1. Math 6330: Statistical Consulting Class 9
Tony Cox
University of Colorado at Denver
Course web site:

2. Course schedule April 14: Draft of project/term paper due
April 18, 25, May 2, (May 9): In-class presentations
May 4: Final project/paper due by 8:00 PM

3. Agenda Prescriptive (decision) analytics (Cont.) Decision psychology
Decision trees (wrap-up)
Simulation-optimization
Newsvendor problem and applications
Decision rules, optimal statistical decisions
Quality control, SPRT
Decision psychology
Heuristics and biases

4. Advanced decision tree analysis
Game trees
Different decision-makers
Monte Carlo tree search (MCTS) in games with risk and uncertainty
Generating trees
Apply rules to expand and evaluate nodes
Learning trees from data
Sequential testing

5. Summary on decision trees
Decision trees show sequences of choices, chance nodes, observations, and final consequences.
Mix observations, acts, optimization, causality
Good for very small problems; less good for medium-sized problems; unwieldy for large problems  use IDs instead
Can view decision trees and other decision models as simple c(a, s) models
But need good optimization solvers!

6. Example: Influence diagrams

7. Influence diagram algorithms
Goal: Calculate/infer the settings of decision variables that will maximize expected utility
Extend Bayesian Network inference algorithms
Distributed message-passing/updating on graphs
CPTs at chance nodes, EU-maximizing decisions at choice nodes, EU at value node
One approach: “Marginalize out” chance nodes, “optimize out” decision nodes until none left
Ross Schachter, Evaluating Influence Diagrams, 1986
ID algorithms are very mature

8. Simulation-optimization Example: Optimal R&D effort
Each new employee a company hires has a 10% probability (independently of anyone else) of solving a certain R&D problem in the next year 
If solution is obtained in the next year, it is worth $1M (else $0).
Each new employee costs $0.05M. 
To maximize EMV, how many new employees should the company hire to work on this R&D problem?
Approach: Simulation-optimization in R

9. Simulation-Optimization (SO)
Randomly or adaptively* sample act a from choice set A.
Sample state s from Pr(s | a) and sample consequence c from Pr(c | a, s)
Assess Pr(s | a) and Pr(c | a, s) by simulation, analysis, or statistics
Evaluate u(c) or EMV(c)
Repeat steps 2-3 many times, average results to estimate EU(a) and uncertainty intervals
Repeat steps 1-5 many times to find the act a that gives largest estimated EU(a) with high confidence.
* For the (many) technical details, see the following:
(short overview)
(long review)
(tutorial)
Commercial solver in Excel:

10. Solution Each employee has probability 10% of solving problem.
Solution is worth $1M. Each employee costs $0.05M. 
How many employees to hire to maximize EMV?
N = c(1:20); EMV = 1*( ^N) *N; plot(N, EMV)
EMV[6]; EMV[7]; EMV[8];
[1]
[1]
[1]
Optimal number is 7 employees

11. Example application of SO: Newsvendor problem
How many papers to stock?
Each costs $k
Each sells for $
Number sold = min(stock, demand)
demand is uncertain, with CDF of F

12. Newsvendor problem: Analysis
How many papers to stock?
Each costs $k
Each sells for $
Number sold = min(stock, demand)
demand is uncertain, with CDF of F
Analytic solution:
Profit if stock = a and demand = s is *min(a, s) - a*k
Optimization: Marginal benefit (revenue) = marginal cost
(1 - F(a)) = k  F(a*) = 1 – k/
(1 - F(a)) - k = expected additional (marginal) profit from buying one more paper = 0 at optimum.
F(a) = probability it remains unsold

13. Solution by SO
Template: Choose a to optimize EU(a) or EMV(a), given uncertain consequences with distribution Pr(c | a)
Sample many values of a = order quantity
EMV(a) = cEMV(c)*Pr(c | a) = *smin(a, s)*Pr(s) – ka
Risk profile Pr(c | a) = sPr(c | a, s)*Pr(s | a), s = demand
Pr(s | a) = Pr(s) = F(s) – F(s -1)
Optimize (choose) inventory order to maximize average profit, given uncertain demand
Solution by simulation
Solution by analysis
Optimal order quantity is a*, where
pdist(a*) = ( - k)/ =
pdist(x*) = CDF(a*) = Pr( demand < a*)
k = unit cost,  = selling price

14. DA makes a big difference
In experiments, decision-makers (MBA students at Duke) ordered too few high-profit products and too many low-profit products
Average profits are less than optimal by 5%-61%, depending on experimental conditions (Schweitzer and Cachon, 2000)

15. Examples of newsvendor-like problems (act = single number)
Water reservoirs for wind energy backup
Cash reserves
Number of cars or jets in business fleet
Minutes to buy in cell phone plan
How fast to drive
Reservation price for selling house
Pricing an IPO
How early to leave for class?

16. Unknown unknowns: More realistic decision problems
What to do if probability distribution for demand is unknown, and must be estimated from data?
Bayesian decision theory handles this without difficulty: Update prior based on data, then make optimal decision given posterior
Adaptive learning algorithms
Warren Powell’s “knowledge gradient” approach

17. Unknown demand distribution with random shocks
Solve via machine learning algorithms applied to simulated data
Weighted Majority Newsvendor Shifting
DSE = diversity of statistical experts
META = hybrid of approaches from the literature, e.g.,minimax
Many sensitivity analysis results (O’Neill et al., 2015)

18. Example applications of SO: Supply chain inventory control
Template: Choose a to maximize EU(a) = cu(c)*Pr(c | a)
Choose base stock s and order-up-to level S to minimize average holding cost of supply chain inventory
Optimal decision a has (S, s) form, easily optimized
(theory)
deterministic
stochastic, adaptive

19. Important general concepts
A decision rule or policy maps information (what we see) to action (what we do)
(S, s) policy maps observed inventory level to inventory reorder decisions (when, how much)
Netica influence diagrams
Advanced statistical decision theory is largely about optimizing decision rules
Numerical optimization makes some insightful analysis irrelevant
E.g., how should lead time and demand variability affect optimal (S, s)?

20. Example applications of SO
Optimize stock portfolio to maximize average return with uncertain stock prices
Optimize selling time of asset to maximize profit, given uncertain offers and holding costs
Staff emergency room to minimize average total cost per day (including costs of waiting times), given uncertain arrivals
Optimize location of fire stations or ambulances to achieve undominated distribution of response times

21. Wrap-up on SO Very useful but very technical methods
Requires some understanding of problem environment so that probable consequences of alternative decisions (or policies) can be simulated
Can require searching complicated choice sets efficiently and adaptively