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Adaptive Two-stage Stochastic Programming with an Analysis on Capacity Expansion Planning Problem

Beste Basciftci, Shabbir Ahmed, Nagi Gebraeel

Abstract

Multi-stage stochastic programming is a well-established framework for sequential decision making under uncertainty by seeking policies that are fully adapted to the uncertainty. Often such flexible policies are not desirable, and the decision maker may need to commit to a set of actions for a number of planning periods. Two-stage stochastic programming might be better suited to such settings, where the decisions for all periods are made here-and-now and do not adapt to the uncertainty realized. In this paper, we propose a novel alternative approach, where the stages are not predetermined but part of the optimization problem. Each component of the decision policy has an associated revision point, a period prior to which the decision is predetermined and after which it is revised to adjust to the uncertainty realized thus far. We motivate this setting using the multi-period newsvendor problem by deriving an optimal adaptive policy. We label the proposed approach as adaptive two-stage stochastic programming and provide a generic mixed-integer programming formulation for finite stochastic processes. We show that adaptive two-stage stochastic programming is NP-hard in general. Next, we derive bounds on the value of adaptive two-stage programming in comparison to the two-stage and multi-stage approaches for a specific problem structure inspired by the capacity expansion planning problem. Since directly solving the mixed-integer linear program associated with the adaptive two-stage approach might be very costly for large instances, we propose several heuristic solution algorithms based on the bound analysis. We provide approximation guarantees for these heuristics. Finally, we present an extensive computational study on an electricity generation capacity expansion planning problem and demonstrate the computational and practical impacts of the proposed approach from various perspectives.

Adaptive Two-stage Stochastic Programming with an Analysis on Capacity Expansion Planning Problem

Abstract

Multi-stage stochastic programming is a well-established framework for sequential decision making under uncertainty by seeking policies that are fully adapted to the uncertainty. Often such flexible policies are not desirable, and the decision maker may need to commit to a set of actions for a number of planning periods. Two-stage stochastic programming might be better suited to such settings, where the decisions for all periods are made here-and-now and do not adapt to the uncertainty realized. In this paper, we propose a novel alternative approach, where the stages are not predetermined but part of the optimization problem. Each component of the decision policy has an associated revision point, a period prior to which the decision is predetermined and after which it is revised to adjust to the uncertainty realized thus far. We motivate this setting using the multi-period newsvendor problem by deriving an optimal adaptive policy. We label the proposed approach as adaptive two-stage stochastic programming and provide a generic mixed-integer programming formulation for finite stochastic processes. We show that adaptive two-stage stochastic programming is NP-hard in general. Next, we derive bounds on the value of adaptive two-stage programming in comparison to the two-stage and multi-stage approaches for a specific problem structure inspired by the capacity expansion planning problem. Since directly solving the mixed-integer linear program associated with the adaptive two-stage approach might be very costly for large instances, we propose several heuristic solution algorithms based on the bound analysis. We provide approximation guarantees for these heuristics. Finally, we present an extensive computational study on an electricity generation capacity expansion planning problem and demonstrate the computational and practical impacts of the proposed approach from various perspectives.

Paper Structure

This paper contains 32 sections, 16 theorems, 41 equations, 9 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Adaptive Two-stage Stochastic Programming Formulation, Complexity and Value
  4. Generic Structure of Adaptive Two-stage Approach
  5. Formulation of adaptive two-stage approach
  6. A motivating example: the newsvendor problem
  7. Scenario Tree Formulation
  8. Value of Adaptive Two-stage Stochastic Solutions
  9. Analysis of Capacity Expansion Planning Problem
  10. Problem Formulation
  11. Deriving VATS for Capacity Expansion Planning
  12. VATS for the single-resource problem
  13. Analytical insights on the single-resource problem
  14. Demand Sensitivity
  15. Cost Sensitivity
  16. ...and 17 more sections

Key Result

Theorem 1

Order quantity for the adaptive two-stage problem eq:ATS_DPFormulation can be represented in the following form: where $X_{i,j}$ is the order up to levels from periods $i$ to $j$, $D_{i,j} = \sum_{t=i}^{j} d_t$, $\widetilde{F}_{i,j}$ is the cumulative distribution function of $D_{i,j}$, $s_{t^*}$ is the inventory level at time $t^{*}$, and $c_{T+1} = 0$.

Figures (9)

  • Figure 1: Decision dynamics in different methodologies in terms of the duration of the here-and-now decisions determined at the beginning of the planning.
  • Figure 2: Objective values under different revision times.
  • Figure 3: Scenario tree structure.
  • Figure 4: Decision structures for $\{x_n\}_{n \in {\mathcal{T}}}$ in different stochastic programming approaches.
  • Figure 5: Value of Adaptive Two-stage on Instances with Different Variability based on RVATS (%)
  • ...and 4 more figures

Theorems & Definitions (18)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • Theorem 3
  • Proposition 4
  • ...and 8 more