Dampening the Curse of Dimensionality Decomposition Methods for - - PowerPoint PPT Presentation

Dampening the Curse of Dimensionality

Decomposition Methods for Stochastic Optimization Problems Vadim Gorski Adriana Kiszka Nils Löhndorf Goncalo Terca David W. Conference on Computational Management Science Mathematical Methods in Industry and Economics 03/2019

Outline

Multi-Stage Stochastic Programming

Discretization & complexity The curse of dimensionality

Stochastic Optimization with a Markovian Structure

Scenario trees and scenario lattices

Decomposition algorithms

Solving stochastic optimization problems on scenario lattices

Illustrative problems

Dynamic newsvendor models Aggregate production planning

The QUASAR framework

1/28

Optimization Under Uncertainty

Question 1

How does uncertainty enter the problem? As a discrete or continuous random variable As a discrete or continuous random process As a set of scenarios without probabilities

2/28

Optimization Under Uncertainty

Question 1

How does uncertainty enter the problem? As a discrete or continuous random variable As a discrete or continuous random process As a set of scenarios without probabilities

Question 2

What is the nature of the decision problem? Optimizing a worst-case outcome Optimizing a (possibly risk adjusted) expectation Problem with or without recourse decisions Almost sure or probabilistic constraints Discrete time, continuous time problems Finite horizon or infinite horizon Convex or non-convex problems

2/28

Setting of this Talk

Problem Class

In this talk we will consider convex, multi-stage decision problems in discrete time with a finite planning horizon with relative complete recourse where and constraints have to hold almost surely randomness is modeled by a Markov process.

3/28

Setting of this Talk

Problem Class

In this talk we will consider convex, multi-stage decision problems in discrete time with a finite planning horizon with relative complete recourse where and constraints have to hold almost surely randomness is modeled by a Markov process.

Standard Approach: Scenario Trees

Model stochastic process as a scenario tree Solve problem as deterministic equivalent

3/28

Multi-Stage Stochastic Optimization

Multi-stage problem with T stages ξ = (ξ1, . . . , ξT) with ξt(ω) ∈ Rm and ξt = (ξ1, . . . , ξt) x = (x1, . . . , xT) decisions with xt = (x1, . . . , xt) and xt ∈ Xt xt measurable w.r.t. σ(ξt)

4/28

Multi-Stage Stochastic Optimization

Multi-stage problem with T stages ξ = (ξ1, . . . , ξT) with ξt(ω) ∈ Rm and ξt = (ξ1, . . . , ξt) x = (x1, . . . , xT) decisions with xt = (x1, . . . , xt) and xt ∈ Xt xt measurable w.r.t. σ(ξt) Define the value functions VT(xT−1, ξT) = max

xT ∈XT (xT−1,ξT ) RT(xT, ξT)

Vt(xt−1, ξt) = max

xt∈Xt(xt−1,ξt) Rt(xt, ξt) + E

• Vt+1(xt, ξt+1)|ξt

, ∀t < T

4/28

Multi-Stage Stochastic Optimization

Multi-stage problem with T stages ξ = (ξ1, . . . , ξT) with ξt(ω) ∈ Rm and ξt = (ξ1, . . . , ξt) x = (x1, . . . , xT) decisions with xt = (x1, . . . , xt) and xt ∈ Xt xt measurable w.r.t. σ(ξt) Define the value functions VT(xT−1, ξT) = max

xT ∈XT (xT−1,ξT ) RT(xT, ξT)

Vt(xt−1, ξt) = max

xt∈Xt(xt−1,ξt) Rt(xt, ξt) + E

• Vt+1(xt, ξt+1)|ξt

, ∀t < T

Note

Functions xt → E

• Vt+1(xt, ξt+1)|ξt

have to be evaluated Closed form expressions are rarely available Discretization of ξ required for numerical solutions

4/28

Complexity of Stochastic Programming

ξ ξ3 ξ2

Number of required points grows exponentially in dimension of ξ

5/28

Complexity of Stochastic Programming

ξ ξ3 ξ2

Number of required points grows exponentially in dimension of ξ Even if solution algorithm well behaved problems are intractable

5/28

Complexity of Stochastic Programming

ξ ξ3 ξ2

Number of required points grows exponentially in dimension of ξ Even if solution algorithm well behaved problems are intractable Curse of Dimensionality in Stochastic Optimization

Swamy [2005] Shapiro and Nemirovski [2005] Shmoys and Swamy [2006] Dyer and Stougie [2006] Hanasusanto et al. [2016]

5/28

Complexity of Stochastic Programming

ξ ξ3 ξ2

Number of required points grows exponentially in dimension of ξ Even if solution algorithm well behaved problems are intractable

Drivers of the Curse of Dimensionality

Number of random variables per stage Number of stages

5/28

Scenario Tree Models

Nodes represent values, arcs the possible transitions

6/28

Scenario Tree Models

Nodes represent values, arcs the possible transitions Tree model represents conditional probability allowing to solve Vt(xt−1, ξt) = min

xt∈Xt Rt(xt, ξt) + E

• Vt+1(xt, ξt+1)|ξt

6/28

Scenario Tree Models

ξ3 ξ2 ξ3 ξ2

Restriction

Branching constrains scenario locations Discretization becomes harder

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Scenario Tree Models

Sufficient branching is important for realistic models

Finance: to avoid arbitrage n assets require n + 1 successors

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Scenario Tree Models

Sufficient branching is important for realistic models Tree where every node ≥ 2 successor has at least 2T nodes

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Scenario Tree Models

Sufficient branching is important for realistic models Tree where every node ≥ 2 successor has at least 2T nodes Alternative: trees where some nodes only have one successor

Sub-problems on the nodes are deterministic Decision maker is partially clairvoyant Overly optimistic planning and flawed policies

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Scenario Tree Models

Sufficient branching is important for realistic models Tree where every node ≥ 2 successor has at least 2T nodes Alternative: trees where some nodes only have one successor

Scenario generation literature

Monte Carlo

Shapiro [2003, 2008]

Probability Metrics

Pflug [2001, 2009], Pflug and Pichler [2012] Dupacová et al. [2003], Heitsch and Römisch [2003]

Moment Matching

Høyland and Wallace [2001], Høyland et al. [2003], Kaut and Wallace [2003]

Integration Quadratures

Pennanen [2005, 2009]

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Lattices: A Compressed Representation

Definition

A scenario lattice is a scenario tree, where a node can have multiple predecessors.

Scenariotree Scenariolattice

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Lattices: A Compressed Representation

Definition

A scenario lattice is a scenario tree, where a node can have multiple predecessors.

Scenariotree Scenariolattice

No history → only works for Markov processes

9/28

Lattices: A Compressed Representation

Definition

A scenario lattice is a scenario tree, where a node can have multiple predecessors.

Scenariotree Scenariolattice

No history → only works for Markov processes Tree represents |NT| scenarios Lattice (potentially) represents |N1| × |N2| × · · · × |NT| scenarios

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Optimal Lattice Generation

Aim

Given a Markov process ξ = (ξ1, . . . , ξT) find a small scenario lattice ˆ ξ such that |Vt(x, ξt) − Vt(x, ˆ ξt)| is small for all x.

10/28

Optimal Lattice Generation

Aim

Given a Markov process ξ = (ξ1, . . . , ξT) find a small scenario lattice ˆ ξ such that |Vt(x, ξt) − Vt(x, ˆ ξt)| is small for all x. Bally and Pagès [2003] reduces a GBM by minimizing the Wasserstein metric using stochastic gradient descent.

10/28

Optimal Lattice Generation

Aim

Given a Markov process ξ = (ξ1, . . . , ξT) find a small scenario lattice ˆ ξ such that |Vt(x, ξt) − Vt(x, ˆ ξt)| is small for all x. Bally and Pagès [2003] reduces a GBM by minimizing the Wasserstein metric using stochastic gradient descent. Löhndorf and Wozabal [2018] generalize these ideas to general processes using a second order stochastic gradient method.

Fast parameter free method

10/28

Optimal Lattice Generation

Aim

Given a Markov process ξ = (ξ1, . . . , ξT) find a small scenario lattice ˆ ξ such that |Vt(x, ξt) − Vt(x, ˆ ξt)| is small for all x. Bally and Pagès [2003] reduces a GBM by minimizing the Wasserstein metric using stochastic gradient descent. Löhndorf and Wozabal [2018] generalize these ideas to general processes using a second order stochastic gradient method.

Fast parameter free method

Kiszka and Wozabal [2018] adapt ideas from Pflug and Pichler [2012] to lattices with randomness in the constraints.

Theoretically superior to Löhndorf and Wozabal [2018] Not (yet) suitable for large models

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Solving Stochastic Optimization on Trees

Assign a decision xn to every node n ∈ Nt, 1 ≤ t ≤ T Set up deterministic equivalent formulation Find optimal decisions x∗

n

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Stochastic Optimization on Lattices

Generally: decision dependent on path leading to n ∈ Nt xn = xn(xn−1, ξt)

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Stochastic Optimization on Lattices

Generally: decision dependent on path leading to n ∈ Nt xn = xn(xn−1, ξt) Define the resource state at the beginning of period t St(x1, ξ1, . . . , xt−1, ξt−1) = St(St−1, xt−1, ξt−1)

12/28

Stochastic Optimization on Lattices

Generally: decision dependent on path leading to n ∈ Nt xn = xn(xn−1, ξt) Define the resource state at the beginning of period t St(x1, ξ1, . . . , xt−1, ξt−1) = St(St−1, xt−1, ξt−1)

Example: Hydro storage optimization

Storage level St is given by St = S0 +

t−1

• i=1

(xi + ξi) = St−1 + (xt−1 + ξt−1) where xi are decisions and ξi are the natural inflows in period i.

12/28

Stochastic Optimization on Lattices

Generally: decision dependent on path leading to n ∈ Nt xn = xn(xn−1, ξt) Define the resource state at the beginning of period t St(x1, ξ1, . . . , xt−1, ξt−1) = St(St−1, xt−1, ξt−1) Decision are based on current state of the system xt(x1, ξ1, . . . , xt−1, ξt−1, ξt) = xt(St, ξt)

12/28

Stochastic Optimization on Lattices

Generally: decision dependent on path leading to n ∈ Nt xn = xn(xn−1, ξt) Define the resource state at the beginning of period t St(x1, ξ1, . . . , xt−1, ξt−1) = St(St−1, xt−1, ξt−1) Decision are based on current state of the system xt(x1, ξ1, . . . , xt−1, ξt−1, ξt) = xt(St, ξt)

Markov Decision Process (MDP)

If ξ is Markov and the Vt only depend on the current state, i.e., E

• Vt(xt−1, ξt)|ξt−1

= E [Vt(St, ξt)|ξt−1] , then the problem has a Markov structure.

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Solving MDPs

Discretization Strategy

Discretization of events as lattice (environmental state) Continuous actions and resource states St

13/28

Solving MDPs

Discretization Strategy

Discretization of events as lattice (environmental state) Continuous actions and resource states St Solving MDP equivalent to finding the functions S → Vt(S, ξt)

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Solving MDPs

Discretization Strategy

Discretization of events as lattice (environmental state) Continuous actions and resource states St Solving MDP equivalent to finding the functions S → Vt(S, ξt) L-Shaped Method: Van Slyke and Wets [1969] Benders decomposition: Benders [1962], Louveaux [1980] Stochastic Dual Dynamic Programming: Pereira and Pinto [1991] Approximate Dual Dynamic Programming (ADDP): Löhndorf et al. [2013], Löhndorf and Wozabal [2018]

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How to find Value Functions?

Assume that for each realization ˆ ξT of the discretized Markov process VT(ST, ˆ ξT) =

• maxxT

c⊤

ˆ ξT xT

s.t. Wˆ

ξT xT ≤ bˆ ξT + Bˆ ξT ST

with VT(ST, ˆ ξT) > −∞ for every ST.

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How to find Value Functions?

Assume that for each realization ˆ ξT of the discretized Markov process VT(ST, ˆ ξT) =

• maxxT

c⊤

ˆ ξT xT

s.t. Wˆ

ξT xT ≤ bˆ ξT + Bˆ ξT ST

with VT(ST, ˆ ξT) > −∞ for every ST. S → VT(S, ˆ ξT) is a piecewise linear concave function Supergradient of S → VT can be extracted from dual solution

Idea

Approximate S → VT(S, ˆ ξt) by (minimum of) supergradients.

14/28

How to find Value Functions?

Assume that for each realization ˆ ξT of the discretized Markov process VT(ST, ˆ ξT) =

• maxxT

c⊤

ˆ ξT xT

s.t. Wˆ

ξT xT ≤ bˆ ξT + Bˆ ξT ST

with VT(ST, ˆ ξT) > −∞ for every ST. S → VT(S, ˆ ξT) is a piecewise linear concave function Supergradient of S → VT can be extracted from dual solution

Idea

Approximate S → VT(S, ˆ ξt) by (minimum of) supergradients.

s1 V S s2 s1 V S s2 s1 V S s3

14/28

How to find Value Functions?

Assume that for each realization ˆ ξT of the discretized Markov process VT(ST, ˆ ξT) =

• maxxT

c⊤

ˆ ξT xT

s.t. Wˆ

ξT xT ≤ bˆ ξT + Bˆ ξT ST

with VT(ST, ˆ ξT) > −∞ for every ST. S → VT(S, ˆ ξT) is a piecewise linear concave function Supergradient of S → VT can be extracted from dual solution

Idea

Approximate S → VT(S, ˆ ξt) by (minimum of) supergradients.

Problem

How to determine for which S to calculate supergradients?

14/28

How to find Value Functions?

Assume that for each realization ˆ ξT of the discretized Markov process VT(ST, ˆ ξT) =

• maxxT

c⊤

ˆ ξT xT

s.t. Wˆ

ξT xT ≤ bˆ ξT + Bˆ ξT ST

with VT(ST, ˆ ξT) > −∞ for every ST. S → VT(S, ˆ ξT) is a piecewise linear concave function Supergradient of S → VT can be extracted from dual solution

Idea

Approximate S → VT(S, ˆ ξt) by (minimum of) supergradients.

Problem

How to determine for which S to calculate supergradients?

Solution: Sampling

Sample S using current value function approximations.

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ADDP: Approximating Value Funcions

Given an approximation S → ˆ Vt(S, ˆ ξt), ∀t, ∀ˆ ξt. Forward-Pass

Sample path (ˆ ξ1, . . . , ˆ ξT) from lattice Decide on xt using ˆ Vt(S, ˆ ξt) and store resource path (s1, . . . , sT)

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ADDP: Approximating Value Funcions

Given an approximation S → ˆ Vt(S, ˆ ξt), ∀t, ∀ˆ ξt. Forward-Pass

Sample path (ˆ ξ1, . . . , ˆ ξT) from lattice Decide on xt using ˆ Vt(S, ˆ ξt) and store resource path (s1, . . . , sT)

Backward-Pass (from t = T, . . . , 1)

At t calculate tangent l to S → ˆ Vt(S, ˆ ξt) at point st ˆ Vt(st, ˆ ξt) =

• maxxt

c⊤

ˆ ξt xt + E

• Vt+1(St+1, ˆ

ξt+1)|ˆ ξt

• s. t.

W ˆ

ξt xt ≤ b ˆ ξt + B ˆ ξt st

ˆ Vt(S, ˆ ξt) ← min

• l, ˆ

Vt(S, ˆ ξt)

• .

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ADDP: Approximating Value Funcions

Given an approximation S → ˆ Vt(S, ˆ ξt), ∀t, ∀ˆ ξt. Forward-Pass

Sample path (ˆ ξ1, . . . , ˆ ξT) from lattice Decide on xt using ˆ Vt(S, ˆ ξt) and store resource path (s1, . . . , sT)

Backward-Pass (from t = T, . . . , 1)

At t calculate tangent l to S → ˆ Vt(S, ˆ ξt) at point st ˆ Vt(st, ˆ ξt) =

• maxxt

c⊤

ˆ ξt xt + E

• Vt+1(St+1, ˆ

ξt+1)|ˆ ξt

• s. t.

W ˆ

ξt xt ≤ b ˆ ξt + B ˆ ξt st

ˆ Vt(S, ˆ ξt) ← min

• l, ˆ

Vt(S, ˆ ξt)

• .

V(S,ξ) S Value function V(S,ξ) S Value function st

15/28

ADDP: Approximating Value Funcions

Given an approximation S → ˆ Vt(S, ˆ ξt), ∀t, ∀ˆ ξt. Forward-Pass

Sample path (ˆ ξ1, . . . , ˆ ξT) from lattice Decide on xt using ˆ Vt(S, ˆ ξt) and store resource path (s1, . . . , sT)

Backward-Pass (from t = T, . . . , 1)

At t calculate tangent l to S → ˆ Vt(S, ˆ ξt) at point st ˆ Vt(st, ˆ ξt) =

• maxxt

c⊤

ˆ ξt xt + E

• Vt+1(St+1, ˆ

ξt+1)|ˆ ξt

• s. t.

W ˆ

ξt xt ≤ b ˆ ξt + B ˆ ξt st

ˆ Vt(S, ˆ ξt) ← min

• l, ˆ

Vt(S, ˆ ξt)

• .

Exploitation

Enumeration of all possible states (exploration) too expensive Sampling makes use (exploits) structure of the problem Approximations ˆ V only tight at relevant points

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ADDP Convergence

Proposition

Approximate value functions S → ˆ Vt(S, ξt)

are upper bounds for true value functions; converge to the true value function.

Linear problems → finitely many iterations

Differences to SDDP

Works with Markov processes, no restriction to stage-wise independence. Randomness can occur everywhere in the problem, not only in the right hand side of the constraints.

16/28

ADDP: Convergence Check

Lower Bound

Simulate N trajectories (ˆ ξ1, . . . , ˆ ξT) on the lattice Use ˆ Vt to compute cumulative profits ˆ Πi, i = 1, . . . , N

17/28

ADDP: Convergence Check

Lower Bound

Simulate N trajectories (ˆ ξ1, . . . , ˆ ξT) on the lattice Use ˆ Vt to compute cumulative profits ˆ Πi, i = 1, . . . , N Define a lower bound LB = N−1 N

i=1 ˆ

Πi Calculate α-quantile of LB F −1(α) by bootstrapping Check if ˆ V1(S1, ξ1) × (1 − gap) ≤ F −1(α)

17/28

ADDP: Convergence Check

Lower Bound

Simulate N trajectories (ˆ ξ1, . . . , ˆ ξT) on the lattice Use ˆ Vt to compute cumulative profits ˆ Πi, i = 1, . . . , N Define a lower bound LB = N−1 N

i=1 ˆ

Πi Calculate α-quantile of LB F −1(α) by bootstrapping Check if ˆ V1(S1, ξ1) × (1 − gap) ≤ F −1(α)

17/28

Value of a Policy

Out of Sample Value

Average profit for original model of process Simulate N trajectories (ξ1, . . . , ξT) for environmental state

18/28

Value of a Policy

Out of Sample Value

Average profit for original model of process Simulate N trajectories (ξ1, . . . , ξT) for environmental state Use ˆ Vt of closest lattice node for decision

18/28

Value of a Policy

Out of Sample Value

Average profit for original model of process Simulate N trajectories (ξ1, . . . , ξT) for environmental state Use ˆ Vt of closest lattice node for decision N simulated profits ˆ Πi, i = 1, . . . , N

18/28

Value of a Policy

Out of Sample Value

Average profit for original model of process Simulate N trajectories (ξ1, . . . , ξT) for environmental state Use ˆ Vt of closest lattice node for decision N simulated profits ˆ Πi, i = 1, . . . , N OV = N−1 N

i=1 ˆ

Πi

18/28

Value of a Policy

Out of Sample Value

Average profit for original model of process Simulate N trajectories (ξ1, . . . , ξT) for environmental state Use ˆ Vt of closest lattice node for decision N simulated profits ˆ Πi, i = 1, . . . , N OV = N−1 N

i=1 ˆ

Πi OV is the value under the original model If |Objective − OV| small → lattice sufficiently sized Not (easily) possible with scenario trees!

18/28

Numerical Examples

Idea

Benchmark ADDP against state of the art tree based modeling.

19/28

Numerical Examples

Idea

Benchmark ADDP against state of the art tree based modeling.

Approach

For a given problem: Solve in a rolling stochastic fashion N times Common random numbers (scenarios) to build trees and lattices Use same out of sample scenarios to evaluate decisions Use SCENRED2 (GAMS) to build scenario trees Solve tree based problems using deterministic equivalents Test whether observed differences are significant

19/28

Rolling Stochastic Optimization

20/28

Flower Girl

max E T

t=1 stpR t − btpW t

• s.t.

st ≤ max(Dt, It), ∀t It = γIt−1 + bt−1 − st, ∀t 0 ≤ It ≤ ¯ I, ∀t Newsvendor with storage It with capacity ¯ I Optimal buying (bt) and selling (st) decisions of flowers Deterministic wholesale and retail prices pW

t , pR t

Random demand Dt following a driftless GBM

21/28

Flower Girl

max E T

t=1 stpR t − btpW t

• s.t.

st ≤ max(Dt, It), ∀t It = γIt−1 + bt−1 − st, ∀t 0 ≤ It ≤ ¯ I, ∀t Newsvendor with storage It with capacity ¯ I Optimal buying (bt) and selling (st) decisions of flowers Deterministic wholesale and retail prices pW

t , pR t

Random demand Dt following a driftless GBM

T Profit Tree Profit ADDP Diff % t-paired test 5 3,919 3,925 0.15% – 10 5,743 5,945 3.51% – 15 17,297 18,940 9.50% 4.1% 20 21,303 23,423 9.95% 4.1% 50 57,343 72,543 26.51% 2.0%

21/28

Flower Girl

max E T

t=1 stpR t − btpW t

• s.t.

st ≤ max(Dt, It), ∀t It = γIt−1 + bt−1 − st, ∀t 0 ≤ It ≤ ¯ I, ∀t Assume random sales price pR

t following a driftless GBM

Sales prices and demand are assumed to be independent

T Profit Tree Profit ADDP Diff % t-paired test 5 2,973 3,180 6.97% – 10 10,508 11,813 12.42% – 15 57,389 66,122 15.22% – 20 82,332 102,433 24.41% 5.1% 50 120,321 155,312 29.08% 1.0%

22/28

Flower Girl

max (1 − λ)E T

t=1 stpR t − btpW t

• + λ CVaRα

T

t=1 stpR t − btpW t

• s.t.

st ≤ max(Dt, It), ∀t It = γIt−1 + bt−1 − st, ∀t 0 ≤ It ≤ ¯ I, ∀t Use CVaR-expectation mixture in the objective Sales deterministic

T Profit Tree Profit ADDP Diff % t-paired test 5 2,936 3,143 7.04% – 10 9,284 10,675 14.98% – 15 57,893 71,050 22.73% 4.5% 20 79,871 103,212 29.22% 2.1%

23/28

Aggregate Production Planning

min E T

t=1

I

i=1 citXit + hitIit + rtR1 + otOt

• s.t.

It − It+1 + Xit = Dit I

i=1 miXit ≤ R1 + Ot

R1 ≤ b X, I, R, O ≥ 0 Aggregate production planning, n products, Bitran et al. [1981] where cit . . . unit production costs excluding labor hit . . . inventory carrying cost mi . . . hours required to produce one unit rt, ot . . . regular and overtime labor costs Dit . . . Random demand for each of the i products R1 . . . Regular labor hiring decision Ot . . . Overtime labor hiring decision Xit . . . Production decision for each of the i products Iit . . . Inventory decision for each of the i products

24/28

Aggregate Production Planning

min E T

t=1

I

i=1 citXit + hitIit + rtR1 + otOt

• s.t.

It − It+1 + Xit = Dit I

i=1 miXit ≤ R1 + Ot

R1 ≤ b X, I, R, O ≥ 0

Characteristics

Stochastic problem with scalable dimension (number of products) Two types of decisions Inventory and production: in anticipation of the next stage Regular labor: first-stage decision only

24/28

Aggregate Production Planning

Random demand modeled as GBM Correlation between products modeled by random covariance matrix sampled according to Lewandowski et al. [2009] Test different levels of dependency Lattice and trees built based on 50,000 to 1,000,000 scenarios. Problem parameters similar to Bitran et al. [1981]

25/28

Results APP

I T correlation Cost Tree Cost Lattice Diff % p-value 1 5 medium 12,474 11,253 10.85 <1% 3 5 medium 3,084 2,739 12.57 <1% 5 5 medium 7,974 7,040 13.26 <1% 7 5 medium 16,275 14,315 13.69 <1% 9 5 medium 17.558 11,745 49.49 <1% 5 5 very high 12,432 9,957 19.91 <1% 5 5 high 80,095 70,307 12.22 <1% 5 5 medium 79,129 70,254 11.22 <1% 5 5 low 79,263 70,230 11.40 <1% 5 3 medium 4,466 4,146 7.15 2.8% 5 5 medium 8,009 7,066 11.78 <1% 5 10 medium 15,843 10,382 34.47 <1%

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QUASAR: An ADDP Implementation

ADDP implementation in JAVA

Generation of lattices Parallelized solution algorithm

iPython and MATLAB interfaces

Easy algebraic formulation of problems and stochastic processes Analysis and exporting facilities

Free for academic use: http://www.quantego.com

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QUASAR Flower Girl Implementation

1

T = 10; orderPrice = 2; sellPrice = 3; loss = 0.2; storageSize = 1000;

2 3

model = quasarDecisionProblem(’FlowerGirl’);

4

prevStorage = 0; lastOrder = 0;

5

profit = quasarExpression();

6

for t = 0:T-1

7

storage = model.addVariable(t, ’storage’); % amount in storage

8

storeDelta = model.addVariable(t, ’storeD’); % change in storage

9

discard = model.addVariable(t, ’discard’); % flowers discarded

10

• rder = model.addVariable(t, ’order’);

% flowers ordered

11

sell = model.addVariable(t, ’sell’); % amount of flowers sold

12 13

model.addConstraint(storage == gamma*prevStorage + storeDelta - discard);

14

model.addConstraint(storeDelta == lastOrder - sell + discard);

15

model.addConstraint(storage <= storageSize);

16

model.addConstraint(sell <= quasarRandomVariable(’D’));

17 18

profit = profit - orderPrice * order + sellPrice * sell;

19

lastOrder = order; prevStorage = storage;

20

end

21

model.maximize(profit);

22 23

vol = 0.1; rho = 1.0; D0 = 1000;

24

errors = quasarLognormalDist.createUnivariate(0, vol);

25

process = quasarAutoregressiveModel.createUnivariate(’D’, D0, rho, errors);

26

process.geometric;

27 28

• ptBuilder = quasarDynamicOptimizerBuilder.create(model, process, 100);

29

• ptProblem = optBuilder.build; optProblem.solve;

28/28

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