The Internet Shopping Optimization Problem with Multiple Item Units (ISHOPU): Formulation, Instances, NPCompleteness, and Evolutionary Optimization
Abstract
:1. Introduction
2. Problem StateoftheArt
2.1. The original Internet Shopping Optimization Problem (ISHOP)
2.2. Internet Shopping Optimization Problem Sensitive to Price Discounts
2.3. Budgeted Internet Shopping Optimization Problem
2.4. Internet Shopping Optimization Problem with Delivery Constraints
2.5. Trusted Internet Shopping Optimization Problem
2.6. Internet Shopping Problem with Shipping Costs
2.7. Multiple Item Units in the StateoftheArt
3. Proposed ISHOP with Item Units (ISHOPU)
3.1. ISHOPU Definition
3.2. ISHOPU Is NPHard
 Show that ISHOPU ∈ NP;
 Select a known ${\pi}^{\prime}\in $ NPC;
 Construct a polynomial transformation such that ${\pi}^{\prime}{\le}_{p}$ISHOPU;
 Prove that the transformed instances of ${\pi}^{\prime}$ to ISHOPU remain as yes instance if the original instance is a yes instance;
 Prove that the ${\le}_{p}$ can be computed in polynomial time.
Algorithm 1 Nondeterministic polynomial certificate construction. 

Algorithm 2 Polynomial certificate verification. 

4. ISHOPU Numeric Example
5. Proposed Evolutionary Algorithms
5.1. Evolutionary Operators
5.1.1. Crossover
5.1.2. Mutation
Algorithm 3 Proposed mutation operator for the ISHOP with item units. 

5.1.3. Solution Repair
Algorithm 4 Proposed solution repair for the ISHOP with item units. 

5.2. Genetic Algorithm
5.3. Cellular Genetic Algorithm
6. Experimental Setup
6.1. Proposed Synthetic Benchmark
 The number of products and their required units;
 The number of stores;
 The maximum and minimum values for product costs;
 The maximum and minimum for the delivery costs.
6.2. Evolutionary Algorithms Tuning
6.3. Sensitivity Analysis
6.4. Convergence Analysis
6.5. Algorithms for Comparison
6.6. Statistical Validations
 $left\ge 0.95$: in this case, algorithm A outperforms B because their differences have a significant probability of being negative.
 $right\ge 0.95$: in this case, algorithm B outperforms A because their differences have a significant probability of being positive.
 $rope\ge 0.95$: in this case, algorithm A and B are equivalent.
 When none of the above cases are true, the test does not have enough information, making an undecided statement.
7. Results and Discussion
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
CGA  Cellular Genetic Algorithm 
EA  Evolutionary Algorithm 
GA  Genetic Algorithm 
ISHOP  Internet Shopping Optimization Problem 
ISHOPU  Internet Shopping Optimization Problem with item Units 
WCA  Water Cycle Algorithm 
§  Statistically superior to WCA 
?  Statistical indecision to WCA 
${\le}_{p}$  Polynomial transformation 
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A  B  C  D  E  Delivery  

Store 1  18  39  29  48  59  10 
Store 2  24  45  23  54  44  15 
Store 3  22  45  23  53  53  15 
Store 4  28  40  17  57  46  10 
Store 5  24  42  24  47  59  10 
Store 6  27  48  20  55  53  15 
A  B  C  D  E  

Store 1  3  3  2  5  8 
Store 2  4  6  8  7  4 
Store 3  2  4  5  4  7 
Store 4  8  7  8  6  2 
Store 5  6  2  3  8  3 
Store 6  7  8  9  3  2 
A  B  C  D  E  

Units  4  6  8  7  2 
A  B  C  D  E  

Store 1  3  3  0  0  0 
Store 2  0  0  8  0  2 
Store 3  0  0  0  0  0 
Store 4  0  3  0  0  0 
Store 5  1  0  0  7  0 
Store 6  0  0  0  0  0 
GA  CGA  

Population size:  100  100 
MaxEvaluations:  25,000  25,000 
Selection:  Binary Tournament  Binary Tournament 
Neighborhood:  –  L5 
Recombination:  ${p}_{r}=1.0$, See Section 5.1.1  ${p}_{r}=1.0$, See Section 5.1.1 
Mutation:  ${p}_{m}=0.05$, See Algorithm 3  ${p}_{m}=0.05$, See Algorithm 3 
GA  CGA  WCA  

Instance  $\tilde{\mathit{x}}$  IQR  $\tilde{\mathit{x}}$  IQR  $\tilde{\mathit{x}}$  IQR 
Small${}_{1}$  467.04 ${}^{\S}$  0.0  467.04 ${}^{\S}$  0.0  580.17  47.03 
Small${}_{2}$  457.64 ${}^{\S}$  0.41  458.05 ${}^{\S}$  0.69  552.1  41.57 
Small${}_{3}$  585.75 ${}^{\S}$  0.0  587.85 ${}^{\S}$  2.84  726.41  52.11 
Small${}_{4}$  479.69 ${}^{\S}$  1.47  483.07 ${}^{\S}$  4.43  576.01  41.8 
Small${}_{5}$  520.69 ${}^{\S}$  0.97  522.88 ${}^{\S}$  3.35  605.35  35.26 
Medium${}_{1}$  3505.7 ${}^{\S}$  128.35  3709.77 ${}^{\S}$  133.44  4107.11  244.68 
Medium${}_{2}$  4728.01 ${}^{\S}$  140.84  4943.86 ${}^{\S}$  134.12  5383.6  307.62 
Medium${}_{3}$  5234.51 ${}^{\S}$  169.26  5486.86 ${}^{\S}$  166.02  5866.06  349.54 
Medium${}_{4}$  4861.51 ${}^{\S}$  149.79  5111.59 ${}^{\S}$  152.31  5497.51  278.68 
Medium${}_{5}$  5646.67 ${}^{\S}$  182.27  5934.77 ${}^{\S}$  168.39  6333.05  336.3 
Large${}_{1}$  26,415.55 ${}^{?}$  522.62  26,886.7 ${}^{?}$  521.84  25,711.6  1042.82 
Large${}_{2}$  22,002.78 ${}^{?}$  469.4  22,462.82 ${}^{?}$  494.01  21,675.88  1047.13 
Large${}_{3}$  21,983.9 ${}^{?}$  523.42  22,487.28 ${}^{?}$  457.89  21,869.36  731.65 
Large${}_{4}$  24,172.27 ${}^{?}$  479.09  24,620.75 ${}^{?}$  542.36  24,202.27  875.79 
Large${}_{5}$  24,039.28 ${}^{?}$  536.15  24,543.6 ${}^{?}$  475.8  23,875.34  1000.51 
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Ornelas, F.; Santiago, A.; Martínez, S.I.; PonceFlores, M.P.; TeránVillanueva, J.D.; Balderas, F.; Rocha, J.A.C.; García, A.H.; LariaMenchaca, J.; TreviñoBerrones, M.G. The Internet Shopping Optimization Problem with Multiple Item Units (ISHOPU): Formulation, Instances, NPCompleteness, and Evolutionary Optimization. Mathematics 2022, 10, 2513. https://doi.org/10.3390/math10142513
Ornelas F, Santiago A, Martínez SI, PonceFlores MP, TeránVillanueva JD, Balderas F, Rocha JAC, García AH, LariaMenchaca J, TreviñoBerrones MG. The Internet Shopping Optimization Problem with Multiple Item Units (ISHOPU): Formulation, Instances, NPCompleteness, and Evolutionary Optimization. Mathematics. 2022; 10(14):2513. https://doi.org/10.3390/math10142513
Chicago/Turabian StyleOrnelas, Fernando, Alejandro Santiago, Salvador Ibarra Martínez, Mirna Patricia PonceFlores, Jesús David TeránVillanueva, Fausto Balderas, José Antonio Castán Rocha, Alejandro H. García, Julio LariaMenchaca, and Mayra Guadalupe TreviñoBerrones. 2022. "The Internet Shopping Optimization Problem with Multiple Item Units (ISHOPU): Formulation, Instances, NPCompleteness, and Evolutionary Optimization" Mathematics 10, no. 14: 2513. https://doi.org/10.3390/math10142513