restaurant_tv.md

Regression and Other Stories: Restaurant

Andrew Gelman, Jennifer Hill, Aki Vehtari 2021-04-20

• A Computing in R
  • A.7 Some R programming
    • Opimization

Tidyverse version by Bill Behrman.

Demonstration of using Stan for optimization. See Appendix A in Regression and Other Stories.

---

# Packages
library(tidyverse)
library(rstan)

# Parameters
  # Common code
file_common <- here::here("_common.R")

#===============================================================================

# Run common code
source(file_common)

A Computing in R

A.7 Some R programming

Opimization

Restaurant pricing

Net profit for restaurant.

net_profit <- function(x) {
  (5000 / x^2) * (x - 11)
}

Restaurant net profit.

v <- 
  tibble(
    x = seq_range(c(10, 100)),
    y = net_profit(x)
  )

v %>% 
  ggplot(aes(x, y)) +
  geom_line() +
  geom_point(data = tibble(x = 22, y = net_profit(x))) +
  scale_x_continuous(breaks = scales::breaks_width(10)) +
  labs(
    title = "Restaurant net profit",
    subtitle = 
      str_glue("Maximum is at x = 22, y = {format(net_profit(22), digits = 2, nsmall = 2)}"),
    x = "Price of dinner",
    y = "Net profit"
  )

Stan model for net profit.

model_code =
"
parameters {
  real <lower = 0, upper = 100> x;
}
model {
  target += (5000 / x^2) * (x - 11);
}
"

Compile the Stan function and optimize it.

set.seed(327)

fit <- 
  stan_model(model_code = model_code) %>% 
  optimizing()

fit

#> $par
#>  x 
#> 22 
#> 
#> $value
#> [1] 114
#> 
#> $return_code
#> [1] 0
#> 
#> $theta_tilde
#>       x
#> [1,] 22

fit$par - 22

#>         x 
#> -3.06e-07

fit$value - net_profit(22)

#> [1] -1.42e-14

The optimization returns values close to the true values. If the return code had not been zero, that would have indicated a problem with the optimization.