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

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## Abstract

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## 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

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**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 |

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## 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