1. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
A discussion on the performance of the
Coordinate Exchange Algorithm
Tian Tian
Department of Mathematics, Statistics and Computer
Science
University of Illinois at Chicago
09 February 2016
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

2. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Background of DCEs
Background
A product/service of interest
K attributes each with lk levels
L = K
k=1 lk alternatives/profiles
Each choice set consists of J profiles
Q = L
J choice sets
Each design consists of S choice sets
M = Q
S candidate designs (design space)
Goal
–to estimate the appeal of each attribute
–to predict the behaviors of consumers
–to emulate market decisions and forecast market demands
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

3. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Background of DCEs
Two critical components
Parameter setup in the probabilistic model
(we model the probability a consumer chooses profile j in choice
set s — pjs)
An efficient algorithm that finds the optimal discrete choice
design(s)
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

4. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Challenges and motivation
Challenges
First challenge:
The probabilistic choice models are nonlinear ⇒ efficiency of the
design depends on the unknown parameters
Possible solutions:
Linear optimal designs where parameters are set to zeros
Locally optimal designs where parameters are assigned with
nonzero values
Bayesian optimal designs where parameters are introduced with
a prior distribution
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

5. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Challenges and motivation
Challenges (Cont.)
Second challenge:
Difficult (impossible) to theoretically construct locally/Bayesian op-
timal designs under a nonlinear multi-dimensional probabilistic choice
model.
Existing algorithms:
The mFA (modified Federov algorithm)
The RSA (relabeling and swapping algorithm)
The RSCA (relabeling, swapping, and cycling algorithm)
The quadrature scheme
The CEA (coordinate-exchange algorithm)
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

6. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Challenges and motivation
Motivation
Question:
Is the resulting design generated from the CEA indeed optimal, or
at lease highly efficient?
An example:
3 × 3 × 2/3/8 ⇒ (3×3×2
3 )
8
> 4 × 1018
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

7. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Review of the MNL model
The MNL model
The MNL (multinormial logit) probability pjs, j = 1, ..., J, s =
1, ..., S, is modeled as follows,
pjs =
exp(xjsβ)
J
i=1
exp(xisβ)
– ˜K = K
k=1(lk −1) is the total number of main effect levels subject
to the ”zero-sum” constraint
– xjs is a ˜K × 1 vector denoting the attribute levels of profile j in
the sth choice set
– β is a ˜K × 1 vector of parameter representing the main effects of
the attribute levels
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

8. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Review of the CEA
The CEA
1 Start with a randomly chosen starting design
2 Exchange levels for each attribute in each profile in each choice
set in the design
3 A level is updated when the new design results in a smaller
criterion value
4 A complete cycle of iteration terminates when the algorithm has
reached the last attribute in the last profile in the last choice
set
5 The algorithm goes back to the very first attribute and contin-
ues another round
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

9. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Review of the CEA
The CEA (Cont.)
6 A complete try of iteration terminates when no substitution is
made in a whole cycle or other stopping rule is met
7 To avoid poorly local optima, T tries of the algorithm are re-
peated each with a different starting design
8 The optimal design is chosen from the T resulting designs.
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

10. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Discrete and approximate design
To take advantage of the structure under the continuous/approxi-
mate design framework.
Approximate/continuous: ˜ξ = {(Cq, wq)|q = 1, ..., Q}.
Discrete/exact: ξ = {Cs|s = 1, ..., S} = {(Cq, wq)|q = 1, ..., Q}
with wi = 1/S for i = 1, ..., S and wi = 0 for i = S + 1, ..., Q.
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

11. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Rationale
Eff(ξCEA
S ) = Obj(ξ∗
S)
Obj(ξCEA
S )
≥ Obj(ξ∗
)
Obj(ξCEA
S )
where Obj(·) is the objective criterion function.
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

12. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Simulation study
3 × 3 × 2/2/12 3 × 3 × 2/2/12
Locally Bayesian Locally Bayesian
D-optimal 99.65% 98.71% 99.80% 98.82%
A-optimal 99.04% 96.83% 99.11% 97.15%
V -optimal 98.69% 96.75% 97.52% 96.85%
Table: Lower bound of the CEA efficiency under different design setups
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

13. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Simulation study (Cont.)
3 × 3 × 2 × 2 × 2/3/15 4 × 3 × 3 × 2/3/12
Locally Bayesian Locally Bayesian
D-optimal 96.76% 98.67% 96.90% 97.78%
A-optimal 92.84% 97.20% 95.11% 96.03%
V -optimal 94.21% 97.30% 94.12% 96.77%
Table: Lower bound of the CEA efficiency under different design setups
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm

14. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you
Tian Tian MSCS, UIC
A discussion on the performance of the Coordinate Exchange Algorithm