restaurant_tv.Rmd

---
title: "Regression and Other Stories: Restaurant"
author: "Andrew Gelman, Jennifer Hill, Aki Vehtari"
date: "`r Sys.Date()`"
output:
  github_document:
    toc: true
---
Tidyverse version by Bill Behrman.

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

-------------

```{r, message=FALSE}
# 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.

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

Restaurant net profit.

```{r, fig.asp=0.75}
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.

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

Compile the Stan function and optimize it.

```{r, message=FALSE, warning=FALSE, error=FALSE, results=FALSE}
set.seed(327)

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

```{r}
fit

fit$par - 22
fit$value - net_profit(22)
```

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.