The Multi-objective Optimization by the Restricted Area Method to Determine the Technological Mode of Cold Drying Process of Carrot Product

Finding the technological mode of cold drying process of carrot product was the major aim of this study. The experiments were carried out according to experimental plannings. Results obtained were to build the multiobjective optimization problem to describe relationships between objective functions with technological factors (temperature of moisture condensation, temperature of cold drying chamber, velocity air (or drying agents) and time of cold drying) of cold drying process of carrot product. By the Restricted Area Method (RAM), solving the multiobjective optimization problem was found out the technological mode of the cold drying process of carrot product as follows: temperature of moisture condensation was Z1 opt = 15.62°C, temperature of cold drying chamber was Z2 opt = 35.79°C, velocity air (or drying agents) was Z3 opt = 11.74m/s and the time of cold drying process was Z4 opt = 16.05h. Corresponding to these optimal factors, the objective functions reached the minimum value in terms of the final product, including the energy consumption of y1P R = 1.62kWh/kg, the residual water content of y2P R = 4.52%, the anti-rehydration capacity of y3P R = 6.43% (Correspondingly IR = 93.57%) and the loss of total β-caroten inside carrot of y4P R = 4.45%.


INTRODUCTION
The carrots are a kind of vegetable, they have been grown very popularly for thousands of years.Originally, carrots have been cultivated in central Asian, Middle Eastern countries and along with parts of Europe.These original carrots are only the bright orangre and look different from common carrots.They have many colour such as featuring red, purple and yellow that we find them in supermarkets.Carrots were cultivated widely in Europe during the 15th and 16th centuries.Firstly they were brought over to North America to grow during this same general time period (Benjamin et al., 1997).Currently, carrots are grown popularly in Southeast Asia (such as Malaysia, Indonesia, Laos, Cambodia, Thailand, Myanmar and Vietnam), tropical countries, China and Brazil.In general, in Vietnam, carrots are very pupolarly planted from north to south.
The carrots have many important nutritional substances for human's health, including: protein, lipid, Fig. 1: Carrots were harvested sugar, carbohydrate, dietary fiber and mineral salts.In addition, they contain many bioactive compounds that have extremely good effect on human's health such as β-carotene, vitamins and enzymes (Rubatsky et al., 1999;Ross, 2005;Bradeen and Simon, 2007;Simon et al., 2008), carrot product in Viet Nam can see in Fig. 1.The ratio of β-carotene components of dry weight in carrot is very high.According to analytical results of Lab room at HCMC University of Technology and Education, the basic chemical composition of carrot product in Viet Nam is presented in Table 1 to 3.
From Table 1 to 3, they are obvious that carrot product in Viet Nam contains many important bioactive compounds and they have high ratio inside carrot product but the most improtant compound inside carrot product is still β-caroten.Because carrot's characteristic has bright orange colour from β-carotene.
On the other hand, β-carotene is not only easily metabolized but also antioxidized.For this reason, βcarotene is very good for human's health.However, carrot product is a very advantageous environment in order that microorganism grows up and develops.If carrots (or carrot product) are not preserved, they will be easily decomposed or hydrolyzed and oxidized, they will be no longer value of use (Ross, 2005;Simon et al., 2008;Sharma et al., 2012).
In the fact that, there are two methods to apply for preserving carrot product, those are the cooling preservation method and the drying method.Fristly the cooling preservation method, carrot product must be preserved in suitable environment.Temperature of preservation environment is maintained from 0 0 C to 10°°°°C during use time and export time.As a result, it makes to increase the expenditure of preservation carrot product.Secondly, the drying method are used the most popular.The carrot product after the drying placed in nylon bags and seaming, it is preserved in usual environment of 25°°°°C.Thus, it will be not lost the expenditure for preservation process (Dzung and Ba, 2007;Haugvalstad et al., 2005).Currently, there are many different drying methods to preserve carrot product, quality carrot produc after drying depend on very much temperature of drying chamber.Therefore, the aim of this research work is study to apply the cold drying method to preserve carrot product because this method can reduce temperature of cold drying chamber as well as reduce the loss of quality carrot product, (Holman, 1986).
According to research results of Luikov (1975), Holman (1986), Gebhart (1993), Heldman and Lund (1992), Dzung et al. (2012) and Dzung (2014), they were obvious that these researches established and solved the mathematical models about heat and mass transfer in the cold drying process of many different types of drying materials.Results obtained were used to describe the kinetics and set up the technological mode of the cold drying process, but the assessment of the qualified products via the cold drying mode reaching the objectives such as minimum energy consumption or residual water content or the anti-rehydration capacity or the loss of total β-carotene in carrot of cold-dried product (final product) still remained unsolved.
According to Dzung and Dzung (2011), Dzung et al. (2011aDzung et al. ( , 2011b)), Dzung (2011Dzung ( , 2012b) ) and Dzung and Ba (2007), the cold drying process is very complicated, it depends on very technological factors such as: temperature of moisture condensation (Z 1 , °C), temperature of cold drying chamber (Z 2 , °C), velocity drying agents (Z 3 , m/s) and time of cold drying process (Z 4 , h).The problem posed here is how to determine the technological mode for the cold drying process of carrot product in order that carrot after cold drying have the best quality, which mean we determine optimal technological factors in order that the outputs reach the minimal level (Fig. 2), including: the energy consumption per weight (y 1 , kWh/kg), the residual water content (y 2 , %), the anti-rehydration capacity (y 3 , %) and the loss of total β-carotene in carrot (y 4 , %) of the final product), (Dzung, 2011).
From Fig. 2, it can be obvious that problem determine the technological mode of cold drying process, which mean we need to solve the multiobjective optimization problem.This is problem that appears regularly in reality and in different fields.In this study, the multi-objective optimization problem for the cold drying process of carrot product was solved by the RAM.The rsults obtained were used to establish the technological mode of cold drying process of carrot product which was the closest to the utopian point but the furthest from the restricted area C, (Dzung, 2012a(Dzung, , 2012b(Dzung, , 2014;;Dzung et al., 2012Dzung et al., , 2015;;Luc et al., 2013).

Materials:
• The materials used for the cold drying experiments were nature carrot, mainly grown in Viet Nam (Dzung and Ba, 2007).• Before the cold drying process, carrots were separated skin and washed, put on shells to remove water, after that cutting thin slice of carrot, It's the water content was 79.06% (Dzung and Ba, 2007). Apparatus: • The cold drying system DSL-02 controlled by computer was used to dry carrot product (Fig. 3a, 3b and 4).• Determining the weight of samples by Satoriusbasic Type BA310S and mass sensor with the range of 0 to 300g and the error of 0.1g.• Determining the volume of samples by Cylinders with the range of 0 to 500ml and the error of 0.1g.• Dual digital thermometer (T.P.34-23) and temperature sensor were used to determine the temperature of moisture condensation, the temperature of cold drying chamber during the cold drying process with the range of 0 to 100 0 C and the error of 0.5 °C.• Determining time of the cold drying process of carrot product by timer.• Determining velocity drying agents by veloccity sensor (DMK-045) with the error of 0.01m/s.• The equipments of High Performance Liquid Chromatography (HPLC) were used to determine the content of β-carotene inside carrot product.
The ideal rehyration capacity of the product means that the in-water content is equal to the out-water content of the product, i.e.G 1 = G i and IR max = 1 = 100%, y 3min = 0.In fact, y 3 >0, IR<100%.

RESULTS AND DISCUSSION
Develop the mathematical models of the cold drying process of carrot product: In the fact that, all objective functions of the cold drying process of carrot product as the energy consumption per weight (y 1 , kWh/kg), the residual water content (y 2 , %), the antirehydration capacity (y 3 , %) and the loss of total βcarotene in carrot (y 4 , %) of the cold-dried product always depended on the technological factors, including: temperature of moisture condensation (Z 1 , °C), temperature of cold drying chamber (Z 2 , °C), velocity drying agents (Z 3 , m/s) and time of cold drying process (Z 4 , h).Therefore, these all objective functions were established by the experimental planning method with the quadratic orthogonal experimental matrix (k = 4, n 0 = 1).In addition, the experimental factors were established by conditions of the technological cold drying of carrot product (Dzung and Dzung, 2011;Dzung et al., 2011bDzung et al., , 2015;;Dzung, 2011Dzung, , 2012b)), they were summarized in Table 4.
The experiments were carried out with all of the factor levels in Table 4 and all of the experimental planning in Table 5 to determine the value of the objective functions according to technological factors in the cold drying process of carrot product, y j = f j (x 1 , x 2 , x 3 , x 4 ) with j = 1 to 4, (Dzung and Dzung, 2011;Dzung et al., 2011aDzung et al., , 2011b;;Dzung, 2011Dzung, , 2012aDzung, , 2012bDzung, , 2014;;Dzung et al., 2012Dzung et al., , 2015;;Dzung and Du, 2012;Luc et al., 2013).The results were summarized in Table 5.

(13)
The anti-rehydration capacity of final carrot product after cold drying process: The loss of total β-carotene in carrot of final product after cold drying process: One-objective optimization problems for the cold drying process of carrot product: From Fig. 2 (Diagram of subjects of cold drying process) was obvious that all objective functions (y j , j = 1 to 4) for the cold drying process of carrot product depended on the technological factors (x i , i = 1 to 4).If every objective function was individually surveyed, these one-objective functions along with the technological factors would constitute the one-objective optimization problems.Because all the one-objective functions were to find the minimal value, the one-objective optimization problems were restated as follow (Dzung et al., 2011a, Dzung, 2011, 2012b;Dzung et al., 2015): Finding in common the test x jopt = (x 1 jopt , x 2 jopt , x 3 jopt , x 4 jopt ) ∈ Ω x = {-1.414≤ x 1 , x 2 , x 3 , x 4 ≤ 1.414} in order that: ( ) x 1 2 3 4 y f x , x , x , x min f x , x , x , x x 1.414 x , x , x , x 1.414 ; j 1 4 According to the results of Dzung et al. (2011b) and Dzung (2011), if all the one-objective optimization problems ( 16) have the same roots: (x 1 jopt , x 2 jopt , x 3 jopt , x 4 jopt ) = (x 1 kopt , x 2 kopt , x 3 kopt , x 4 kopt ) with k ≠ j, these roots called are utopian roots and also roots of multiobjective optimization problem (17).The optimal plan of utopian roots called is utopian plan.If the utopian roots and the utopian plan do not exist, multi-objective optimization problem (17) will be solved to find the optimal Pareto roots and the optimal Pareto plan.Therefore, solving one-objective optimization problems (16) were found to achieve: y jmin = minf j (x 1 , x 2 , x 3 , x 4 ), j = 1 ÷ 4, with the identified domain Ω x = {-1.414≤x 1 , x 2 , x 3 , x 4 ≤1.414}.
From Table 6, it was also obvious that the utopian root and utopian plan did not exist.Therefore, by the RAM, multi-objective optimization problems ( 17) must be solved to find the optimal Pareto root and the optimal Pareto plan in order that optimal Pareto effect y P R = (y 1P R , y 2P R , y 3P R , y 4P R ) closest to the utopian point f UT (Dzung et al., 2011b;Dzung, 2011Dzung, , 2012b;;Dzung et al., 2015).

Multi-objective optimization problems for cold drying process of carrot product:
It was easilly obvious that all objective functions (y j , j = 1 to 4) always depened on the technological factors (x 1 , x 2 , x 3 and x 4 ) of the cold drying process of carrot product, with the identified domain Ω x = {-1.414≤ x 1 , x 2 , x 3 , x 4 ≤ 1.414}.Consequently, the multi-objective optimization problem to determine the technological mode of the cold drying process of carrot product appeared in this case and it was restated as follow: Finding in common the root x = (x 1 opt , x 2 opt , x 3 opt , x 4 opt ) ∈ Ω x = {-1.414≤ x 1 , x 2 , x 3 , x 4 ≤ 1.414} in order that (Dzung et al., 2011b(Dzung et al., , 2015;;Dzung, 2011Dzung, , 2014;;Luc et al., 2013): where, y 1 <C 1 = 2.5; y 2 <C 2 = 5.2; y 3 <C 3 = 10; y 4 <C 4 = 6.5 (18) The cold drying mode of carrot product established was based on factors including: economic, technicality and quality of the product obtained.Experimental results were obvious that: if the energy consumption for 1 kg final carrot product was over C 1 = 2.5kWh, it would increase the final carrot product price and difficult commercialization.If the residual water content of the final carrot product was over C 2 = 5.2%, the microorganisms would be capable to grow and develope and damage products.Besides, If the antirehydration capacity of the final carrot product was over C 3 = 10%, carrot would be denatured, not be able to recover the original its quality.As a result, quality of product reduced.In addition, if the loss of total βcarotene in carrot of the final product was over C 4 = 6.5%, natural color and flavor of carrot would be destroyed and nutritional value of product reduced.According to Dzung et al. (2011a), if the multiobjective optimization problem was solved by the utopian point method, value of of the objective functions (y 1 , y 2 , y 3 and y 4 ) would not satisfy conditions (17), so the multi-objective optimization problem have to be solved by the RAM (Dzung et al., 2011b(Dzung et al., , 2015;;Dzung, 2011).
The purpose of the experiment was to reach the targets of the cold drying process of carrot product which were expressed by 4 regression Eq. ( 12), ( 13), ( 14) and ( 15), but the tests satisfying all function values (y 1min , y 2min , y 3min , y 4min ) could not be found.Hence, the idea of the four-objective optimization problem was to find the optimal Pareto test for the optimal Pareto effect y(x R ) = y P R = (y 1P R , y 2P R , y 3P R , y 4P R ) closest to the utopian point y UT = (y 1min , y 2min , y 3min , y 4min ) = (0.79, 4.51, 6.31, 4.38).
By choosing R(x) as the objective function, the mobjective optimization problem is restated as: From ( 20), it can be seen: 0≤R(x R )≤1.If R(x R ) = 1, x R = Z UT -the utopian test.If R(x R ) = 0, one of the values of f j violates (21), which means that f j (Z) belongs to the restricted area C (18).The four-objective optimization problem needed to indentify x R = (x 1 R , x 2 R , x 3 R , x 4 R ) ∈ Ω x in order that R(x 1 R , x 2 R , x 3 R , x 4 R ) = Max{R(x 1 , x 2 , x 3 , x 4 )} = Max{R(x)}.The maximum value of (19) was determined by the meshing method programmed in Matlab 7.0, results were pretened in Table 7. Substituting x 1 R , x 2 R x 3 R , x 4 R into these Eq. ( 12), ( 13), ( 14) and ( 15), the optimal Pareto effect was obtained as: y 1P R = 1.62; y 2P R = 4.52; y 3P R = 6.43; y 4P R = 4.45; The rehydration capacity of the cold-dried product was determined as: IR = 100-y 3P R = 100-6.43= 93.57% By the RAM, solving the multi-objective optimization problem with R-optimal combination criterion which satisfied the maximum R-optimal combination criterion (R min = 0.83) was determined the optimal Pareto test (or the technological mode of cold drying process of carrot product) as: temperature of moisture condensation was Z 1 opt = 15.62 0 C, temperature of cold drying chamber was Z 2 opt = 35.79 0C, the velocity drying agents was Z 3 opt = 11.74m/s and the time of cold drying process was Z 4 opt = 16.05h.Corresponding with the optimal Pareto test was also determined the optimal Pareto effect as: the energy consumption per weight of 1 kg final product was y 1P R = 1.62kWh/kg; the residual water content of the final product was y 2P R = 4.52%; the anti-rehydration capacity of the final product was y 3P R = 6.43% (Correspondingly IR = 93.57%)and the loss of total β-carotene in carrot of the final product was y 4P R = 4.45%.Compared with the experimental results from the Table 5, these results above were suitable and satisfying with the objectives of the problem.

Experiment to test the results of multi-objective optimization problem:
The cold drying process of carrot product was carried out at the optimal Pareto test: temperature of moisture condensation of Z 1 opt = 15.62°°°°C,temperature of cold drying chamber of Z 2 opt = 35.79°°°°C, the velocity drying agents of Z 3 opt = 11.74m/s and the time of cold drying process of Z 4 opt = 16.05 hours.The experimental results were determined: The energy consumption per weight of y 1 = 1.65kWh/kg; the residual water content of y 2 = 4.59%; the antirehydration capacity of y 3 = 6.92% (or the rehydration capacity of IR = 100 -y 3 = 93.08%)and the loss of total β-carotene in carrot of y 4 = 4.67% of the colddried product.
Consequently, it was very noticeable that the results from the optimization problems of cold drying process of carrot product had the approximation to the experimental results.
Carrot product was dried by the cold drying method at the optimal Pareto test: Z 1 opt = 15.62°°°°C;Z 2 opt = 35.79°C;Z 3 opt = 11.74m/s;Z 4 opt = 16.05h.The final product obtained could be seen in Fig. 4. It was certain that the optimal Pareto test and the optimal Pareto effect of the multi-objective optimization problem of cold drying process of carrot product be possibly applied to determine the technological mode of cold drying process of carrot product for using in the industry.
The relationship between y 1 , y 2 , y 3 and y 4 with 2 variables x 2 , x 4 , x 1 R = 0.124 and x 3 R = 0.914; was   The energy consumption per weight of 1kg final product y1P R , (kWh/kg) 1.62 6 The residual water content of final product y2P R , (%) 4.52 7 The anti-rehydration capacity of final product y3P R , (%) 6.43 8 The loss of total β-carotene in carrot of the cold-dried product y4P R , (%) 4.45 factors very well described for cold drying process of carrot product.
Determining technological mode of cold drying process of carrot product: From results on above, it allowed to set up the technological mode during the cold drying process of carrot product in Table 8 as follow: The optimal technological mode of cold drying process of carrot product in Table 8 was obvious, when carrot was carried out cold drying at the optimal technological mode, the quality of carrot product after drying had very good quality (Fig. 4).The technological mode of cold drying process of carrot product was found out in Table 8, it can be completely applied for preservation of carrot product in order to be prolonged use time and export time.

CONCLUSION
The mathematical models (12), ( 13), ( 14) and ( 12) which were established from the experiments, they have quite well described the relationship between temperature of moisture condensation, temperature of cold drying chamber, the velocity drying agents and the time of cold drying process with the energy consumption of 1 kg final product; the residual water content of final product; the anti-rehydration capacity of final product (or the rehydration capacity of final product) and the loss of total β-carotene in carrot of the cold-dried product.The system of Eq. ( 17) and ( 18) was the multi-objective optimization problems of the cold drying process of carrot product.These mathematical models were suitably used for calculating and setting up the technological mode of the cold drying process of carrot product.Solving the multiobjective optimization problems (17) and ( 18) determined the technological mode of the cold drying process of carrot product.The results were presented in Table 8.

Fig. 2 :
Fig. 2: Diagram of subjects of cold drying process ) : Weight of the final product (weight of carrot product after cold drying) U (V) : Number of Voltmeter I (A) : Number of Amperemeter τ (s) : Second; cosϕ-power factor

Fig. 4 :
Fig. 4: Final carrot product of cold drying process

Table 1 :
The basic chemical composition of carrot product in Viet Nam

Table 2 :
The vitamins composition of carrot product in Viet Nam

Table 3 :
The minerals composition of carrot product in Viet Nam

Table 4 :
The technological factors levels design

Table 5 :
The orthogonal experimental matrix level 2

Table 6 :
Minimum roots of each one-objective optimization problems j Value of roots of one-objective optimization problems -

Table 7 :
Minimum roots of multi-objective optimization problemsValue of optimal Pareto roots of multi-objective optimization problem

Table 8 :
The technological mode of the cold drying process of carrot product