Sequential Service Region Design with Capacity-Constrained Investment and Spillover Effect
Authors: Tingting Chen, Feng Chu, Jiantong Zhang
Organizations: School of Economics and Management, Tongji University, Shanghai, 200092, China · Laboratoire IBISC, Univ-Evry, Université Paris-Saclay, Evry, 91000, France
Service region design determines the geographic coverage of service networks, shaping long-term operational performance. Capital and operational constraints preclude simultaneous large-scale deployment, requiring expansion to proceed sequentially. The resulting challenge is to determine when and where to invest under demand uncertainty, balancing intertemporal trade-offs between early and delayed investment and accounting for network effects whereby each deployment reshapes future demand through inter-regional connectivity. This study addresses a sequential service region design (SSRD) problem incorporating two practical yet underexplored factors: a k-region constraint that limits the number of regions investable per period and a stochastic spillover effect linking investment decisions to demand evolution. The resulting problem requires sequencing regional portfolios under uncertainty, leading to a combinatorial explosion in feasible investment sequences. To address this challenge, we propose a solution framework that integrates real options analysis (ROA) with a Transformer-based Proximal Policy Optimization (TPPO) algorithm. ROA evaluates the intertemporal option value of investment sequences, while TPPO learns sequential policies that directly generate high option-value sequences without exhaustive enumeration. Numerical experiments on realistic multi-region settings demonstrate that TPPO converges faster than benchmark DRL methods and consistently identifies sequences with superior option value. Case studies and sensitivity analyses further confirm robustness and provide insights on investment concurrency, regional prioritization, and the increasing benefits of adaptive expansion via our approach under stronger spillovers and dynamic market conditions.
Figures & tables
Figure 1: Illustration of the problem scenario
Demand
Uncertain
Reference
Type of planning
Deterministic
Stationary
Nonstationary
Investment limitation
Uncertain spillover effect
Evaluation indicator
Modeling
Method
Carlsson et al. (2013)
✓
SE
MINLP
WVD
Wang and Lin (2017)
✓
C
MILP
CSA
Huang et al. (2018)
✓
C
MILP
ALNS
Qi et al. (2018)
✓
C
CA
NA
Table 1: Comparison with related works
Figure 2: Framework of the TPPO
Figure 3: Training curves of TPPO vs. alternative DRL approaches
Table 2: Comparison with the enumeration approach (4-6 region map)
TPPO
Enumeration
Instances
Num of seqs
Option value
Gap
Runtime(s)
Option value
Runtime(s)
Ins.6-2
2970
4626.01
0.00%
309.32
4626.01
997.60
Ins.6-3
3830
4521.48
2.26%
278.01
4626.01
1232.83
Ins.7-2
15120
7558.69
0.06%
324.47
7563.24
6892.42
Ins.7-3
25410
7274.70
2.93%
282.83
7494.53
11164.09
Average
5995.22
1.31%
298.66
6077.45
5071.74
Table 3: Option value results for TPPO vs enumeration approach Pretrained (500 episodes)
Instances
Option value
TPPO
Myopia-L
Imv.
Myopia-H
Imv.
Ins.7-3
7568.45
5813.13
30.20%
4862.98
55.63%
Ins.7-4
7401.21
5802.00
27.56%
5161.72
43.39%
Ins.7-5
7213.03
5915.72
21.93%
4142.68
74.12%
Ins.8-3
7211.72
6874.55
4.90%
4068.75
77.25%
Ins.8-4
7452.72
7307.80
1.98%
5444.05
36.90%
Table 4: Option value results for TPPO vs alternative benchmarks
Figure 4: Correlation analysis between investment order and regional attributes
Option value
k=2
k=3
k=4
k=5
k=6
Instances
TPPO
Myopia-H
Myopia-L
TPPO
Myopia-H
Myopia-L
TPPO
Myopia-H
Myopia-L
TPPO
Myopia-H
Myopia-L
TPPO
Myopia-H
Myopia-L
Ins.9-LL
5719.03
4525.69
5158.77
5643.06
4523.64
3645.08
5779.15
3614.57
4699.59
5712.58
3878.46
3594.28
5143.12
4446.70
4889.88
Ins.9-LM
5739.52
3945.11
879.29
5603.96
4266.21
4625.47
5717.22
4114.15
4835.90
6246.70
4863.29
3067.16
5274.35
4309.31
4244.96
Ins.9-LH
5674.93
3989.11
3113.27
584.04
3927.68
2907.94
5917.70
4415.96
4743.75
5997.32
2759.30
4997.32
6201.23
3780.50
4084.02
Ins.9-ML
6051.71
4607.05
5485.16
6267.57
4718.12
4917.96
6260.23
4022.52
3482.12
6464.91
4542.25
3745.13
5920.48
4875.61
3760.32
Table 5: Sensitivity analysis of k-region constraint)
Figure 5: Heatmap analysis of k-region constraint
Stationary = False
Stationary = True
Distribution
Strength
TPPO
Myopia-H
Myopia-L
TPPO
Myopia-H
Myopia-L
Gamma
0.8
11059.43
8811.84
13837.79
5319.36
4437.03
4741.43
1.0
18857.02
15381.02
29330.18
7114.86
5621.10
6223.50
1.2
34817.72
31848.86
32686.78
9360.42
7209.27
8314.89
Lognormal
0.8
14171.83
11892.32
12931.20
5404.62
4444.51
4748.73
1.0
49639.97
45135.96
49848.64
7516.41
5623.14
6386.78
Table 6: Sensitivity analysis of spillover effect
Figure 6: Multi-region scenarios in Brooklyn (KingsCounty), NYC
7-region (NYC)
8-region (NYC)
Problem setting
Strategy
E[NPV]
Profitability
E[NPV]
Profitability
All-in deployment
4806.26
0.51
8099.36
0.51
No k-region constraint & spillover effect
TPPO
6474.12
2.03
10421.04
1.92
Myopia-H
7360.76
2.05
12133.52
1.42
Myopia-L
7537.53
1.54
12419.61
0.88
With k-region constraint & spillover effect
TPPO
8133.27
2.95
13816.40
2.93
Table 7: Comparison of strategies (7-region & 8-region scenarios in NYC)
Figure 7: Sensitivity analyses of dynamic cost& scale effect (7-region & 8-region scenarios in NYC)
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
City name
Region index
Region name
Area (km²)
Population density (/km²)
r1
Changning
37.16
18164
r2
Xuhui
55.16
20358
r3
Jingan
36.77
26193
r4
Huangpu
20.5
28451
r5
Yangpu
60.55
30443
r6
Hongkou
23.41
22069
Appendix
Table A1: Region attributes in Shanghai & Beijing, China
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