A TwoStage Location Problem with Order Solved Using a Lagrangian Algorithm and Stochastic Programming for a Potential Use in COVID19 Vaccination Based on SensorRelated Data
Abstract
:1. Introduction
2. TwoStage Stochastic Linear Programming
MP 1 Twostage stochastic mathematical programming formulation with fixed resources. 
$$\begin{array}{ccc}\hfill min\phantom{\rule{1.em}{0ex}}Z& =& {\mathit{c}}^{\top}\mathit{y}+{\mathbb{E}}_{\mathbf{\xi}}[min\mathit{q}{\left(\omega \right)}^{\top}\mathit{x}]\hfill \\ \hfill \mathrm{subject}\mathrm{to}\phantom{\rule{1.em}{0ex}}\mathit{A}\mathit{y}& =& \mathit{b},\hfill \\ \hfill \mathit{T}\left(\omega \right)\mathit{y}+\mathit{W}\mathit{x}& =& \mathit{h}\left(\omega \right),\hfill \\ \hfill \mathit{y}& \ge & \mathbf{0},\hfill \\ \hfill \mathit{x}& \ge & \mathbf{0}.\hfill \end{array}$$

3. Simple Plant Location Problem with Order
MP 2 SPLPO mathematical programming formulation. 
$$\begin{array}{ccc}\hfill min\phantom{\rule{1.em}{0ex}}Z& =& \sum _{i\in I}\sum _{j\in J}{c}_{ij}{x}_{ij}+\sum _{j\in J}{f}_{j}{y}_{j},\hfill \end{array}$$
$$\begin{array}{ccc}\hfill \mathrm{subject}\mathrm{to}\phantom{\rule{1.em}{0ex}}\sum _{j\in J}{x}_{ij}& =& 1,\phantom{\rule{1.em}{0ex}}\forall i\in I,\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {x}_{ij}& \le & {y}_{j},\phantom{\rule{1.em}{0ex}}\forall i\in I,j\in J,\hfill \end{array}$$
$$\begin{array}{ccc}\hfill \sum _{k\in \overline{{W}_{ij}}}{x}_{ik}& \ge & {y}_{j},\phantom{\rule{1.em}{0ex}}\forall i\in I,j\in {J}^{i},\phantom{\rule{42.67912pt}{0ex}}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {x}_{ij}& \ge & 0,\phantom{\rule{1.em}{0ex}}\forall i\in I,j\in J,\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {y}_{j}& \in & \{0,1\},\phantom{\rule{1.em}{0ex}}\forall j\in J.\hfill \end{array}$$

4. Stochastic Formulation for the SPLPO
MP 3 2SSPLPO mathematical programming formulation. 
$$\begin{array}{ccc}\hfill min\phantom{\rule{1.em}{0ex}}Z& =& \sum _{j\in J}{f}_{j}{y}_{j}+\sum _{\omega \in \Omega}{\alpha}^{\omega}\sum _{i\in I}\sum _{j\in J}{c}_{ij}{x}_{ij}^{\omega}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill \mathrm{subject}\mathrm{to}\phantom{\rule{1.em}{0ex}}\sum _{j\in J}{x}_{ij}^{\omega}& =& 1,\phantom{\rule{1.em}{0ex}}\forall i\in I,\omega \in \Omega ,\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {x}_{ij}^{\omega}& \le & {y}_{j},\phantom{\rule{1.em}{0ex}}\forall i\in I,j\in J,\omega \in \Omega ,\hfill \end{array}$$
$$\begin{array}{ccc}\hfill \sum _{k\in \overline{{W}_{ij}^{\omega}}}{x}_{ik}^{\omega}& \ge & {y}_{j},\phantom{\rule{1.em}{0ex}}\forall i\in I,j\in {J}^{i},\omega \in \Omega ,\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {x}_{ij}^{\omega}& \ge & 0,\phantom{\rule{1.em}{0ex}}\forall i\in I,j\in J,\omega \in \Omega ,\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {y}_{j}& \in & \{0,1\},\phantom{\rule{1.em}{0ex}}\forall j\in J.\hfill \end{array}$$

5. A Lagrangian Algorithm for the 2SSPLPO
 Step 1:
 Lagrangian relaxation
Algorithm 1: Subgradient method. 
Algorithm 2: Upper bound heuristic for the 2SSPLP. 
 Step 2:
 SemiLagrangian relaxation
 (i)
 $\mathit{SLR}\left(\mathbf{\lambda}\right)$is concave, nondecreasing on its domain, and$\mathit{b}\mathit{A}\mathit{x}$is a subgradient at the point $\mathit{\lambda}$.
 (ii)
 There is an interval $[{\mathit{\lambda}}^{*},+\infty )$ where for each multiplier, we obtain the optimal solution of $\mathit{SLR}\left(\mathit{\lambda}\right)$.
 (iii)
 $\mathit{LP}\left(\mathrm{P}\right)\le \mathit{LD}\left(\mathit{\lambda}\right)\le \mathit{SLD}\left(\mathit{\lambda}\right)=\left(\mathrm{P}\right)$, that is, $\mathit{SLR}\left(\mathit{\lambda}\right)$ closes the duality gap.
 By using the sorted costs ${\alpha}^{\omega}{c}_{i}^{\left(1\right)}\le \cdots \le {\alpha}^{\omega}{c}_{i}^{(n=J\left\right)}$, each component ${\gamma}_{i}^{\omega}$ of ${\mathbf{\gamma}}^{\omega}$ can be either in an interval of the form $({\alpha}^{\omega}{c}_{i}^{\left(j\right)},{\alpha}^{\omega}{c}_{i}^{(j+1)}]$ or out of it. For the first case, there are infinite values of ${\gamma}_{i}^{\omega}$ that can belong to a single interval $({\alpha}^{\omega}{c}_{i}^{\left(j\right)},{\alpha}^{\omega}{c}^{(j+1)}]$. Each one of them has the same effect in the solution of $\mathrm{SLR}\left({\mathbf{\gamma}}^{\omega}\right)$ since only a change from ${y}_{j}=1$ to ${y}_{j+1}=1$ and ${x}_{ij}^{\omega}=1$ to ${x}_{i,j+1}^{\omega}=1$ modifies the solution.
 We just need a single ${\gamma}_{i}^{\omega}$ representative of the intervals. As we get closer to solving the SLR, the values of the components of ${\mathbf{\gamma}}^{\omega}$ increase. Hence, solving the SLR becomes more and more difficult. Then, it is always convenient to choose a ${\gamma}_{i}^{\omega}\in {I}_{i}$ being as small as possible, that is, at an $\epsilon $ distance from the lower bound of an interval.
Algorithm 3: Dual ascent method. 
 Step 3:
 Variable fixing heuristic
Algorithm 4: Variable fixing heuristic. 
Algorithm 5: Accelerated dual ascent algorithm. 
6. Sensing Patients’ and Simulated Data for the Proposed Methodology
Algorithm 6: Approach for updating patients’ records in a central data warehouse using sensors. 

Algorithm 7: Approach to generate and solve instances of the 2SSPLPO with a sensor. 

7. Computational Experiments
 Prob: Name of the problem.
 Opt: Optimal value of the problem.
 LP(P): Linear relaxation value for a problem (P).
 GAP${}_{\mathrm{X}}=(\mathrm{Opt}\mathrm{LP})/\mathrm{Opt}\times 100\%$: Relative gap between Opt and LP of a problem by using the XPRESS software.
 t: Time in seconds
 H2Sub: Best upper bound with the H2S.
 y: Number of opened facilities.
 SGMlb: Lower bound with the SGM.
 ADAub: Best upper bound with the ADA algorithm.
 DAMlb: Lower bound with the DAM (without the VFH).
 GAP${}_{\mathrm{V}}=(\mathrm{bestUB}\mathrm{LB})/\mathrm{bestUB}\times 100\%$: Relative gap between the best upper bound and lower bound of a problem by using the DAM with VHF.
 Tt: Total time in seconds.
 imp t: $(\left({\mathrm{t}}_{\mathrm{opt}}{\mathrm{t}}_{\mathrm{ADA}}\right)/{\mathrm{t}}_{\mathrm{opt}})\times 100\%$
 GAP: $\left(\right(\mathrm{ADAub}\mathrm{Opt})/\mathrm{Opt})\times 100\%$.
8. Conclusions, Limitations, and Future Research
 (i)
 Formulations for SPLPO and 2SSPLPO with partial preferences were proposed.
 (ii)
 Lagrangian and semiLagrangian structures for the 2SSPLPO with partial preferences were introduced.
 (iii)
 A theoretical analysis of properties of the Lagrangian and semiLagrangian structures for the 2SSPLPO was presented to combine them in a procedure that approximates its solution.
 (iv)
 Theorems 2 and 3 were stated as extensions for the 2SSPLPO of those given in [15] for the uncapacitated facility location problem.
 (v)
 To the best of our knowledge, there have been no algorithms proposed to solve the stochastic version of the SPLPO. The proposed algorithm is a novel approach that uses a relatively new optimization technique known as semiLagrangian relaxation.
 (vi)
 The computational experiments suggested that the ADA algorithm performed satisfactorily on large instances in both search of the optimum and execution time.
 (vii)
 Possible applications in real cases are described in the context of the COVID19 vaccination process and B2C ecommerce.
 (i)
 Since the last step of the ADA algorithm is a heuristic, the optimal is not guaranteed.
 (ii)
 The ADA algorithm is studied in the context of the 2SSPLPO, so its use is limited to it. We suggest studying the same ideas in other location problems.
 (iii)
 A cost–benefit evaluation should be carried out in the use of the proposed algorithm to answer the following question: is the improvement in execution times worth it, with respect to the savings obtained in the value of the objective function?
 (iv)
 Studies and experiments with deeper parameter settings and more scenarios must be performed on larger instances.
 (v)
 We suggest carrying out a computational experiment that allows us to determine the parameter values of the mathematical optimization model from which it is necessary to design and implement a heuristic or metaheuristic algorithm. High computational complexity problems can be solved using the heuristic approach. Some examples of this can be found in [30,31,32].
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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m  n  SGM  DAM  VFH  

sgm_iter  $\mathbf{\beta}$  k  q  dam_iter  $\mathbf{\epsilon}$  vfh_iter  ps  
75  50  50  2  30  0.005  3  0.01  2  0.25 
100  75  100  2  30  0.005  7  0.01  2  0.25 
125  100  170  2  30  0.005  10  0.01  2  0.25 
150  100  170  2  30  0.005  12  0.01  2  0.25 
Prob  XPRESS  H2S  SGM  DAM with VFH  ADA  

Opt  LP  GAP${}_{\mathbf{X}}$  t  H2Sub  y  t  SGMlb  t  ADAub  DAMlb  GAP${}_{\mathbf{V}}$  t  Tt  imp t  GAP  
10a7550_1  1,648,853  1,200,745  27%  67  1,787,955  1  0  840,831  7  1,698,650  1,279,041  25%  60  67  0%  3.02% 
10a7550_2  1,602,647  1,193,378  26%  48  1,738,071  3  0  763,664  6  1,602,647  1,286,736  20%  50  56  −17%  0.00% 
10b7550_1  1,226,979  901,051  27%  48  1,335,244  10  0  334,589  5  1,226,979  953,159  22%  57  62  −28%  0.00% 
10b7550_2  1,263,465  906,280  28%  53  1,299,259  10  0  372,981  7  1,271,641  970,770  24%  70  77  −46%  0.65% 
10c7550_1  1,290,291  900,033  30%  46  1,388,280  9  0  591,938  7  1,290,291  971,785  25%  62  69  −48%  0.00% 
10c7550_2  1,248,312  876,053  30%  47  1,365,717  13  0  557,308  6  1,248,312  936,483  25%  65  71  −51%  0.00% 
25a7550_1  1,670,734  1,199,273  28%  119  1,761,050  5  0  830,624  6  1,670,734  1,284,052  23%  53  59  50%  0.00% 
25a7550_2  1,606,360  1,195,482  26%  67  1,730,389  3  0  770,870  7  1,606,360  1,269,850  21%  53  60  11%  0.00% 
25b7550_1  1,256,755  892,106  29%  58  1,329,801  11  0  360,026  6  1,259,217  947,822  25%  61  67  −15%  0.20% 
25b7550_2  1,324,578  907,308  32%  153  1,348,842  7  0  380,921  6  1,324,578  988,646  25%  98  104  32%  0.00% 
25c7550_1  1,364,933  907,957  33%  166  1,413,489  9  0  579,362  8  1,373,542  978,857  29%  96  104  37%  0.63% 
25c7550_2  1,271,029  873,435  31%  84  1,291,327  13  0  578,313  7  1,271,029  958,058  25%  77  84  0%  0.00% 
50a7550_1  1,689,428  1,212,042  28%  113  1,778,238  6  0  804,221  7  1,689,428  1,302,236  23%  60  67  41%  0.00% 
50a7550_2  1,637,084  1,196,440  27%  86  1,744,372  5  0  760,256  6  1,637,084  1,273,267  22%  51  58  33%  0.00% 
50b7550_1  1,331,207  900,654  32%  165  1,388,062  12  0  294,443  7  1,331,207  1,247,451  6%  381  388  −136%  0.00% 
50b7550_2  1,307,706  896,181  31%  182  1,364,303  11  0  305,591  7  1,307,706  990,369  24%  94  101  45%  0.00% 
50c7550_1  1,347,482  911,370  32%  145  1,446,345  12  0  565,472  6  1,347,482  976,268  28%  77  84  42%  0.00% 
50c7550_2  1,287,982  878,852  32%  112  1,439,845  17  0  546,735  7  1,299,952  952,655  27%  71  78  30%  0.93% 
100a7550_1  1,787,955  1,208,842  32%  267  1,787,955  1  0  824,731  8  1,787,955  1,293,098  28%  89  97  64%  0.00% 
100a7550_2  1,683,058  1,204,739  28%  194  1737,,924  5  0  827,668  7  1,720,482  1,268,026  26%  66  72  63%  2.22% 
100b7550_1  1,451,139  935,113  36%  363  1,496,648  6  6  311,032  6  1,453,678  989,340  32%  86  92  75%  0.17% 
100b7550_2  1,400,184  916,167  35%  271  1,449,128  9  0  358,259  7  1,400,205  1,013,362  28%  106  113  58%  0.00% 
100c7550_1  1,360,674  920,342  32%  169  1,430,298  9  0  562,495  8  1,360,674  998,681  27%  106  114  32%  0.00% 
100c7550_2  1,402,514  918,696  34%  196  1,462,990  16  0  570,058  8  1,404,147  989,592  30%  102  110  44%  0.12% 
Prob  XPRESS  H2S  SGM  DAM with VFH  ADA  

Opt  LP  GAP${}_{\mathbf{X}}$  t  H2Sub  y  t  SGMlb  t  ADAub  DAMlb  GAP${}_{\mathbf{V}}$  t  Tt  imp t  GAP  
100a10075_1  2,469,439  1,811,464  27%  561  2,476,632  1  1  1,644,492  31  2,476,632  1,978,083  20%  265  296  47%  0.29% 
100a10075_2  2,458,870  1,805,025  27%  685  2,476,632  1  1  1,627,278  32  2,458,870  1,971,011  20%  237  270  61%  0.00% 
100b10075_1  2,132,719  1,364,985  36%  17,935  2,270,467  6  1  1,112,051  37  2,132,719  1,558,555  27%  3531  3568  80%  0.00% 
100b10075_2  2,163,818  1,367,450  37%  27679  2,218,215  7  1  1,160,875  43  2,163,818  1,564,516  28%  4380  4422  84%  0.00% 
100c10075_1  1,978,807  1,271,848  36%  14,835  2,072,702  6  1  1,052,860  49  1,988,903  1,496,210  25%  1027  1076  93%  0.51% 
100c10075_2  1,987,757  1,261,290  37%  11,567  2,118,928  9  1  1,066,388  51  1,987,757  1,452,609  27%  3152  3202  72%  0.00% 
100a125100_1  3,070,535  2,416,518  21%  918  3,070,535  1  2  2,237,118  93  3,070,535  2,619,571  15%  573  666  27%  0.00% 
100a125100_2  3,070,535  2,388,054  22%  1088  3,070,535  1  1  2,239,726  106  3,070,535  2,587,669  16%  702  808  26%  0.00% 
100b125100_1  2,800,573  1,815,018  35%  53,666  2,850,413  5  2  1,601,481  118  2,850,413  2,078,979  27%  20,837  20,955  61%  1.78% 
100b125100_2  2,820,883  1,820,001  35%  78,669  3,019,740  4  1  1,592,305  143  2,820,883  2,016,632  29%  8230  8373  89%  0.00% 
100c125100_1  2,702,169  1,698,737  37%  239,967  2,866,218  10  2  1,488,148  157  2,702,169  1,990,068  26%  23,717  23,874  90%  0.00% 
100c125100_2  2,716,252  1,705,149  37%  204,007  2,829,945  5  1  1,477,796  168  2,717,597  2,007,831  26%  27,442  27,610  86%  0.05% 
100a150100_1  3,768,087  2,924,250  22%  1735  3,768,087  1  1  2699949  109  3,768,087  3239975  14%  905  1014  42%  0.00% 
100a150100_2  3,768,087  2,918,397  23%  1819  3,768,087  1  1  2702231  111  3,768,087  3,251,719  14%  117  228  87%  0.00% 
100b150100_1  3,412,417  2,179,897  36%  169,739  3,637,438  1  1  1,923,456  141  3,412,417  2,599,980  24%  21,111  21,252  87%  0.00% 
100b150100_2  3,388,309  2,196,284  35%  69,508  3,637,438  1  2  1,924,300  169  3,388,309  2,679,093  21%  14792  14,962  78%  0.00% 
100c150100_1  3,287,595  2,010,587  39%  502,354  3,413,288  4  2  1,768,341  185  3,413,288  N/A  N/A  N/A  N/A  N/A  3.82% 
100c150100_2  3,229,424  2,012,045  38%  494,721  3,307,459  5  3  1,776,475  132  3,300,341  2,474,515  25%  119,700  119,832  76%  2.20% 
100a150100_1  3,768,087  2,924,250  22%  1735  3,768,087  1  1  2,635,877  94  3,768,087  3,229,888  14%  1159  1253  28%  0.00% 
100a150100_2  3,768,087  2,918,397  23%  1819  3,768,087  1  1  2,637,900  109  3,768,087  3,242,679  14%  1255  1364  25%  0.00% 
100b150100_1  3,412,417  2,179,897  36%  169,739  3,637,438  1  2  1,876,775  147  3,445,585  2,607,746  24%  21,590  21,738  87%  0.97% 
100b150100_2  3,388,309  2,196,284  35%  69,508  3,637,438  4  4  1,863,975  155  3,388,309  2,599,780  23%  8984  9139  87%  0.00% 
100c150100_1  3,287,595  2,010,587  39%  502,354  3,413,288  4  2  1,750,243  179  3,288,348  2,457,997  25%  78,577  78756  84%  0.02% 
100c150100_2  3,229,424  2,012,045  38%  494,721  3,307,459  5  3  1,718,731  185  3,230,261  2,470,331  24%  144,729  144,729  71%  0.03% 
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Cabezas, X.; García, S.; MartinBarreiro, C.; Delgado, E.; Leiva, V. A TwoStage Location Problem with Order Solved Using a Lagrangian Algorithm and Stochastic Programming for a Potential Use in COVID19 Vaccination Based on SensorRelated Data. Sensors 2021, 21, 5352. https://doi.org/10.3390/s21165352
Cabezas X, García S, MartinBarreiro C, Delgado E, Leiva V. A TwoStage Location Problem with Order Solved Using a Lagrangian Algorithm and Stochastic Programming for a Potential Use in COVID19 Vaccination Based on SensorRelated Data. Sensors. 2021; 21(16):5352. https://doi.org/10.3390/s21165352
Chicago/Turabian StyleCabezas, Xavier, Sergio García, Carlos MartinBarreiro, Erwin Delgado, and Víctor Leiva. 2021. "A TwoStage Location Problem with Order Solved Using a Lagrangian Algorithm and Stochastic Programming for a Potential Use in COVID19 Vaccination Based on SensorRelated Data" Sensors 21, no. 16: 5352. https://doi.org/10.3390/s21165352