cost. The idea is to be able to tackle such a problem by evaluating the effect of uncertainty on the formulation of the product and the profit so as to provide a robust formulation for the product to be in the safe side reducing the investment risk. Furthermore, most of the work on optimization under uncertainty focuses on one objective; fewer examples address multiobjective optimization problems.37−39 In this case, we simultaneously consider the design of the production process and the final product involving competing objectives such as production cost or environmental impact to obtain profitable, efficient, and sustainable formulated products. We also cope with other objectives such as customer satisfaction by means of evaluating the performance of the product. However, in this example, we are using product performance as a metric to meet this objective. As a result, the utopia point is computed, which provides the optimal detergent composition.

of the interactions among ingredients. Therefore, understanding the interactions between the different components within the product and with the production process is key.31 In a way, this example shares mathematical similarity with planning problems in the literature.28,31 Examples closer to the presented case can be found in the pharmaceutical industry where the planning of trials is expensive and time-consuming. Colvin and Maravelias32−34 produced a series of papers presenting the resolution on endogenous uncertainty and algorithms. In those cases, the endogenous uncertainty is of the kind where the decision maker can alter the probability distribution so that the uncertainty is presented in the outcome of the clinical trials. 1.2. Exogenous. This source of uncertainty has received the most attention because we can only identify it or, in the best case, influence it (i.e., marketing studies), but we cannot minimize it. On the basis of the paper by Weck et al.,35 we can identify three sources of uncertainty that can be reapplied to formulated products. 1.2.1. Context. There is often huge uncertainty in the way a product is used by the customer and the conditions under which it has been designed to operate or to be used. If the customer does not follow the indications, the performance of the product may not meet the expectations. 1.2.2. Markets. In the customer products industry, the lifetime of the product can be very short, seasonal, or follows trends. The customer may change the demand of a particular product because something else is released by a competitor or simply because it is no longer of interest. That is not the same for all the products. In some cases, the economic situation may force to change from better quality products to cheaper ones. Marketing and product research professionals play here an important role. Being the first into the market is also important so that your product obtains the share before anyone else. At

2. UNCERTAINTY RESOLUTION Process and product design under uncertainty can be formulated as multiperiod or stochastic mixed integer nonlinear 5991

DOI: 10.1021/acs.iecr.5b00792 Ind. Eng. Chem. Res. 2015, 54, 5990−6001

Article

Industrial & Engineering Chemistry Research programming problems (MINLPs)12,13 where the objective is to minimize the expected value:

number of deterministic cases. However, when the original probability distribution is discretized, the number of scenarios is really large. The complexity of the formulation of the problem suggests the need for different approaches to address the problem such as scenario reduction techniques. The idea in Karuppiah et al.40 was to come up with a minimum reduced set that approximates the complete distribution minimizing the error of approximation. In a multiscenario problem, let θ = {θi}i=1,...,I be the vector of uncertain parameters. Let the uncertain parameter θi take on a finite set of values given by {θjii }ji = 1,...,Ji. The probability associated with the uncertain parameter θi taking on a value θjii is pjii . With multiple uncertain parameters, these can be combined together by considering the corresponding Cartesian product of all the values of the uncertain parameters to yield the set with |S| scenarios. The scenario s involves the following vector of uncertain parameters θs̅ ={θjii }i=1,...,I, and there is a total of |S| = Πi I= 1Ji scenarios. Assuming independent distributions, the probability associated with a scenario s in the original set of scenarios is given by ps = p1j1,j2,...,IjI = Πi = 1pjii. To select a minimum subset from the original set of scenarios, Karuppiah et al.40 proposed that the sum of the probabilities of the new scenarios in which the uncertain parameter value θjii appears is equal to pjii . To obtain a set of scenarios that satisfies the premises, we solve the problem given in eq 3, which corresponds with a linear relaxation to the original MILP formulation.40 In this reformulation, the objective function involves the known probabilities of the scenarios so that in essence it tries to maintain the original scenarios with larger probabilities. As a result, the model provides a reduced set of scenarios with aggregated probabilities. It is not guaranteed that it is the minimum number of scenarios due to the relaxation:

ν* = min Eθ [f (x , y , zi , θi)] x, y, z s. t. gj(x , y , zi , θi) ≤ 0 ∀ j ∈ J x ∈ X , z ∈ Z , y ∈ {0, 1} θ∈Θ

(1)

where y is a vector of binary 0−1 variables denoting the choice of the units or the existence of the streams; x a vector of design variables, in our case, pools or product composition; z a vector of control/state variables, which can vary over periods/ scenarios; and θ a vector of uncertain parameters. The objective is often to minimize the expected value, Eθ, of costs or maximize the expected value of profit. Furthermore, we also consider the environmental impact generated by the product, as well as the customer satisfaction, and thus the problem becomes a multiobjective optimization one. g corresponds to the process and performance constraints. Sahinidis19 presented an interesting review on the mathematical approaches to deal with uncertainty. To deal with probability distributions for the uncertain parameters, typically the expected value function is approximated by a deterministic counterpart by means of a number of scenarios. Sampling-based approximation methods include two basic philosophies: internal sampling and external sampling of the probability distribution of the uncertain parameters. The internal sampling methods perform sampling over the probability distribution inside an algorithm. New samples are generated and accumulated over iterations, and the entire history of samples is used in computation at each iteration. External sampling, which is also called sample-path optimization, sample average approximation (SAA), or stochastic counterpart method, approximates the true problem by the sample average approximation: vN = min x , y , zi

1 N

J1

min f =

JI

1

j1 = 1 j2 = 1

N

∑ f (x , y , zi , θi)

2

I

1

jI = 1

2

I

s. t. J2

i=1

J3

JI I

= p1j1 j1 = 1, ..., J1

I

= p2j2 j2 = 1, ..., J2

∑ ∑ ... ∑ p1̂ j ,2j ,...,Ij

s. t. g (x , y , zi , θi) ≤ 0, ∀ i ∈ Ik ; ∀ j ∈ J ; x ∈ X; z ∈ Z

J2

∑ ∑ ... ∑ (1 − p1j p2j ...pIj ). p1̂ j ,2j ,...,Ij

j2 = 1 j3 = 1

(2)

J1

Hence, once the sample is generated, the stochastic problem becomes a deterministic one, which can be solved by existing deterministic algorithms. In this paper, we consider two case studies whether or not there is a known probability distribution. In both cases, we use a scenario-based approach to account for the uncertainty. However, if the probabilities of occurrence are known, a scenario reduction technique based on the literature will be used due to the size and mathematical complexity of the problem. In case the probabilities of occurrence are not known, we propose an internal sampling algorithm to select a number of scenarios that allow a robust solution without using an average value that has more risk. Both methods are described in the following sections. 2.1. Scenario Reduction Techniques for External Sampling. In some cases, data regarding the probability distribution are available based on historical information on the volatility of the variable. Instead of using a continuous distribution, which has a complex mathematical formulation, we can discretize it into a number of scenarios and deal with a

jI = 1

J3

1

2

JI

∑ ∑ ... ∑ p1̂ j ,2j ,...,Ij j1 = 1 j3 = 1

jI = 1

1

2

⋮ J1

J2

JI − 1

∑ ∑ ... ∑ j1 = 1 j2 = 1 J1

p1̂ j ,2j ,..., Ij = pIjI jI = 1, ..., JI 1

jI − 1 = 1

J2

I

2

JI

∑ ∑ ... ∑ p1̂ j ,2j ,..., Ij j1 = 1 j2 = 1

jI = 1

1

2

=1

I

0 ≤ p1̂ j ,2j ,..., Ij ≤ 1 ∀ j1 , j2 , ..., jI 1

2

I

(3)

2.2. Internal Sampling. On the basis of the difficulty of solving a discritized problem involving a large number of samples,41 since it may be computational prohibitive, and we propose the use of the distance between the objective function at two consecutive iterations as a finalization criteria following an internal sampling based method, see Figure 2. The reason 5992


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∑

Min

θsc(Cost Production + Pools)

Sc = Scenarios

(4)

S.t. Feed availability AiL ≤

∑ xi ,l + ∑ zi ,l ≤ AiU X

∀i (5)

Z

Pool capacity

∑ xi ,l ≤ Sl

∀l (6)

X

Product demand DjL ≤

∑ yl ,j + ∑ zi ,j ≤ DUj Y

∀j (7)

Z

Material balance

∑ xi ,l − ∑ yl ,j ≤ 0 X

∀l (8)

Y

Component balance

∑ CCi ,kxi ,l − pk ,l ∑ yl ,j ≤ 0 X

∀ l, k (9)

Y

Product composition bounds

∑ zi ,j − ∑ pk ,l yl ,j ≤ PUj ,k Z

∑ zi ,j − ∑ pk ,l yl ,j ≥ PjL,k Z

∀ l, k

Y

∀ l, k (10)

Y

Hard bounds

Figure 2. Algorithm.

0 ≤ xi , l ≤ min{AiU , Sl ,

for using this approximation is the complexity of the model, which makes it difficult or even not possible to solve a stochastic optimization problem with a medium to large number of scenarios. We understand that the number of scenarios may become large for a sufficiently small error, δ. Initially, we consider 0.03%. In that case, we would allow a larger δ to obtain a solution for the problem or use the objective value obtained using the mean value of the uncertain variables.

∑ DUj }

∀ Tx

Y

0 ≤ yi , l ≤ min{Sl , DUj ,

∑ AlU }

∀ Ty

X

0 ≤ zi , l ≤ min{DUj , AlU }

∀ Tz

(11)

The data for the bounds can be found in the Supporting Information. Furthermore, process, performance, and environmental constraints for the different formulas are included. For details on the development of the models for these constraints, see Martı ́n and Martı ́nez.11 3.2. Product Design Constraints. In this section, we describe the specific additions to the model so as to design formulated detergents. Bear in mind that if a different product is to be designed such as food, perfumes, or cosmetics, we will change these constraints to others more specific. For instance, instead of product performance, we will be talking about flavor, odor, or skins softness. Environmental constraints remain the same but applied to the specific set of chemicals used. Finally, process constraints are again specific of the type of product, particle, or liquid and the flowsheet to process the ingredients. 3.2.1. Product Performance. Actually, product performance is another objective that somehow measures the customer satisfaction. We consider that a customer that buys product AAA expects top performance. Those selecting BBB expect high performance but a little lower price. Finally, those selecting CCC are on a tighter budget but still expect a good performance:

3. MATHEMATICAL MODELING 3.1. Formulated Product Design Constraints. The problem of the optimization of the formulation of a detergent is presented in a previous paper by the authors.11 In that work, a number of ingredients were mixed to obtain the formulation required for a family of finished products to be sold in the market. We consider surfactants, builders, bleaches, fillers, antifoam, enzymes, polymers, and water as basic ingredients grouping the different types in these classes. This mixture, but those finishing ingredients, is fed to the system and processed through the crutcher followed by the spray drier. However, we consider that the water added here as ingredient is that remaining in the final product. The model has the objective of minimizing the production costs of a set of detergents, AAA, BBB, and CCC, from top quality and high price to medium quality and lower price, subjected to typical pooling problem constraints.42−45 The problem is formulated as follows in eqs 4−17: 5993


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Industrial & Engineering Chemistry Research Performance(j) = (107PQ (j, Surfactants) + 1872PQ (j, Enzyme)

Particle(j) = 224.5 + 1509.78PQ (j, Water) + 1000PQ (j, Filler) − 31PQ (j, Water) × PQ (j, Filler);

+ 53.9PQ (j, Builder) + 134PQ (j, Polymer) + 119PQ (j, Bleach))

(12)

Particle(j) ≥ 400;

Performance(AAA) ≥ 0.95

Performance(CCC) ≥ 0.70

(13)

Cakest(j) = 2.98PQ (j, ′Water′) + 2.69PQ (j, ′Polymer′) + 0.08PQ (j, ′Polymer′) × PQ (j, ′Water′);

3.2.2. Environmental Constraints. The environmental burden is defined as eq 14 as the summation of the burden that the produced detergents generate on the environment. However, since we are not dealing with actual chemicals for proprietary reasons, for which we can easily find the environmental impact in different databases, we propose weights for the environmental burden generated by each of the families of ingredients as presented in Table 1, from 0−

Cakest(j) ≥ 0;

ingredient

CEnv

reason

surfactant builder bleaches fillers antifoam enzymes polymers water

9 7 10 5 5 2 8 0

the more you have, the highest impact affects water by binding ions oxidant

3.3. Identification of the Uncertainty. 3.3.1. Endogenous Uncertainty. It represents the uncertainty related to the performance of finished products due to, for instance, a new ingredient or how a new formulation is going to behave across the different stages involved in its processing through the crutcher or the spray drier tower. Alternatively, we can develop models to relate the processability of the mixture as a function of the composition. Most of these uncertainties, including the error in the fitting, of the performance model, or process constraints, can be resolved at lab or pilot plant scale where the formulations are tested before production. Therefore, in a way, we can be talking about a gradual uncertainly resolution similar to the proposed approach by Tarhan and Grossmann.46 Thus, in this work, we do not address endogenous uncertainty within the problem formulation. 3.3.2. Exogenous Uncertainty. In this paper, we focus on the uncertainty due to the market because we can monitor the volatility in the prices, product demand, and ingredients availability, but we cannot actually control how the market is going to behave in terms of the prices of the raw materials or the demand for the different finished products. Furthermore, the market is what drives the consumer goods industry, and thus we can see how this methodology can help to minimize losses. Thus, we first solve internally the endogenous uncertainty, and next we use the approaches presented in section 2 to deal with the exogenous type of uncertainty. From now on, only exogenous uncertainty will be considered. The exogenous uncertainty is addressed using a scenario-based approximation where the probability distribution is discretized into scenarios. Two cases of study are evaluated based on the availability of information regarding the probability distribution, whether the information is available based on historic trends or if there is only a typical range of values. Each case presents particular challenges, and thus the solution procedure is different as described above.

reduce operation temperature and dose organic

10.11 The reasons for those values are provided in the same table considering the benefits or impacts that each family can have when the detergents are released to the environment after the washing process. We consider that the use of wastes is beneficial for the environment since there is no need to produce further chemicals, and therefore in eq 14, the contribution of the use of wastes is negative: np i = indiv . ing ning

∑ ∑ ∑ CEnv,kCCi ,kzi ,j +

i k n pools i = indiv . ing ning

∑ l

−

∑ ∑ CEnv,kCCi ,kxi ,l i

k

ning

np

∑ ∑ ∑ CEnv,kCCi ,kzi ,j j

−

wastes n pools

k

ning

∑ ∑ ∑ CEnv,kCCi ,kxi ,l l

wastes

k

(14)

4. CASES OF STUDY The main parameters used in the base problem, eqs 3−12, such as raw material and final product prices, ranges for the raw material purity or composition (including the composition of the wastes) finished product composition, availability of raw materials, product demands, can be seen in Martiń and ́ 11 We consider up to three different wastes to evaluate Martinez. their recycling assuming bounds for their availability, up to five pools, the addition of the environmental constraint, eq 14, transforming the problem into a multiobjective optimization one, as well as process constraints, eqs 16 and 17. The wastes are considered to be available at no cost to favor their use, while

We use the ε-constraint method to deal with this as a second objective: EnvBurden ≤ ε

Cakest(j) ≤ 1.4 (17)

Table 1. Weights for Environmental Function

j

(16)

Cake strength is given by eq 17:

Performance(BBB) ≥ 0.80

EnvBurden =

Particle(j) ≤ 500

(15)

3.2.3. Process Constraints. As monitoring properties of the final product, we consider particle size, responsible for detergent dissolution among others, and cake strength, which defines flowability of the powder. By using data from open literature, Martı ́n and Martı ́nez11 developed simple models to relate them to some of the ingredients so as to impose them as process constraints for the product formulation. The particle size (μm) is computed as in eq 16: 5994


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amount as the environmental burden decreases and with low enzyme and antifoam amount. Actually, the problem is a two−stage stochastic problem with recourse, even though as presented in the previous paper, Martı ́n and Martı ́nez,11 the small number of ingredients reduces the need for pools. In this section, we present two cases of study. The first one assumes that, on the basis of data from historical trends and market studies, we have a probability distribution with known probabilities of occurrence for the prices of the ingredients. In this case, we use a scenario reduction technique from the literature40 to solve the problem. However, in many cases, it is not possible to have the information related to the probability distribution of the prices but only a range. Thus, we propose a simple algorithm to address product design under uncertainty. 4.1. Estimated Probabilities. In the first case of study, we consider that the uncertain variable is the ingredient price, and we assume that we can estimate the probabilities of occurrence of these prices based on previous information. Therefore, the probability of distribution of the prices is assumed to be as follows. We consider three levels of prices: low, medium, and high for each ingredient. Since we have a distribution, we can also compute an average price for each of the ingredients. Table 3 presents the values for the prices, the corresponding probability, and the weighted average price. We assume that the use of wastes is for free since otherwise one has to pay to treat them or to dispose them. With this distribution, there are 2187 different scenarios to be considered within the stochastic optimization framework. The large problem size resulting from the number of scenarios together with the mathematical complexity of the problem results in the fact that we cannot find a global optimum solution for the stochastic problem using BARON 9.0.6. However, we can use Karuppiah et al.40 algorithm to determine a reduced set of scenarios that optimally approximates the distribution. In this way, instead of considering the full space of 2187 scenarios, the optimal reduced set consists of only 10 with aggregated scenarios whose probabilities are given in Table 4. In Figure 4, we compare the effect of the environmental burden on the profit we obtain from the detergent formulas using the composition obtained when we use the best case

the contribution to their environmental burden is negative since we recycle those materials instead of generating waste streams. On the basis of the comments in section 3.2, we focus on exogenous uncertainty. Data in the from the literature show that detergent demand depends on the population, and within a season or a year time, there is small variability.47 On the other hand, ingredients price, our raw materials, are more volatile. Thus, the main problem is the large number of ingredients, and therefore, the number of scenarios may explode. We consider here prices for our example based on Martı ́n and Martı ́nez.11 We have as ingredients surfactants, builder, bleaches, filler, antifoam, enzymes, polymers, and water, and three wastes whose composition can be seen in the appendix of the previous paper. The range of prices for each one is given in Table 2. Table 2. Range for Raw Material Cost ingredient

price range (€/kg)

surfactant builder bleach filler antifoam enzymes polymer water

0.07−0.12 0.005−0.015 0.08−0.12 0.008−0.012 0.35−0.6 0.8−1.3 0.08−0.13 0.01

We first present the detergent composition for the best case scenario, in other words, for minimum price of all ingredients and different environmental burdens, see Figure 3. In Figure 3, we can see that the AAA product has the higher amount of enzymes and antifoam agents, while the composition is quite stable with the environmental burden. Therefore, we can say that it is a robust product since the formulation is stable over a range of environmental burdens. The BBB product suffers the larger changes in composition to meet the constraints by reducing the builder fraction and increasing the surfactants. As a result, the particle size and Cake strength increase as the environmental burden is forced to be lower. CCC product is also quite stable in composition reducing slightly the builder

Figure 3. Composition for the best case scenario. 5995


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product design. The stochastic solution when using 10 scenarios is close to the one using the average scenario representing the 2187 scenarios, but it is more conservative. In Figure 4, we need to be aware of the fact that the worst case scenario is not completely informative. While it is simple to say that the best case is obtained when the prices of the raw materials are the lowest, the worse case does not corresponds to the highest prices. The reason is that not all the ingredients are involved in the performance and the processability constraints. Therefore, for the highest price of the raw materials, the model will look for a composition that meets the constraints while reducing the cost. Thus, in Figure 4, we look for how close the optimal case considering raw material price uncertainty (stochastic solution) is to that using the average cost of the raw materials (average) and also the losses in benefits with respect to the best case (best). Finally, we see from Figure 4 that we can compute the utopia point, at around an Environmental burden of 1200, which can provide an interesting result for the optimal detergent composition with a reduced environmental burden and still profitable. In terms of composition, in Figure 5, we see the optimal formulation of the three detergents using averaged prices for the ingredient, and in Figure 6, we see the composition obtained from the stochastic optimization of the set of 10 scenarios that approximate the distribution. In both cases, the compositions are similar, and thus either by using the average price of the ingredients or optimizing the reduced set of scenarios, we are on the safe side when facing price variability. The main difference can be seen when reducing the EnvBurden since the composition in builder and filler is modified. The advantage of the reduced set of scenarios is the fact that we have more information, and the solution is more robust. In Figure 7, we compare, for two values of the environmental burden, 1500 and 1100, the optimal composition of the formulas for each of the 10 scenarios that optimally represent the distribution, the composition resulting from the optimization using the average price for the ingredients, as scenario (AP), and the composition resulting from the stochastic optimization as scenario (AS). We see that there are some differences from scenario to scenario among the reduced set of 10 scenarios. 4.2. Unknown Probabilities. In this case, we again consider exogenous uncertainty in the prices of the raw materials. However, sometimes the actual probability of occurrence of a certain scenario is not accessible due to the lack of detailed information and the distribution cannot be easily discretized. We actually can compute a mean price, but in general that mean may not be representative of the market, see Table 5. To address this problem, the algorithm presented in Figure 2 is used. The idea is to determine a random set of scenarios with fixed prices so that we can use that as representative. With this reduced set, as in the previous case, we can solve the stochastic problem directly using commercial software. Following the algorithm described before, we see that as the number of scenarios of the sample increases, the profit at the higher environmental burden comes close to the one obtained for a single average scenario. Furthermore, with 10 scenarios, the difference in the profit with nine is below the error and that it is easy to match the profit obtained for the average values. We can compute the optimal product composition as the weighted average among the optimal products obtained from the scenarios assuming that each scenario has the same probability

Table 3. Ingredient Price for the Different Scenarios ingredient

scenario

price (€/kg)

probability

surfactant

A B C A B C A B C A B C A B C A B C A B C

0.07 0.09 0.12 0.005 0.01 0.015 0.08 0.1 0.12 0.008 0.01 0.012 0.6 0.45 0.35 0.8 1 1.3 0.08 0.1 0.13 0.01 0 0 0

0.3 0.4 0.3 0.2 0.5 0.3 0.3 0.4 0.3 0.2 0.7 0.1 0.4 0.3 0.3 0.15 0.45 0.4 0.25 0.35 0.4 1 1 1 1

builder

bleach

filler

antifoam

enzymes

polymer

water waste 1 waste 2 waste 3

weighted average 0.093

0.0105

0.1

0.0098

0.48

1.09

0.107 0.01 0 0 0

Table 4. Aggregated Scenarios and Probabilities aggregated scenario

aggregated probability

A.A.A.C.C.A.A A.A.C.A.B.C.A A.A.C.A.C.A.A A.C.C.A.C.C.B A.C.C.B.B.C.B C.B.A.B.C.B.B C.B.C.B.C.C.B C.C.A.B.B.C.B C.C.C.A.B.C.A B.B.B.B.A.B.C

0.10 0.05 0.05 0.05 0.05 0.05 0.05 0.15 0.05 0.40

Figure 4. Comparison of lower and upper bound for the solution with the average and stochastic solutions.

scenario, lowest price, an average price for each ingredient, and an averaged composition computed from the reduce set of 10 scenarios. We see that by evaluating 10 scenarios, the solution is more robust, which prevents over optimistic or pessimistic 5996


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Figure 5. One Scenario. Average price for the complete distribution.

Figure 6. Composition computed by the stochastic optimization.

of occurrence, see Figure 8. In this figure, we include the best and worst case using the lowest and highest price, respectively. As in previous cases, the worst case scenario is not fully informative. We see that by using only three scenarios, the small sample cannot represent the whole spectrum. However, with a larger sample of nine or 10, we reach an asymptotic objective value within our expected error, and therefore the algorithm stops. To determine the minimum number of scenarios to consider, we resample to produced groups of three, five, nine, and 10 scenarios, and in all cases, the algorithm stops at 10 scenarios. For the sake of the length of the paper, we do not include further figures like Figure 8. Thus, we consider 10 as a representative number of scenarios. In Figure 9, we compare the objective value for different sets of 10 scenarios versus the objective obtained for the average price. We see that for higher environmental burdens, when the problem is less constrained, the value for the objective function using average prices is higher. For higher environmental burdens, there is not much of a difference between both solutions. Therefore, the use of a stochastic approach is more conservative, as expected. From these results, we also compute the gap between the values for the objective function. Less than

0.15% difference in the worst case was found, and typically around 0.05% difference between the largest and the smallest objective function value was found. In Figures 10 and 11, we present the composition of the detergent formulas that result from the stochastic solution using 10 scenarios (sample 1) and the average price, respectively. We see that there are some differences in the product formulation, in particular, surfactant and builder content and, for the CCC product, the enzyme content as well. As a result, especially the cake strength of final product, CCC is affected, while for products AAA and BBB, the formulation mitigates the effect on the process constraints and the product performance.

5. CONCLUSIONS AND FUTURE WORK The use of mathematical programming techniques for designing the optimal formulation of detergents is a powerful technique that allows simultaneously including process, legal, and performance constraints to the typical pooling problem constraints for the design of economical and environmentally friendly formulations, especially in design under uncertainty. 5997


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Figure 7. Composition for the optimal set of scenarios, mean and averaged scenarios. AP: Composition for the averaged price of each raw material. AS: Composition of the optimal product from stochastic optimization.

Table 5. Scenarios for Unknown Probabilities ingredient

price (€/kg)

mean

surfactant builder bleach filler antifoam enzymes polymer water waste 1 waste 2 waste 3

0.07−0.12 0.005−0.015 0.08−0.12 0.008−0.012 0.35−0.06 0.8−1.3 0.08−0.13 0.01 0 0 0

0.093 0.01 0.1 0.01 0.47 1.03 0.103 0.01 0 0 0

Figure 8. Some results of the iteration.

profit is similar. In the second case, we propose an algorithm to obtain a reduced set so that we optimize the profit. Again, good agreement in terms of profit is obtained by comparing the reduced set of scenarios and the one using an averaged value. The stochastic or averaged solutions provide a profit in between the best and worst case scenarios so that it allows a more robust product composition. The proposed approach allows dealing with uncertainty depending on the availability of the data. Finally, the utopia point allows us to design a competitive product with reduced environmental burden.

We have considered two different approaches evaluating the case when, on the basis of historical trend or market studies, we can associate the scenarios with a probability of occurrence, and in case we only have a typical range of prices. A scenario reduction technique is used in the first case to determine a reduced set of scenarios so that we can compute an averaged product composition and compare to the mean one for the whole set. It turns out that the composition is different, but the 5998


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Figure 9. Objective function for different sets of 10 scenarios versus the one using average prices.

Figure 10. Optimal composition for the stochastic optimization using 10 scenarios.

Figure 11. Composition for the mean value of the price.

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ASSOCIATED CONTENT

S Supporting Information *

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Information regarding the bounds of the pooling problem. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b00792.

Notes

The authors declare no competing financial interest. 5999


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ACKNOWLEDGMENTS M.M. thanks the University of Salamanca for software licenses. NOMENCLATURE Ai = availability of ingredient (i) (kg) CEnv,k = environmental burden weight due to ingredient k CCi,k = composition of ingredient k in the flow i Dj = demand of product j (kg) x(i, l) = flow from raw material (i) to intermediate pool (l) y(l, j) = flow from pool (l) to product (j) z(i, j) = flow from raw material (i) to product (j) p(l, k) = composition in component (k) of pool (l) Pj,k = bound for the product quality PQ(j, k) = composition in component (k) of product (J) S(l) = maximun tank size (kg) X = set of ingredients Y = set of intermediates at the pools Z = set of final products performance(j) = performance of product (j) Envburden = variable to evaluate the burden to the environment of a certain sets of products gamma_pool(l) = cost coeffiecient of pool (l) objval θ = vector of uncertain parameters

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