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Case report

Unlocking weight loss potential: Investigating the impact of personalized nutrigenetic-based diet in an Indian population

  • Obesity and its related complications have become a pressing public health issue, requiring personalized nutritional and lifestyle interventions. Nutrigenetic diets utilize genetic information to tailor dietary recommendations based on an individual's genetic variations. This case-control study aimed to evaluate the impact of a nutrigenetic diet on weight loss and clinical parameters. Three groups were included: obese individuals following a nutrigenetic diet (n = 27), obese individuals following a generic diet (n = 23), and a control group of individuals with a normal body mass index (BMI) (n = 19). Based on polygenic risk scoring, personalized diet plans were developed that considered various genetic traits such as the impact of high amounts of protein on weight loss, the impact of low amounts of carbohydrates on weight loss, the risk of a high body fat percentage, the impact of a calorie restriction on weight loss, lactose intolerance, and gluten intolerance. By assessing a subject's risk scores, a personalized diet was created. Measurements taken at baseline and after four months included weight, BMI, body fat, lean mass, fasting blood sugar levels, total cholesterol, triglycerides, thyroid-stimulating hormone (TSH), triiodothyronine (T3), thyroxine (T4), and uric acid. Results showed significant differences favouring the nutrigenetic group in weight (p < 0.001), BMI (p < 0.001), and body fat percentage (p = 0.05) when compared to the control and the generic diet groups. Additionally, the nutrigenetic group exhibited significant improvements in triglycerides (p = 0.003). Moreover, the within-group effect among nutrigenetic subjects showed a significant weight reduction (p < 0.001), BMI (p < 0.001), body fat percentage (p < 0.001), fat mass (p < 0.001), fasting blood sugar level (p = 0.019), and uric acid (p = 0.042). These findings suggest that a nutrigenetic diet may yield more effective weight loss and improved clinical parameters compared to a generic diet.

    Citation: Duraimani Shanthi Lakshmi, Sati Bhawna, Ahmed Khan Ghori Junaid, Selvanathan Abinaya, Saikia Katherine, Lote Ishita, Ahluwalia Geetika, Gosar Hetal, Dharmaraj Swetha, Bhatt Dhivya, Kocharekar Akshada, Salat Raunaq, Ramesh Aarthi, AR Balamurali, Ranganathan Rahul. Unlocking weight loss potential: Investigating the impact of personalized nutrigenetic-based diet in an Indian population[J]. AIMS Molecular Science, 2024, 11(1): 21-41. doi: 10.3934/molsci.2024002

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  • Obesity and its related complications have become a pressing public health issue, requiring personalized nutritional and lifestyle interventions. Nutrigenetic diets utilize genetic information to tailor dietary recommendations based on an individual's genetic variations. This case-control study aimed to evaluate the impact of a nutrigenetic diet on weight loss and clinical parameters. Three groups were included: obese individuals following a nutrigenetic diet (n = 27), obese individuals following a generic diet (n = 23), and a control group of individuals with a normal body mass index (BMI) (n = 19). Based on polygenic risk scoring, personalized diet plans were developed that considered various genetic traits such as the impact of high amounts of protein on weight loss, the impact of low amounts of carbohydrates on weight loss, the risk of a high body fat percentage, the impact of a calorie restriction on weight loss, lactose intolerance, and gluten intolerance. By assessing a subject's risk scores, a personalized diet was created. Measurements taken at baseline and after four months included weight, BMI, body fat, lean mass, fasting blood sugar levels, total cholesterol, triglycerides, thyroid-stimulating hormone (TSH), triiodothyronine (T3), thyroxine (T4), and uric acid. Results showed significant differences favouring the nutrigenetic group in weight (p < 0.001), BMI (p < 0.001), and body fat percentage (p = 0.05) when compared to the control and the generic diet groups. Additionally, the nutrigenetic group exhibited significant improvements in triglycerides (p = 0.003). Moreover, the within-group effect among nutrigenetic subjects showed a significant weight reduction (p < 0.001), BMI (p < 0.001), body fat percentage (p < 0.001), fat mass (p < 0.001), fasting blood sugar level (p = 0.019), and uric acid (p = 0.042). These findings suggest that a nutrigenetic diet may yield more effective weight loss and improved clinical parameters compared to a generic diet.



    Dealing with optimization problems is maximizing or minimizing objective functions by selecting optimal parameters and schemes under given constraints. From the perspective of objective function, there are two branches of optimization problems, multi-objective problems and single-objective problems. The single-objective problems have a unique objective function and the result is an undisputed optimal solution. Multi-objective problems usually have multiple objective functions, and the result is usually a pareto optimal solution set, which usually requires trade-offs to select the relatively better solution.

    From the perspective of decision variables, optimization problems are classified into continuous problems and discrete problems. In continuous problems, the decision variables belong to the field of real numbers. The range of continuum problems is vast, and the literature is rich, covering such hot fields as engineering, medical science, and machine learning [1,2,3]. However, the decision variables are often the elements of an integer set in discrete problems. There are many practical optimization problems in the field of discrete optimization, e.g., the Traveling Salesman Problem (TSP) [4,5,6], Graph Coloring Problem (GCP) [5] and DNA Sequence Design Problem (DSDP) [7]. TSP is classified as an Np problem because its time complexity is O (N!) [8]. Although the solution of TSP is very time-consuming [9], it still has many practical applications in many areas, e.g., DNA Fragment Assembly Problem (DFAP) [10], Job Shop Scheduling Problem (JSP) [11,12] and Vehicle Routing Problem (VRP) [13].

    The methods for solving TSP are roughly classified into two categories in literature: deterministic algorithms and nondeterministic algorithms. The deterministic algorithms include, Branch and bound (BnB) [14], Dynamic Programming (DP) [15] and Lagrangian Dual (LD) [16], etc. However, as the size of TSP increases, the performance of such algorithms declines significantly [17]. nondeterministic algorithms are generally referring to approximation algorithms and meta-heuristic algorithms, where meta-heuristic algorithms can solve TSP well in controllable time cost. Numerous meta-heuristic algorithms for solving TSP were put forward in the existing literature. These algorithms usually use discrete operators to reconstruct the original algorithms. For example, Karuna Panwar reconstructed the Gray Wolf Optimizer (GWO) by 2-optimization (2-opt) and hamming distance to solve TSP [6]. Mesut Gunduz proposed a Discrete JAYA algorithm (DJAYA) to solve TSP. In DJAYA, roulette is used to control the behavior of the transformation operator, and two search trend parameters ST1 and ST2 are added to enhance comprehensive optimization ability [18]. Huang et al. put forward a nearest neighbor heuristic information mechanism and obtained Discrete Shuffled Frog-leaping Algorithm (DSFLA). In DSFLA, population diversity is maintained by adopting a reverse roulette strategy, and exploration ability is enhanced by utilizing a separate elite set mechanism [19]. Akhand et al. realized the discreteness of the Discrete Spider Monkey Optimizer (DSMO) by utilizing two new cross operators [20]. To deal with TSP, Cinar et al. put forward Discrete Tree Seed Algorithm (DTSA). In DTSA, a combination of multiple transformation operators is introduced to increase exploration ability, and the final solution is improved by 2-opt [21]. Yongquan Zhou et al combined 3-Opt and 2-Opt to propose a discrete invasive weed optimization algorithm (DIWO) to solve the TSP problem [22], and the team also proposed discrete flower pollination algorithms based on the order-based crossover, pollen discarding behavior and partial behaviors [23]. Although these algorithms have good performance, they still have room for improvement in robustness and time cost. In particular, The TSP is a practical problem with more stringent requirements for robustness and time cost. We hope to propose a new discrete algorithm with stable performance and fast running speed to solve TSP.

    We put forward a Discrete Salp Swarm Algorithm (DSSA) for solving TSP. Salp Swarm Algorithm (SSA) is a swarm-based algorithm [24]. It was originally used to solve benchmark and real problems of continuous optimization, and SSA also has satisfactory performance in solving engineering design problems. There are two main reasons for proposing a discrete version of SSA to solve the TSP problem:

    (1) There are few pieces of literature on the discretization of SSA, especially on TSP.

    (2) The exploration mode of the SSA algorithm is that the leader leads the follower to move, which makes the approach of the whole population towards the optimal solution gradually, which is similar to the common neighborhood search idea in the TSP problem.

    Therefore, this paper proposes an improved DSSA based on the properties of TSP, and compares it with many advanced meta-heuristic algorithms on 23 benchmark instances. Experimental results reflect DSSA the effectiveness and robustness in solving TSP. In addition, the application results of DSSA on TSP also illustrate the application prospect of this algorithm in solving discrete optimization problems. The rest is arranged as follows: In Section 2, TSP and the corresponding mathematical model are briefly described. The original SSA is briefly described in Section 3. In section 4, the d-opt operator, TPALS operator and DSSA are introduced. Section 5 proves that DSSA has good performance and robustness in solving TSP through several experiments. Finally, in Section 6, the conclusion is presented.

    We can describe TSP as follows, a salesman should pass multiple cities and towns to sell goods. The salesman starts in one city, passes through all the planned cities along the way, and ends his trip in the starting city. And to cut down time costs, the salesman should choose the shortest travel path as far as possible. The main difficulty of the TSP is that there are too many potential travel routes: for symmetric TSP of n cities, there is a total of (n-1)/2! Possible paths. The distance from the ith city to the jth city is calculated using Euclidean distance, as shown in Formula Eq (1) [6]:

    di,j=(xixj)2+(yiyj)2 (1)

    The distance from the ith city to the jth city is defined di, j. xi and yi are x coordinate and y coordinate of the ith city, xj and yj are x coordinate and y coordinate of the jth city. To calculate the length of the travel path, we use the f function in Eq (2) [6]:

    min(f)=dn,1+n1k=1dk,k+1 (2)

    Where n is city numbers. If dj, i and di, j are equivalent(i = 1, 2, …, n), then it is called symmetric TSP. We need to find a least cost Hamiltonian path on a weighted graph in TSP [6].

    SSA is a very efficient algorithm which is put forward by Seyedali Mirjalili in 2017, which is inspired by the phenomenon that salps move in a chain when foraging. There is one leader and many followers in salps, where the leader leads swarm to the food position, while followers directly or indirectly follow the leader [24].

    The leader position is renew using Formula Eq (3):

    x1,j={Fj+c1((ubjlbj)c2+lbj)c3>0.5Fjc1((ubjlbj)c2+lbj)c30.5 (3)

    Where x1, j represent the jth dimension leader position and Fj represents the jth dimension food position. ubj represents the jth dimension upper bound, lbj represents the jth dimension lower bound. c2 and c3 are random numbers in [0, 1]. c1 is the key coefficient used to balance exploration and exploitation, its calculation formula is shown in Eq (4):

    c1=2e(4tT) (4)

    Where T is maximum iterations and t is current iteration. The followers position is renewed using formula Eq (5):

    xi,j=12(xi,j+xi1,j)i2 (5)

    Where xi, j represents the jth dimension position of follower i. See 1 for pseudocode about SSA.

     

    Algorithm 1 The Classical SSA
    Initialize the salp population xi (i = 1, 2, ..., n) considering ub and lb
    while (end condition is not satisfied)
      calculate the fitness of each salp
      F = the best salp
      update c1 by (4)
      for each salp (xi)
         if (i = = 1)
          update the position of the leading salp by (3)
         else
          update the position of the follower salp by (5)
         end if
      end for
      amend the salps based on the upper and lower bounds of variables
    end while
    return F

    To cut down the time cost of 2-optimization (2-opt), we improved it to obtain a d-optimization (d-opt). 2-opt is a local search algorithm proposed by Croes [25] which is broad applied by researchers to deal with various discrete problems. The core idea of 2-opt is to select i and j, and then reverse the subsequence from i to j. If the path cost becomes smaller, perform this operation; otherwise, keep the original solution. For example, the original solution is A−D−C−B−E. If i = 2 and j = 5 are selected, the original solution is converted to A−E−B−C−D. The algorithm continuously improves the solution by repeating these steps.

    The time cost of 2-opt is high for two main reasons: 1) 2-opt traverses all possible i and j (i ≠ j), 2) and calculates the total length of the path in each iteration. d-opt improves the above two disadvantages respectively. Firstly, parameter d is introduced to reduce the traversal scale. Where d represents the minimum distance from i to j selected, only subsequences with a length greater than d are selected to attempt inversion. Secondly, d-opt only calculates the change of path length after inversion, not the total path length. See Algorithm 2 and Algorithm 3 for pseudocode about d-opt.

    In order to better integrate the Problem Aware Local Search (PALS) algorithm into DSSA to solve TSP. We improve it to obtain Tsp Problem Aware Local Search (TPALS). PALS is a heuristic algorithm proposed by Alba and Luque in 2007 to solve DNA Fragment Assembly Problem (DFAP) [26]. The solutions of PALS are defined as sequences of ordinal numbers of DNA fragments and replaced the current solution with neighborhood information in each iteration. The neighborhood solution set is obtained by reversing the subsequence from given i to j in the current solution. Unlike 2-opt, the termination condition of PALS is that no better solution exists in the domain solution set.

     

    Algorithm 2 d-opt (tour, d)
    tour: Initial solution
    for i = 1: n-d
      for j = i+d:n
        deltaFCalculateDeltaF(tour, i, j)
        if (deltaF < 0)
          ApplyMovement (tour, I, j)
        end if
      end for
    end for
    return tour

     

    Algorithm 3 CalculateDeltaF(tour, i, j)
    calculate the path length len
    if (i!= 1 and j!= len)
      deltaFdtour[i-1], tour[j]dtour[i-1], tour[i] + dtour[i], tour[j+1]dtour[j], tour[j+1]
    else if (i == 1 and j!= len)
      deltaFdtour[i], tour[j+1]dtour[j], tour[j+1] + dtour[len], tour[j] – dtour[len], tour[i]
    else if (i!= 1 and j == len)
      deltaFdtour[i-1], tour[j]dtour[i-1], tour[i] + dtour[i], tour[1]dtour[j], tour[1]
    else if (i == 1 and j == len)
      deltaFdtour[i], tour[j]dtour[j], tour[i]
    end if
    return deltaF

    Two problems need to be overcome in applying PALS to TSP. Firstly, the indicator for evaluating the neighborhood solution set in PALS is contig, so we delete the calculation about contig and take the solution with a shorter path length as the better solution among the neighborhood solutions. Secondly, there is no need to consider the overlap score between the last fragment and the first fragment in DFAP, so we add the calculation of the last path back to the starting point in TPALS. See Algorithm 4 for pseudocode about TPALS, where deltaF is also calculated with Algorithm 3.

    SSA was originally put forward to solve continuous problems, while TSP is discrete. Therefore, we use discrete operators to reconstruct the original SSA to solve the TSP. The discrete operators used are d-opt and TPALS mentioned above. To increase the exploration ability, we verify the effectiveness of five classical discrete crossover operators respectively, and introduced the operator with the best performance into DSSA. Each slap in the swarm represents a viable solution to TSP.

     

    Algorithm 4 TPALS (tour)
    tour: Initial solution
    repeat
        L ← ∅
        for i = 1:n-1
          for j = i+1:n
            deltaFCalculateDeltaF(tour, i, j)
            if (deltaF < 0)
              L ← L ∪ (i, j, deltaF)
            end if
         end for
        end for
        if L ≠ ∅
           (i, j) ← SelectMovement (L)
           ApplyMovement(tour, i, j)
        end if
    until no changes;
    return tour

    In the proposed method, d-opt is used to update leaders, and the update formula is shown in Eq (6):

    Touri=dopt(Touri,d) (6)

    Where Touri represents ith slap. d is a parameter used to control the search intensity of d-opt, and its updating formula is shown in Eq (7):

    d=[n×(dMax(dMaxdMin)×tT)] (7)

    Where n is city numbers in TSP. The minimum and maximum value of d are defined as dMax and dMin. T is the maximum iterations and t is the current iteration. [.] is integer function.

    We use TPALS and crossover operators to renew the followers position. There are 6 alternative crossover operators in this paper, and the best crossover operator is determined in Experiment 5.1. The update formula of followers is shown in Eq (8):

    Touri=operator(Touri,Touri1) (8)

    Where operator is the best crossover operator determined in the experiment.

    To enhance the exploration ability, DSSA introduced a mechanism named Second Leader Principle (SPL), whose core idea is that in each iteration, a second leader will appear among followers, and the second leader will also use TPALS to renew the position. The formula of second leader is shown in Eq (9):

    Touri=TPALS(Touri) (9)

    The formula for calculating the probability of followers becoming the second leader is shown in Eq (10):

    pi=iNi=2,,n (10)

    Note that there is only one second leader in each iteration. In Algorithm 5, the pseudo code about DSSA can be obtained.

     

    Algorithm 5 DSSA
    Initialize the salp population Touri(i = 1, 2, ..., n)
    while (end condition is not satisfied)
        calculate the fitness of each salp
        bestTour = the best salp
        update c1 by (8)
        for each salp (Touri)
          if (i = = 1)
            update the position of the leading salp by (6)
          else
            update the position of the follower salp by (8) and (9)
          end if
        end for
    end while
    return bestTour

    We tested it on 23 benchmark instances of small, medium, and large symmetric TSP to verify the effectiveness of the DSSA [18]. Table 1 provides the relevant information of benchmark instances that appeared in the article. Where the number after the instance name represents the number of cities, for example, Oliver30 indicates that the benchmark instance has 30 cities. All the benchmark functions used in the paper are from TSPLIB.

    Table 1.  TSP instance used in the experiments.
    Instance Dimension size Optimal
    oliver30 30 420
    att48 48 33522
    eil51 51 426
    berlin52 52 7542
    st70 70 675
    eil76 76 538
    pr76 76 108159
    kroA100 100 21282
    kroB100 100 22141
    kroC100 100 20749
    kroD100 100 21294
    kroE100 100 22068
    eil101 101 629
    lin105 105 14379
    pr124 124 59030
    pr136 136 96772
    kroB150 150 26130
    pr152 152 73682
    u159 159 42080
    pr226 226 80369
    pr264 264 49135
    pr299 299 48191
    pr439 439 107217

     | Show Table
    DownLoad: CSV

    All methods run 20 times for a comprehensive comparison. Each was run with the set parameters: population number N = 50, iterations number T=D+ΣDi=1, where D is the scale of the problem. The results were analyzed by Best, Avg, Standard deviation (Std) and Relative error (Re). The calculation formula of Re is shown in Eq (10) [18]:

    Re=Avgoptimaloptimal×100% (10)

    Where Avg is the average value of the best path cost obtained by running the algorithm 20 times, and Optimal is the Optimal path cost of the benchmark instance. All experiments were carried out under the same experimental environment: Intel(R) Core (TM) I510500 3.10 GHz CPU and 16.00 GB RAM, and were programmed on MATLAB R2020b. The parameters of the algorithm used in this article are shown in Table 2.

    Table 2.  Parameter settings.
    Algorithms Parameters Values
    DSSA Population size 50
    Crossover function SEC
    C1 [0.1, 0.9]
    DGWO Population size 50
    Crossover function 2-opt
    ESA Population size 50
    Successor functions 2-opt & insertion
    Temperature −supΔf/ln(p)
    Cooling constant 0.95
    GA Population size 50
    Crossover function OX
    Mutation functions Insertion & 3-opt
    Cross. prob. 0.95
    Mut. prob 0.25
    Selection function Binary tournament
    Survior function Binary tournament
    FDA Population size 50
    Movement functions 2-opt & 3-opt
    Initial A0i Random number in [0.7, 1.0]
    Initial r0i Random number in [0.0, 0.4]
    α & γ 0.98
    IDGA Population size 50
    Crossover function OB & OBX
    Mutation functions Insertion & 3-opt
    Cross. prob. [0.95, 0.9, 0.8, 0.75]
    Mut. prob [0.05, 0.1, 0.2, 0.25]
    Selection function Binary tournament
    Survior function Binary tournament
    Migration strat. Best–Replace–Worst

     | Show Table
    DownLoad: CSV

    In order to determine the influence of crossover operators and parameter c1 on DSSA, this experiment studied five crossover operators and five parameter combinations on the 19 benchmark instances. The alternative Crossover operators are Partial-Mapped Crossover (PMX) [27], Order Crossover (OX) [28], position-based Crossover (PBX) [29], Order Based Crossover (OBX) [29] and Subtour Exchange Crossover (SEC) [30]. Table 3 shows the influence of different crossover operators on DSSA performance, in which bold numbers represent best results. From Table 3, When the SEC crossover operator is used, the Re of 19 instances is the smallest among all algorithms, and is less than one percent. In addition, see Table 4 for the Friedman test results of Best and Avg. From Table 4, the rank of DSSA-SEC is the smallest on both Best and Avg, and the p-value is much less than 0.05, which indicates that DSSA-SEC is obviously superior to other algorithms in performance and robustness. Therefore, SEC is regarded as the best crossover operator of DSSA.

    Table 3.  Performance analysis of crossover operators.
    Instance Algorithm Best Avg Re Time
    oliver30 DSSA-PMX 420 420.1 0.02% 0.43
    DSSA-OX 420 420.75 0.18% 0.36
    DSSA-PBX 420 420.05 0.01% 0.47
    DSSA-OBX 420 421.1 0.26% 0.41
    DSSA-SEC 420 420 0.00% 0.46
    att48 DSSA-PMX 33522 33738.3 0.65% 1.63
    DSSA-OX 33522 33817.7 0.88% 1.41
    DSSA-PBX 33522 33706.65 0.55% 1.74
    DSSA-OBX 33522 33866.1 1.03% 1.64
    DSSA-SEC 33522 33564.7 0.13% 1.78
    eil51 DSSA-PMX 426 430.65 1.09% 1.95
    DSSA-OX 426 431.8 1.36% 1.70
    DSSA-PBX 428 433.6 1.78% 2.08
    DSSA-OBX 428 435.4 2.21% 1.97
    DSSA-SEC 426 427.55 0.36% 2.04
    berlin52 DSSA-PMX 7542 7566.5 0.32% 2.07
    DSSA-OX 7542 7708.3 2.20% 1.79
    DSSA-PBX 7542 7577.75 0.47% 2.27
    DSSA-OBX 7542 7646.8 1.39% 2.14
    DSSA-SEC 7542 7542 0.00% 2.14
    st70 DSSA-PMX 675 679.5 0.67% 4.93
    DSSA-OX 675 684.35 1.39% 4.48
    DSSA-PBX 675 680.45 0.81% 5.29
    DSSA-OBX 675 684.85 1.46% 5.33
    DSSA-SEC 675 675.05 0.01% 5.29
    eil76 DSSA-PMX 541 549 2.04% 6.45
    DSSA-OX 542 553.3 2.84% 5.82
    DSSA-PBX 545 554.05 2.98% 6.84
    DSSA-OBX 538 555.25 3.21% 6.91
    DSSA-SEC 538 540.5 0.46% 7.02
    pr76 DSSA-PMX 108159 108479.2 0.30% 6.38
    DSSA-OX 108280 109228.9 0.99% 5.79
    DSSA-PBX 108159 109075.8 0.85% 6.77
    DSSA-OBX 108309 109317.6 1.07% 6.83
    DSSA-SEC 108159 108159 0.00% 6.71
    kroA100 DSSA-PMX 21282 21334.5 0.25% 14.77
    DSSA-OX 21343 21714 2.03% 14.31
    DSSA-PBX 21378 21512.55 1.08% 16.31
    DSSA-OBX 21282 21581.35 1.41% 16.90
    DSSA-SEC 21282 21287.8 0.03% 16.08
    kroB100 DSSA-PMX 22141 22252.2 0.50% 14.83
    DSSA-OX 22283 22629.25 2.21% 14.41
    DSSA-PBX 22141 22507.3 1.65% 16.17
    DSSA-OBX 22258 22543.7 1.82% 16.82
    DSSA-SEC 22141 22159 0.08% 15.91
    kroC100 DSSA-PMX 20749 20835.95 0.42% 14.71
    DSSA-OX 20853 21179.15 2.07% 14.43
    DSSA-PBX 20749 20972.65 1.08% 16.89
    DSSA-OBX 20749 21084.1 1.62% 16.85
    DSSA-SEC 20749 20758 0.04% 15.76
    kroD100 DSSA-PMX 21294 21378.1 0.39% 15.19
    DSSA-OX 21294 21742.25 2.11% 14.44
    DSSA-PBX 21309 21744.3 2.11% 16.12
    DSSA-OBX 21343 21684.8 1.84% 17.13
    DSSA-SEC 21294 21323.45 0.14% 15.45
    kroE100 DSSA-PMX 22068 22192.5 0.56% 15.04
    DSSA-OX 22111 22438 1.68% 14.42
    DSSA-PBX 22117 22423.75 1.61% 16.09
    DSSA-OBX 22068 22452.7 1.74% 17.08
    DSSA-SEC 22068 22100.35 0.15% 15.97
    eil101 DSSA-PMX 633 643.65 2.33% 15.55
    DSSA-OX 635 653.55 3.90% 14.89
    DSSA-PBX 645 654.95 4.13% 17.11
    DSSA-OBX 647 655.7 4.24% 17.47
    DSSA-SEC 629 633.95 0.79% 16.87
    lin105 DSSA-PMX 14379 14430.4 0.36% 17.08
    DSSA-OX 14379 14539.95 1.12% 16.85
    DSSA-PBX 14379 14461.35 0.57% 19.24
    DSSA-OBX 14379 14531.85 1.06% 19.95
    DSSA-SEC 14379 14381.2 0.02% 17.90
    pr124 DSSA-PMX 59030 59080.4 0.09% 28.45
    DSSA-OX 59030 59563.4 0.90% 29.50
    DSSA-PBX 59030 59399.75 0.63% 33.46
    DSSA-OBX 59030 59362.8 0.56% 34.25
    DSSA-SEC 59030 59042.15 0.02% 30.52
    pr136 DSSA-PMX 96956 97529.15 0.78% 40.27
    DSSA-OX 97609 99776.55 3.10% 39.50
    DSSA-PBX 97270 98644.85 1.94% 44.66
    DSSA-OBX 97889 99695.8 3.02% 46.27
    DSSA-SEC 96772 97048.7 0.29% 44.97
    kroB150 DSSA-PMX 26141 26285.75 0.60% 54.36
    DSSA-OX 26324 26905.75 2.97% 54.69
    DSSA-PBX 26246 26679.6 2.10% 60.04
    DSSA-OBX 26411 26732.1 2.30% 63.88
    DSSA-SEC 26130 26176.45 0.18% 59.66
    pr152 DSSA-PMX 73682 73866.4 0.25% 54.95
    DSSA-OX 74373 74739.95 1.44% 57.20
    DSSA-PBX 73818 74193.55 0.69% 63.69
    DSSA-OBX 73888 74405 0.98% 64.94
    DSSA-SEC 73682 73763.6 0.11% 60.20
    u159 DSSA-PMX 42080 42339.7 0.62% 62.77
    DSSA-OX 42535 43654.4 3.74% 66.44
    DSSA-PBX 42080 42998.9 2.18% 73.35
    DSSA-OBX 42080 43154.8 2.55% 75.62
    DSSA-SEC 42080 42132.8 0.13% 65.58

     | Show Table
    DownLoad: CSV
    Table 4.  Friedman test of DSSA with different crossover operators.
    Rank & p DSSA-PMX DSSA-OX DSSA-PBX DSSA-OBX DSSA-SEC
    rank (Best) 2.39 3.63 3.34 3.47 2.16
    p 5.999E-05
    rank (Avg) 2.11 4.37 3.21 4.32 1.00
    p 3.774E-13

     | Show Table
    DownLoad: CSV

    In this experiment, DSSA was compared with several classical or advanced algorithms (DGWO, DFA, DICA, ESA, GA, IBA, IDGA), which have been taken from recently published work [6,31]. Table 5 shows the performance of eight algorithms on 23 benchmark instances, with the best results in bold. Where "\" indicates that the algorithm has not been tested on related problems in the original literature. As can be seen from Table 5, DSSA achieved the best results in all indicators. On the one hand, from the Best index, DSSA can get the theoretical optimal value in 23 instances, which shows that DSSA has satisfactory performance in solving TSP problems. According to the analysis, this may be due to the SEC operator and d-opt operator which gradually reduces the search range, which to some extent improves the accuracy of the optimal solution. On the other hand, from the Avg index, DSSA can win in all 23 instances, which shows that DSSA has satisfactory robustness in solving TSP problems. This shows that the combination of the TPALS operator and the SPL mechanism provides a certain degree of randomness to the algorithm and effectively avoids the algorithm falling into the local optimal solution. In addition, DSSA is superior to DGWO and slightly inferior to the other six algorithms in the Time index, this indicates that DSSA takes a little longer to solve the TSP problem, which may be due to the fact that both d-opt operator and TPALS operator contain the behavior of searching for the optimal solution of the neighborhood. But as a whole, the Re of these six algorithms is much larger than DSSA. If more than 1% of the instance of Re were considered as failures, 53.8% (7/13) of DGWO, 82.4% (14/17) of DFA, 88.2% (15/17) of DICA, 76.5% (13/17) of ESA, and 88.2% (15/17) of GA failed, 64.7% (11/17) of IBA and 88.2% (15/17) of IDGA failed. In all DSSA cases, the Re is less than 1%.

    Table 5.  Comparisons with DSSA, DGWO, DFA, DICA, ESA, GA, IBA and IDGA.
    Fun Alg Avg Best Re
    (%)
    Time Fun Alg Avg Best Re
    (%)
    Time
    oliver
    30
    DSSA 420 420 0.00 0.5 eil
    101
    DSSA 633.95 629 0.79 16.87
    DGWO \ \ \ \ DGWO \ \ \ \
    DFA 420 420 0.00 0.4 DFA 659 643 4.77 13.3
    DICA 420 420 0.00 0.5 DICA 663.8 644 5.53 12
    ESA 420 420 0.00 0.7 ESA 658.4 650 4.67 16.3
    GA 422.8 420 0.67 0.2 GA 673.8 655 7.12 10.6
    IBA 420 420 0.00 0.4 IBA 646.4 634 2.77 13.1
    IDGA 421.5 420 0.36 0.2 IDGA 660.7 650 5.04 11.7
    att
    48
    DSSA 33564.7 33522 0.13 1.8 lin
    105
    DSSA 14381.2 14379 0.02 17.90
    DGWO 33600 33523 0.23 3.0 DGWO 14520 14382 0.98 34.3
    DFA \ \ \ \ DFA \ \ \ \
    DICA \ \ \ \ DICA \ \ \ \
    ESA \ \ \ \ ESA \ \ \ \
    GA \ \ \ \ GA \ \ \ \
    IBA \ \ \ \ IBA \ \ \ \
    IDGA \ \ \ \ IDGA \ \ \ \
    eil
    51
    DSSA 427.55 426 0.36 2.04 pr
    124
    DSSA 59042.15 59030 0.02 30.52
    DGWO \ \ \ \ DGWO 59390.9 59030 0.61 44.4
    DFA 430.8 426 1.13 1.6 DFA 59404.3 59030 0.63 18.8
    DICA 432.3 426 1.48 1.8 DICA 59436.9 59030 0.69 19
    ESA 431.6 426 1.31 2.1 ESA 59593.6 59030 0.95 23.1
    GA 440.8 427 3.47 1.7 GA 59901 59030 1.48 17.3
    IBA 428.1 426 0.49 1.7 IBA 59412.1 59030 0.65 18.5
    IDGA 434.4 426 1.97 1.2 IDGA 59912.8 59072 1.50 17.8
    berlin
    52
    DSSA 7542 7542 0.00 2.1 pr
    136
    DSSA 97048.7 96772 0.29 44.97
    DGWO \ \ \ \ DGWO 99310.5 97826 2.62 74.3
    DFA 7542 7542 0.00 2.2 DFA 99683.7 97716 3.01 24.1
    DICA 7542 7542 0.00 2.5 DICA 99583.7 97736 2.91 24
    ESA 7542 7542 0.00 2.3 ESA 99858.3 98499 3.19 29.5
    GA 7542 7542 0.00 2.3 GA 100472.4 98432 3.82 23.8
    IBA 7542 7542 0.00 2.1 IBA 99351.2 97547 2.67 23.4
    IDGA 7542 7542 0.00 2.4 IDGA 99932.7 98532 3.27 23.7
    st
    70
    DSSA 675.05 675 0.01 5.3 krob
    150
    DSSA 26176.45 26130 0.18 59.66
    DGWO \ \ \ \ DGWO 26756.2 26320 2.39 125.2
    DFA 685.3 675 1.53 4.3 DFA \ \ \ \
    DICA 684.7 675 1.44 4.1 DICA \ \ \ \
    ESA 682.1 675 1.05 4.5 ESA \ \ \ \
    GA 709.8 675 5.16 4.2 GA \ \ \ \
    IBA 679.1 675 0.61 3.9 IBA \ \ \ \
    IDGA 690.2 675 2.25 4.1 IDGA \ \ \ \
    eil
    76
    DSSA 540.5 538 0.46 7.0 pr
    152
    DSSA 73763.6 73682 0.11 60.20
    DGWO \ \ \ \ DGWO 74230 73690 0.74 142.8
    DFA 556.8 543 3.49 5.3 DFA 74934.3 74033 1.70 32.1
    DICA 557.6 544 3.64 5.3 DICA 74886.7 74052 1.63 32
    ESA 553.7 546 2.92 5.8 ESA 74969.5 74172 1.75 39.5
    GA 565.4 545 5.09 5.6 GA 75658.3 74520 2.68 33.4
    IBA 548.1 539 1.88 5.1 IBA 74676.9 73921 1.35 31
    IDGA 557.7 545 3.66 5.1 IDGA 75126.7 74249 1.96 32
    pr
    76
    DSSA 108159 108159 0.00 6.7 u159 DSSA 42132.8 42080 0.13 65.58
    DGWO 108900 108159 0.68 13.2 DGWO 42563.3 42142 1.14 142.8
    DFA \ \ \ \ DFA \ \ \ \
    DICA \ \ \ \ DICA \ \ \ \
    ESA \ \ \ \ ESA \ \ \ \
    GA \ \ \ \ GA \ \ \ \
    IBA \ \ \ \ IBA \ \ \ \
    IDGA \ \ \ \ IDGA \ \ \ \
    kroA100 DSSA 21287.8 21282 0.03 16.1 pr
    226
    DSSA 80446.8 80369 0.10 248.4
    DGWO \ \ \ \ DGWO 81153.7 80648 0.95 648.6
    DFA 21483.6 21282 0.95 10.3 DFA \ \ \ \
    DICA 21500.3 21282 1.03 10.8 DICA \ \ \ \
    ESA 21481.7 21282 0.94 14 ESA \ \ \ \
    GA 21812.4 21350 2.49 9.9 GA \ \ \ \
    IBA 21445.3 21282 0.77 10.6 IBA \ \ \ \
    IDGA 21731.8 21345 2.11 10.7 IDGA \ \ \ \
    kroB100 DSSA 22159 22141 0.08 15.9 pr
    264
    DSSA 49166.15 49135 0.06 422.8
    DGWO 22444.6 22159 1.37 34.5 DGWO \ \ \ \
    DFA 22604.8 22183 2.10 11.6 DFA 51837 50491 5.50 93
    DICA 22599.7 22180 2.08 11.3 DICA 51943.6 50553 5.72 94.1
    ESA 22602.2 22202 2.09 13.6 ESA 52198.5 51603 6.23 102.5
    GA 22687.4 22176 2.47 10.7 GA 52499.8 51712 6.85 92.1
    IBA 22506.4 22140 1.65 11.1 IBA 50908.3 49756 3.61 92.5
    IDGA 22712.6 22208 2.59 10.7 IDGA 52290 51653 6.42 94.5
    kroC100 DSSA 20758 20749 0.04 15.8 pr
    299
    DSSA 48286.9 48191 0.20 633.7
    DGWO 21780 20749 1.58 34.4 DGWO \ \ \ \
    DFA 21096.3 20756 1.67 12.8 DFA 49839.7 48579 3.42 149.1
    DICA 21103.9 20756 1.71 11.7 DICA 49880.3 48600 3.51 150.3
    ESA 21170.4 20749 2.03 15.4 ESA 50532.3 49242 4.86 158.7
    GA 21510.4 20861 3.67 10.2 GA 50817.1 49659 5.45 147.6
    IBA 21050 20749 1.45 12 IBA 49674.1 48310 3.08 147.2
    IDGA 21298.7 20830 2.65 11.2 IDGA 50513.3 49572 4.82 149.94
    kroD100 DSSA 21323.5 21294 0.14 15.5 pr
    439
    DSSA 107562.3 107217 0.32 2261
    DGWO \ \ \ \ DGWO 112850.3 110415 5.25 2811
    DFA 21683.8 21408 1.83 12.4 DFA 115558.2 111967 7.78 202.4
    DICA 21666.8 21399 1.75 12.6 DICA 115763.1 111983 7.97 203.7
    ESA 21726.5 21500 2.03 15.9 ESA 116706.9 113497 8.85 206.4
    GA 22184.6 21492 4.18 9.7 GA 116943.4 113576 9.07 208.4
    IBA 21593.4 21294 1.41 11.7 IBA 115256.4 11153 7.50 201.9
    IDGA 21696.9 21582 1.89 12.1 IDGA 116436.1 113207 8.60 205.7
    kroE100 DSSA 22100.4 22068 0.15 16 \ \ \ \ \ \
    DGWO 22131 22410 1.54 34.3 \ \ \ \ \
    DFA 22413 22079 1.56 11.6 \ \ \ \ \
    DICA 22453.3 22083 1.75 11.7 \ \ \ \ \
    ESA 22499.7 22099 1.96 15 \ \ \ \ \
    GA 22741.3 22150 3.05 9.4 \ \ \ \ \
    IBA 22349.6 22068 1.28 11.4 \ \ \ \ \
    IDGA 22721.9 22110 2.96 12.6 \ \ \ \ \

     | Show Table
    DownLoad: CSV

    In this experiment, DSSA was compared with several classical or advanced algorithms (IBA, ESA and DFA), which have been taken from recently published work [6,31]. In Table 6, the average number (in thousands) of objective function evaluations required to reach the final solution for each instance is shown, with the best results in bold. From Table 6, on the one hand, the average evaluations number of DSSA is much smaller than the other algorithms, which indicates that DSSA shows better convergence in all 23 instances, on the other hand, the evaluations number of DSSA does not change significantly by orders of magnitude as the size of the problem rises, suggesting that DSSA has advantages in solving TSP on a larger scale. Finally, despite the longest single run time of the DSSA, the overall time cost of solving TSP can be reduced by determining the appropriate evaluations number. Therefore, the DSSA proposed in the paper is a promising approach to solving TSP.

    Table 6.  Convergence of DSSA, DFA, ESA nad IBA, expressed in thousands of objective function evaluations.
    Instance / Algorithms DSSA DFA ESA IBA
    oliver30 0.89 3.38 23.91 2.17
    eil51 8.51 17.56 85.91 15.37
    berlin52 1.99 23.68 128.26 20.07
    st70 9.82 69.56 216.08 72.67
    eil76 19.93 164.18 262.89 91.53
    kroA100 19.24 812.56 784.84 739.86
    kroB100 20.29 813.68 729.83 461.05
    kroC100 14.81 835.79 726.35 872.51
    KroD100 14.81 875.74 689.49 600.31
    KroE100 20.89 843.72 791.76 602.94
    3il101 32.86 617.83 598.11 512.73
    pr124 16.92 1589.71 1446.91 1602.51
    pr136 47.04 2763.8 2318.2 2866.6
    pr152 23.19 4769.37 3853.91 4853.19
    pr264 47.13 6686.39 6096.45 6375.46
    pr299 113.26 7016.91 6731.23 6597.94
    pr439 278.13 8736.28 8006.91 8346.85

     | Show Table
    DownLoad: CSV

    As a swarm-based algorithm, SSA was put forward to deal with continuous optimization problems of single and multiple objectives. In this paper, we proposed a DSSA for solving TSP. Firstly, we improved 2-opt and PALS into d-opt and TPALS respectively, and added them into DSSA as discrete operators. Secondly, we made a comparative study of five crossover operators, and confirmed that SEC is the best crossover operator of DSSA and introduce it into DSSA. Finally, the proposed DSSA was compared with several advanced algorithms on 23 benchmark examples, and the results showed DSSA possesses satisfactory properties and robustness in solving TSP. In the process of experiment, we found that the SEC operator and d-opt operator which gradually reduced the search range could improve the exploitation ability of the algorithm and help to improve the accuracy of the optimal solution, and the combination of the TPALS operator and the second leader mechanism provided certain randomness to the algorithm, so that the algorithm could avoid falling into the local optimal solution. At the same time, DSSA also exposes the disadvantage of long running time. According to the analysis, it may be because the d-opt operator and TPALS operator both contain the behavior of searching for the optimal solution of the neighborhood. In the future, In the future, we plan to make improvements to address the long running time of DSSA, and we intend to put forward excellent and novel discrete operators for DSSA to deal with DNA fragment assembly problems.

    This work is supported by 111 Project (No.D23006), by the National Natural Science Foundation of China (Nos. 62272079, 61972266), Liaoning Revitalization Talents Program (No. XLYC2008017), Natural Science Foundation of Liaoning Province (Nos. 2021-MS-344, 2021-KF-11-03, 2022-KF-12-14), the Postgraduate Education Reform Project of Liaoning Province (No. LNYJG2022493), the Dalian Outstanding Young Science and Technology Talent Support Program (No. 2022RJ08).

    On behalf of all authors, the corresponding author states that there is no conflict of interest.


    Acknowledgments



    I would like to express my heartfelt gratitude to all our subjects for their invaluable contributions in conducting this research. Additionally, I extend my thanks to the co-workers who assisted in data collection, formulated meal plans, and diligently monitored the participants' progress. Without their dedication and support, this study would not have been possible.

    Conflict of interest



    The authors declare no conflicts of interest.

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