# A Mixed-Integer Linear Formulation for a Dynamic Modified Stochastic p-Median Problem in a Competitive Supply Chain Network Design

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Modeling Process and Methods

#### 3.1. Notation List

#### 3.2. Mathematical Formulation

#### 3.3. Solution Approach

#### 3.3.1. Robust Optimization

#### 3.3.2. Lagrangian Relaxation

- Relax one of the constraints by multiplying it by a Lagrange multiplier and bringing the constraint into the objective function;
- Solve the model to find the optimal values of the relaxed problem;
- Find the feasible solution to the original problem by using the resulting decision variables found in step 2;
- Compute the lower bound using the solution obtained from the relaxed problem in step 2;
- Use the subgradient optimization method to modify the Lagrange multiplier assigned to the violated constraint and return to step 2 after finding the new multiplier(s) for the Lagrange variable.

#### Step 1. Solving the Relaxed Problem

#### Step 2. Finding a Feasible Solution and an Upper Bound

#### Step 3. Finding a Lower Bound and Updating the Lagrange Multipliers

#### Step 4. Termination Criteria

- A predetermined number of iterations have been completed;
- The lower bound is equal to the upper bound ($UB=L{B}^{n}$) or is close enough to the upper bound ($UB-L{B}^{n}<0.1$);
- The value of $\alpha $ becomes small.

## 4. Numerical Experiment

#### 4.1. Case Description

#### 4.2. Data Collection

#### 4.3. Computational Experiments

#### 4.4. Sensitivity Analysis

#### 4.4.1. Time Horizon

#### 4.4.2. Number of Facilities to Be Located (P)

#### 4.4.3. Cost Coefficients

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Fahlevi, M.; Zuhri, S.; Parashakti, R.; Ekhsan, M. Leadership styles of food truck businesses. J. Res. Bus. Econ. Manag.
**2019**, 13, 2437–2442. [Google Scholar] - Nader, L. Contrarian Anthropology: The Unwritten Rules of Academia; Berghahn Books: Oxford, NY, USA, 2018. [Google Scholar]
- Phillips, E. U.S Food Truck Services Market Size & Share Report. 2022. Available online: https://www.grandviewresearch.com/industry-analysis/us-food-truck-services-market-report (accessed on 19 December 2022).
- Hecht, A.A.; Biehl, E.; Barnett, D.J.; Neff, R.A. Urban food supply chain resilience for crises threatening food security: A qualitative study. J. Acad. Nutr. Diet.
**2019**, 119, 211–224. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pelling, M. Natural disasters. In Social Nature: Theory, Practice, and Politics; Blackwell Publishers, Inc.: Malden, MA, USA, 2001; pp. 170–189. [Google Scholar]
- Mohan, S.; Gopalakrishnan, M.; Mizzi, P. Improving the efficiency of a non-profit supply chain for the food insecure. Int. J. Prod. Econ.
**2013**, 143, 248–255. [Google Scholar] [CrossRef] - Esparza, N.; Walker, E.T.; Rossman, G. Trade associations and the legitimation of entrepreneurial movements: Collective action in the emerging gourmet food truck industry. Nonprofit Volunt. Sect. Q.
**2014**, 43, 143S–162S. [Google Scholar] [CrossRef] - Wessel, G. From place to nonplace: A case study of social media and contemporary food trucks. J. Urban Des.
**2012**, 17, 511–531. [Google Scholar] [CrossRef] - Restuputri, D.P.; Fridawati, A.; Masudin, I. Customer Perception on Last-Mile Delivery Services Using Kansei Engineering and Conjoint Analysis: A Case Study of Indonesian Logistics Providers. Logistics
**2022**, 6, 29. [Google Scholar] [CrossRef] - Bourlakis, M.A.; Weightman, P.W. Food Supply Chain Management; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Vahdani, B.; Soltani, M.; Yazdani, M.; Mousavi, S.M. A three level joint location-inventory problem with correlated demand, shortages and periodic review system: Robust meta-heuristics. Comput. Ind. Eng.
**2017**, 109, 113–129. [Google Scholar] [CrossRef] - Tirkolaee, E.B.; Mardani, A.; Dashtian, Z.; Soltani, M.; Weber, G.W. A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design. J. Clean. Prod.
**2020**, 250, 119517. [Google Scholar] [CrossRef] - Handfield, R.; Nichols, E., Jr. Introduction to Supply Chain Management; Prentice-Hall: Hoboken, NJ, USA, 1999; pp. 1–29. [Google Scholar]
- Sahebi-Fakhrabad, A.; Sadeghi, A.H.; Handfield, R. Evaluating State-Level Prescription Drug Monitoring Program (PDMP) and Pill Mill Effects on Opioid Consumption in Pharmaceutical Supply Chain. Healthcare
**2023**, 11, 437. [Google Scholar] [CrossRef] - Fakhrabad, A.S.; Sadeghi, A.H.; Kemahlioglu-Ziya, E.; Handfield, R.B.; Tohidi, H.; Farahani, I.V. The Impact of Opioid Prescribing Limits on Drug Usage in South Carolina: A Novel Geospatial and Time Series Data Analysis. arXiv
**2023**, arXiv:2301.08878. [Google Scholar] - Faghihinejad, F.; Mohammadi Fard, M.; Roshanghalb, A.; Beigi, P. A framework to assess the correlation between transportation infrastructure access and economics: Evidence from Iran. Math. Probl. Eng.
**2022**, 2022, 8781686. [Google Scholar] [CrossRef] - Berman, O.; Krass, D. The generalized maximal covering location problem. Comput. Oper. Res.
**2002**, 29, 563–581. [Google Scholar] [CrossRef] - Marianov, V.; ReVelle, C. Siting emergency services. Facil. Locat. Surv. Appl. Methods
**1995**, 1, 199–223. [Google Scholar] - Beigi, P.; Khoueiry, M.; Rajabi, M.S.; Hamdar, S. Station Reallocation and Rebalancing Strategy for Bike-Sharing Systems: A Case Study of Washington DC. arXiv
**2022**, arXiv:2204.07875. [Google Scholar] - Daskin, M. Network and discrete location: Models, algorithms and applications. J. Oper. Res. Soc.
**1997**, 48, 763–764. [Google Scholar] [CrossRef] - Church, R.; ReVelle, C. The maximal covering location problem. In Papers of the Regional Science Association; Springer: Berlin/Heidelberg, Germany, 1974; Volume 32, pp. 101–118. [Google Scholar]
- Church, R.L.; Meadows, M.E. Location modeling utilizing maximum service distance criteria. Geogr. Anal.
**1979**, 11, 358–373. [Google Scholar] [CrossRef] - Calik, H.; Labbé, M.; Yaman, H. p-Center problems. In Location Science; Springer: Cham, Switzerland, 2015; pp. 79–92. [Google Scholar]
- Davidović, T.; Ramljak, D.; Šelmić, M.; Teodorović, D. Bee colony optimization for the p-center problem. Comput. Oper. Res.
**2011**, 38, 1367–1376. [Google Scholar] [CrossRef] - Daskin, M.S.; Maass, K.L. The p-median problem. In Location Science; Springer: Berlin/Heidelberg, Germany, 2015; pp. 21–45. [Google Scholar]
- Berge, C. Two theorems in graph theory. Proc. Natl. Acad. Sci. USA
**1957**, 43, 842. [Google Scholar] [CrossRef][Green Version] - Miehle, W. Link-length minimization in networks. Oper. Res.
**1958**, 6, 232–243. [Google Scholar] [CrossRef] - Hakimi, S.L. Optimum locations of switching centers and the absolute centers and medians of a graph. Oper. Res.
**1964**, 12, 450–459. [Google Scholar] [CrossRef] - Hakimi, S.L. Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper. Res.
**1965**, 13, 462–475. [Google Scholar] [CrossRef] - ReVelle, C.S.; Swain, R.W. Central facilities location. Geogr. Anal.
**1970**, 2, 30–42. [Google Scholar] [CrossRef] - Moheb-Alizadeh, H.; Sadeghi, A.H.; Jaunich, M.K.; Kemahlioglu-Ziya, E.; Handfield, R.B. Reverse Logistics Network Design to Estimate the Economic and Environmental Impacts of Take-back Legislation: A Case Study for E-waste Management System in Washington State. arXiv
**2023**, arXiv:2301.09792. [Google Scholar] - Balinski, M.L. Integer programming: Methods, uses, computations. Manag. Sci.
**1965**, 12, 253–313. [Google Scholar] [CrossRef] - Toregas, C.; Swain, R.; ReVelle, C.; Bergman, L. The location of emergency service facilities. Oper. Res.
**1971**, 19, 1363–1373. [Google Scholar] [CrossRef] - Louveaux, F.V. Discrete stochastic location models. Ann. Oper. Res.
**1986**, 6, 21–34. [Google Scholar] [CrossRef] - Louveaux, F.V.; Peeters, D. A dual-based procedure for stochastic facility location. Oper. Res.
**1992**, 40, 564–573. [Google Scholar] [CrossRef] - Laporte, G.; Louveaux, F.V.; van Hamme, L. Exact solution to a location problem with stochastic demands. Transp. Sci.
**1994**, 28, 95–103. [Google Scholar] [CrossRef] - Current, J.; Ratick, S.; ReVelle, C. Dynamic facility location when the total number of facilities is uncertain: A decision analysis approach. Eur. J. Oper. Res.
**1998**, 110, 597–609. [Google Scholar] [CrossRef] - Berman, O.; Drezner, Z. The p-median problem under uncertainty. Eur. J. Oper. Res.
**2008**, 189, 19–30. [Google Scholar] [CrossRef] - Sonmez, A.D.; Lim, G.J. A decomposition approach for facility location and relocation problem with uncertain number of future facilities. Eur. J. Oper. Res.
**2012**, 218, 327–338. [Google Scholar] [CrossRef] - Liu, H.; Song, G. Employing an Effective Robust Optimization Approach for Cooperative Covering Facility Location Problem Under Demand Uncertainty. Axioms
**2022**, 11, 433. [Google Scholar] [CrossRef] - Zaferanih, M.; Abareshi, M.; Jafarzadeh, M. A bi-level p-facility network design problem in the presence of congestion. Comput. Ind. Eng.
**2023**, 176, 109010. [Google Scholar] [CrossRef] - Ejlali, M.; Arian, E.; Taghiyeh, S.; Chambers, K.; Sadeghi, A.H.; Cakdi, D.; Handfield, R.B. Developing Hybrid Machine Learning Models to Assign Health Score to Railcar Fleets for Optimal Decision Making. arXiv
**2023**, arXiv:2301.08877. [Google Scholar] - Taghiyeh, S.; Lengacher, D.C.; Sadeghi, A.H.; Sahebifakhrabad, A.; Handfield, R.B. A multi-phase approach for product hierarchy forecasting in supply chain management: Application to MonarchFx Inc. arXiv
**2020**, arXiv:2006.08931. [Google Scholar] - Wang, C.; Han, C.; Guo, T.; Ding, M. Solving uncapacitated P-Median problem with reinforcement learning assisted by graph attention networks. Appl. Intell.
**2022**, 53, 2010–2025. [Google Scholar] [CrossRef] - Sadeghi, A.H.; Bani, E.A.; Fallahi, A. Grey Wolf Optimizer and Whale Optimization Algorithm for Stochastic Inventory Management of Reusable Products in a two-level Supply Chain. arXiv
**2023**, arXiv:2302.05796. [Google Scholar] - Basirati, M.; Akbari Jokar, M.R.; Hassannayebi, E. Bi-objective optimization approaches to many-to-many hub location routing with distance balancing and hard time window. Neural Comput. Appl.
**2020**, 32, 13267–13288. [Google Scholar] [CrossRef] - Babaee Tirkolaee, E.; Abbasian, P.; Soltani, M.; Ghaffarian, S.A. Developing an applied algorithm for multi-trip vehicle routing problem with time windows in urban waste collection: A case study. Waste Manag. Res.
**2019**, 37, 4–13. [Google Scholar] [CrossRef][Green Version] - Ballou, R.H. Dynamic warehouse location analysis. J. Mark. Res.
**1968**, 5, 271–276. [Google Scholar] [CrossRef] - Sweeney, D.J.; Tatham, R.L. An improved long-run model for multiple warehouse location. Manag. Sci.
**1976**, 22, 748–758. [Google Scholar] [CrossRef][Green Version] - Scott, A.J. Dynamic location-allocation systems: Some basic planning strategies. Environ. Plan. A
**1971**, 3, 73–82. [Google Scholar] [CrossRef] - Wesolowsky, G.O. Dynamic facility location. Manag. Sci.
**1973**, 19, 1241–1248. [Google Scholar] [CrossRef] - Warszawski, A. Multi-dimensional location problems. J. Oper. Res. Soc.
**1973**, 24, 165–179. [Google Scholar] [CrossRef] - Cavalier, T.M.; Sherali, H.D. Sequential location-allocation problems on chains and trees with probabilistic link demands. Math. Program.
**1985**, 32, 249–277. [Google Scholar] [CrossRef] - Drezner, Z. Dynamic facility location: The progressive p-median problem. Locat. Sci.
**1995**, 3, 1–7. [Google Scholar] [CrossRef] - Hakimi, S.L.; Labbé, M.; Schmeichel, E.F. Locations on time-varying networks. Netw. Int. J.
**1999**, 34, 250–257. [Google Scholar] [CrossRef] - Wesolowsky, G.O.; Truscott, W.G. The multiperiod location-allocation problem with relocation of facilities. Manag. Sci.
**1975**, 22, 57–65. [Google Scholar] [CrossRef] - Galvão, R.D.; Santibañez-Gonzalez, E.d.R. A Lagrangean heuristic for the pk-median dynamic location problem. Eur. J. Oper. Res.
**1992**, 58, 250–262. [Google Scholar] [CrossRef] - Dias, J.; Captivo, M.E.; Clímaco, J. Efficient primal-dual heuristic for a dynamic location problem. Comput. Oper. Res.
**2007**, 34, 1800–1823. [Google Scholar] [CrossRef][Green Version] - Ahmed, S.; Garcia, R. Dynamic capacity acquisition and assignment under uncertainty. Ann. Oper. Res.
**2003**, 124, 267–283. [Google Scholar] [CrossRef][Green Version] - Romauch, M.; Hartl, R.F. Dynamic facility location with stochastic demands. In Proceedings of the Stochastic Algorithms: Foundations and Applications: Third International Symposium, SAGA 2005, Moscow, Russia, 20–22 October 2005; pp. 180–189. [Google Scholar]
- Marques, M.d.C.; Dias, J.M. Simple dynamic location problem with uncertainty: A primal-dual heuristic approach. Optimization
**2013**, 62, 1379–1397. [Google Scholar] [CrossRef][Green Version] - Bertsimas, D.; Sim, M. The price of robustness. Oper. Res.
**2004**, 52, 35–53. [Google Scholar] [CrossRef][Green Version] - Roosta, S.; Mirnajafizadeh, S.; Harandi, H. Development of a robust multi-objective model for green capacitated location-routing under crisis conditions. J. Proj. Manag.
**2023**, 8, 1–24. [Google Scholar] [CrossRef] - Sun, X.A.; Conejo, A.J. Robust Optimization in Electric Energy Systems; Springer: Berlin/Heidelberg, Germany, 2021. [Google Scholar]
- Ben-Tal, A.; Nemirovski, A. Robust solutions of uncertain linear programs. Oper. Res. Lett.
**1999**, 25, 1–13. [Google Scholar] [CrossRef][Green Version] - Prékopa, A. Stochastic Programming; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 324. [Google Scholar]
- Mladenović, N.; Brimberg, J.; Hansen, P.; Moreno-Pérez, J.A. The p-median problem: A survey of metaheuristic approaches. Eur. J. Oper. Res.
**2007**, 179, 927–939. [Google Scholar] [CrossRef][Green Version] - Diabat, A.; Richard, J.P.; Codrington, C.W. A Lagrangian relaxation approach to simultaneous strategic and tactical planning in supply chain design. Ann. Oper. Res.
**2013**, 203, 55–80. [Google Scholar] [CrossRef][Green Version] - Duong, V.H.; Bui, N.H. A mixed-integer linear formulation for a capacitated facility location problem in supply chain network design. Int. J. Oper. Res.
**2018**, 33, 32–54. [Google Scholar] [CrossRef] - Rafie-Majd, Z.; Pasandideh, S.H.R.; Naderi, B. Modelling and solving the integrated inventory-location-routing problem in a multi-period and multi-perishable product supply chain with uncertainty: Lagrangian relaxation algorithm. Comput. Chem. Eng.
**2018**, 109, 9–22. [Google Scholar] [CrossRef] - Kheirabadi, M.; Naderi, B.; Arshadikhamseh, A.; Roshanaei, V. A mixed-integer program and a Lagrangian-based decomposition algorithm for the supply chain network design with quantity discount and transportation modes. Expert Syst. Appl.
**2019**, 137, 504–516. [Google Scholar] [CrossRef] - Hamdan, B.; Diabat, A. Robust design of blood supply chains under risk of disruptions using Lagrangian relaxation. Transp. Res. Part E Logist. Transp. Rev.
**2020**, 134, 101764. [Google Scholar] [CrossRef] - Kelley, J.E., Jr. The cutting-plane method for solving convex programs. J. Soc. for Ind. Appl. Math.
**1960**, 8, 703–712. [Google Scholar] [CrossRef] - Borghetti, A.; Frangioni, A.; Lacalandra, F.; Nucci, C.A. Lagrangian heuristics based on disaggregated bundle methods for hydrothermal unit commitment. IEEE Trans. Power Syst.
**2003**, 18, 313–323. [Google Scholar] [CrossRef] - Robomart Self-Driving Cars Bring Groceries Home. 2018. Available online: pymnts.com (accessed on 19 December 2022).
- Bradski, G.; Kaehler, A. OpenCV. J. Softw. Tools
**2000**, 3, 2. [Google Scholar] - Bixby, B. The gurobi optimizer. Transp. Res. Part B
**2007**, 41, 159–178. [Google Scholar]

**Figure 1.**Sample picture of a Mobile Grocery Store from [75].

**Figure 2.**University of Waterloo—campus map (Source: www.uwaterloo.ca/map).

**Figure 6.**The impact of opening/closing cost of facilities on the objective functions, the dotted line represents the objective value.

Sets | |

T | Set of time periods in the planning horizon; $t\in \{1,2,\cdots ,|T\left|\right\}$ |

K | Set of categories (groups) in the area |

B | Set of candidate locations; $i,j\in \{1,2,\cdots ,|B\left|\right\}$ |

where, based on our problem, the candidate locations (j) are the same as demand nodes (i). | |

Parameters | |

${c}_{ij}$ | Unit cost of satisfying demand of location i from facility j |

${\gamma}^{o}$ | Mobile store’s opening cost in each location |

${\gamma}^{c}$ | Mobile store’s closing cost in each location |

${d}_{it}$ | Demand of location i at day t |

p | Available number of mobile stores in each day |

${m}_{k}$ | Maximum number of stores allowed in group k |

${n}_{k}$ | Minimum number of stores allowed in group k |

Decision Variables | |

${x}_{ijt}$ | Fraction of demand of i that is supplied from j at day t |

${y}_{jt}$ | Binary variables that is 1 if a mobile store is located at j at day t, and is 0 otherwise |

${a}_{jt}$ | Auxiliary binary variables which 1 if a store is located in j |

at day t and will not be located in j at day $t+1$ (i.e., closing variable), and is 0 otherwise | |

${b}_{jt}$ | Auxiliary binary variable which is 1 if a store is not located in j |

at day t and will be located in j at day $t+1$ (i.e., opening variable), and is 0 otherwise |

Segment | Buildings Included in the Segment (Building Id) |
---|---|

Academic Buildings | COG (1), COM (2), CPH (3), RA2 (4), M3 (5), ML (6), RAC (7), GSC (8), GH (9), BRH (10), SLC (11), OWE (12), HMN (13), MC (14), TC (15), KOC (16), C2 (17), EV3 (18), OPT (19), DC (20), SCH (21), FED (22), QNC (23), AL (24), HS (25), B2 (26), EV2 (27), REN (28), ESC (29), EV1 (30), STJ (31), EIT (32), HH (33), STP (34), Bl (35), PAS (36), CGR (37), E3 (38), EC3 (39), BMH (40), PHY (41), EC1 (42), LHI (43), NHI (44), EC2 (45), UC (46), LIB (47), ECH (48), ERC (49), E2 (50), ES (51), CSB (52), RCH (53), E6 (54). |

Parking Lots | Parking CL (55), Parking A (56), Parking X (57), Parking C (58), Parking W (59), Parking OV (60), Parking V (61), Parking S (62), Parking K (63), Parking J (64), Parking R (65), Parking P (66), Parking T (67), Parking M (68), Parking L (69), Parking D (70), Parking EC (71), Parking HV (72), Parking N (73), Parking UWP (74). |

Residence Buildings | CLN (75), CLV (76), MKV (77), V1 (78), REV (79), TH (80), MHR (81), UWP (82). |

Research Park Buildings | 445 (83), 375 (84), 340 (85), 275 (86), ACW (87), 300 (88). |

Athletic Buildings | CLF (89), PAC (90). |

University Plaza | Plaza (91). |

Segment (1) | Population |
---|---|

Engineering Faculty | 11,000 |

Mathematics Faculty | 9260 |

Science Faculty | 6000 |

Health Faculty | 3643 |

Arts Faculty | 3000 |

Environment Faculty | 3000 |

Others | 1200 |

Total population | 37,103 |

Total facilities | 54 |

Average per facility | 687 |

Segment (2) | Population |
---|---|

Campus parking | 4000 |

Total population | 4000 |

Total facilities | 20 |

Average per facility | 200 |

Segment (3) | Population |
---|---|

Columbia Lake Village—North | 404 |

Columbia Lake Village—South | 400 |

William Lyon Mackenzie King Village | 320 |

Student Village 1 | 1381 |

Ron Eydt Village | 960 |

Tutors’ Houses | 100 |

Minota Hagey Residence | 70 |

University of Waterloo Place | 1650 |

Others | 200 |

Total population | 5285 |

Total facilities | 8 |

Average per facility | 660 |

Segment (4) | Population |
---|---|

David Johnson Research Park | 4000 |

Others | 100 |

Total population | 4100 |

Total facilities | 6 |

Average per facility | 683 |

Segment (5) | Population |
---|---|

Columbia Icefield | 2100 |

Physical Activities Complex | 2100 |

Others | 100 |

Total population | 4300 |

Total facilities | 2 |

Average per facility | 2150 |

Segment (6) | Population |
---|---|

University Shops Plaza | 3200 |

Total population | 3200 |

Total facilities | 1 |

Average per facility | 3200 |

**Table 9.**The Estimated Utilization Rate for Different Functional Buildings for Each Day over a Week.

Functionality | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
---|---|---|---|---|---|---|---|

Academic Buildings | 100 | 90 | 90 | 80 | 90 | 30 | 30 |

Parking Spots | 100 | 100 | 100 | 100 | 100 | 20 | 20 |

Residence Buildings | 50 | 50 | 50 | 50 | 60 | 100 | 100 |

Research Park Buildings | 100 | 90 | 90 | 80 | 90 | 10 | 10 |

Athletic Buildings | 50 | 50 | 50 | 60 | 50 | 100 | 100 |

University Plaza | 100 | 100 | 70 | 80 | 90 | 50 | 50 |

Functionality | Min (${\mathit{n}}_{\mathit{k}}$) | Max (${\mathit{m}}_{\mathit{k}}$) |
---|---|---|

Academic Buildings | 7 | 14 |

Parking Lots | 2 | 6 |

Residence Buildings | 1 | 3 |

Research Park Buildings | 0 | 2 |

Athletic Buildings | 0 | 1 |

University Plaza | 0 | 1 |

t | Id of Opened Buildings |
---|---|

1 (Monday) | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90 |

6 (Saturday) | 0, 2, 3, 5, 10, 17, 27, 34, 40, 44, 48, 54, 63, 76, 77, 81, 89, 90 |

8 (Monday) | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90 |

13 (Saturday) | 0, 2, 3, 5, 10, 17, 27, 34, 40, 44, 48, 54, 63, 76, 77, 81, 89, 90 |

15 (Monday) | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90 |

20 (Saturday) | 0, 2, 3, 5, 10, 17, 27, 34, 40, 44, 48, 54, 63, 76, 77, 81, 89, 90 |

22 (Monday) | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90 |

27 (Saturday) | 0, 2, 3, 5, 10, 17, 27, 34, 40, 44, 48, 54, 63, 76, 77, 81, 89, 90 |

T | Id of Opened Buildings | Id of Closed Buildings | Objective Value | CPU Time (S) |
---|---|---|---|---|

14 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) | 63, 77, 81, (12, 55, 83) | 21570.1 | 12.43 |

21 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) | 63, 77, 81, (12, 55, 83) | 32372.9 | 6.08 |

28 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) | 63, 77, 81, (12, 55, 83) | 43175.7 | 12.99 |

35 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) | 63, 77, 81, (12, 55, 83) | 53978.5 | 17.02 |

42 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) | 63, 77, 81, (12, 55, 83) | 64781.3 | 20.40 |

P | Id of Opened Buildings | Id of Closed Buildings | Objective Value | CPU Time (S) |
---|---|---|---|---|

12 | 2, 5, 10, 17, 27, 35, 44, 54, 65, 76, 83, 90, (63, 89) | 63, 89, (65, 83) | 58,616.4 | 58.07 |

15 | 2, 3, 5, 10, 27, 35, 45, 48, 63, 66, 74, 76, 83, 89, 90 (81, 17, 61, 79) | 81, 17, 61, 79, (83, 3, 66, 76) | 49,402.9 | 7.79 |

18 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) | 63, 77, 81, (12, 55, 83) | 43,175.7 | 12.99 |

21 | 0, 2, 5, 7, 10, 12, 17, 23, 26, 34, 40, 44, 48, 53, 54, 55, 76, 83, 85, 89, 90, (64, 77, 81, 3, 79) | 64, 77, 81, 3, 79, (7, 55, 83, 76, 85) | 38,266.5 | 7.89 |

24 | 0, 2, 3, 5, 6, 10, 12, 17, 23, 34, 40, 44, 48, 53, 54, 61, 67, 76, 80, 81, 83, 87, 89, 90, (64, 78) | 64, 78, (80, 87) | 34,874.4 | 107.60 |

${\mathit{\gamma}}^{\mathit{o}\left(\mathit{c}\right)}$ | Id of Opened Buildings | Id of Closed Buildings | Objective Value | CPU Time (S) |
---|---|---|---|---|

0 | 0, 2, 3, 10, 12, 17, 26, 35, 45, 48, 54, 65, 76, 83, 86, 88, 90, (1, 4, 5, 23, 26, 27, 34, 40, 44, 56, 63, 81, 89) | 4, 5, 27, 34, 40, 44, 56, 63, 81, 89, (23, 26, 35, 45, 65, 86, 88) | 42,900 | 3.90 |

2.5 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 79, 81) | 63, 77, 79, 81, (12, 55, 76, 83) | 43,049.3 | 9.34 |

5 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) | 63, 77, 81, (12, 55, 83) | 43,175.7 | 12.99 |

10 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (80, 81) | 80, 81, (12, 83) | 43,382.3 | 58.45 |

20 | 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (81) | 81, (83) | 43,658.2 | 82.58 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sadeghi, A.H.; Sun, Z.; Sahebi-Fakhrabad, A.; Arzani, H.; Handfield, R.
A Mixed-Integer Linear Formulation for a Dynamic Modified Stochastic p-Median Problem in a Competitive Supply Chain Network Design. *Logistics* **2023**, *7*, 14.
https://doi.org/10.3390/logistics7010014

**AMA Style**

Sadeghi AH, Sun Z, Sahebi-Fakhrabad A, Arzani H, Handfield R.
A Mixed-Integer Linear Formulation for a Dynamic Modified Stochastic p-Median Problem in a Competitive Supply Chain Network Design. *Logistics*. 2023; 7(1):14.
https://doi.org/10.3390/logistics7010014

**Chicago/Turabian Style**

Sadeghi, Amir Hossein, Ziyuan Sun, Amirreza Sahebi-Fakhrabad, Hamid Arzani, and Robert Handfield.
2023. "A Mixed-Integer Linear Formulation for a Dynamic Modified Stochastic p-Median Problem in a Competitive Supply Chain Network Design" *Logistics* 7, no. 1: 14.
https://doi.org/10.3390/logistics7010014