Stochastic Programming Project 1 –Solved

In our first extension, we will assume that if you don’t print enough newspapers to satisfy demand, then 
you can send a rush order to the printers to print all that you need. These rushed newspapers cost more 
to print, we will call the cost per rushed newspaper g, where g>c. Additionally, if you print more than 
the demand you must pay a disposal fee of t dollars per newspaper. You could alternatively think about 
this as receiving money for taking excess newspapers to a recycling center if t<0, but for now we’ll think 
of t>0. Putting this together, the objective function is: 
max 
! 
1
?&$ (??" − ?? − ?(?" − ?)# − ?(? − ?")#) 
"%& 
where (?)# = max (?, 0). Part of this project will be to reformulate this as a linear program. 
In our second extension of the NV model we will assume that price impacts demand linearly with error, 
?" = ?' + ?&? + ?", and jointly solve for the optimal price and quantity to print. This is in the model 
with disposal costs, and quick printing costs. To do this, you will be given a data set with price and 
demand data. You will fit a linear regression to this data, and you will assume the source of randomness 
is the residuals of this regression. This means once you find ?' and ?& you can find lots of demand data 
points for any one price: get all your residuals and plug them into the regression equation for that one 
price. That is, your data becomes just the residuals of your regression, instead of the original 
price/demand data. If you find ?' = 1000 and ?& = −2 and two of the residuals are 15 and -9, then for 
a price of $2 you get 2 demand data points of 1000-2*2+15=1011 and 1000-2*2-9=9987. 
You can see that when demand becomes a function of price like this you must multiply ??" but this 
becomes ?(?' + ?&? + ?"). This is then a quadratic function of ?. 
To solve these problems, you will have to create some dummy variables for each day, ℎ", that represent 
some part of the objective; in class we let ℎ
" represent the entire profit for each day. In this problem, 
you should set ℎ" to be the negative of cost on each day. Then the objective is the average of revenue 
plus negative cost, where the revenue part comes from ??" and the negative cost comes from ℎ". This 
way the quadratic part would stay in the objective instead of going down to a constraint when price 
impacts demand. You still need 2 constraints for the negative cost dummy variables, but these are 
linear constraints instead of quadratic. This is a little tricky to formulate because the problem is a quadratic program, instead of a linear program. The reason we would define ℎ" to be negative cost is 
that we’re maximizing over h, so we want negative costs to be as big as possible; we don’t want big 
costs. Also, the constraints need to be upper bounds on h, not lower bounds. You can see this by 
drawing a similar graph to the one we drew in class when we did the original NV problem. Remember 
that costs are always positive in this model, so negative costs are always negative. Set you lb 
accordingly. 
Specifics 
1) Data on price and demand are included in the assignment. Fit a linear regression model to this 
data set. 
2) Let c=0.5, g=0.75, and t=0.15. Using the residuals, assume the price is p=1 and generate 
demand data. 
3) Solve the optimal quantity to produce when p=1. This is not quadratic at all, just an LP. 
4) Now let price impact demand and solve the resulting QP. What are the optimal price and 
quantity to print? 
5) If you get in a bind and can’t figure out how to formulate the QP you can formulate it as a 
general NLP, but that will result in a 10 point deduction from the project grade. 
6) We are now interested to know how sensitive the optimal price and quantity are to our data set. 
Take a bootstrap sample of the original dataset. Go back and fit new beta’s to the new 
bootstrapped dataset and redo step 4. Find the optimal price and quantity. 
7) Repeat this process of getting new bootstrapped data, fitting new betas to the newly simulated 
data and finding the optimal price/quantity many times. Make histograms of the optimal price 
and quantity. Make a scatterplot with histograms on the x and y axis for this. You can do this 
quite easily using ggplot2. Make a histogram of expectation of profits. 
8) You work for a publishing company. Your boss has been using the standard NV model to set 
printing decisions using the dataset given. Is your boss’ model as good as this one? Could you 
increase revenue by switching models for production/pricing decisions? Describe the 
advantages and disadvantages of both techniques. Your boss is pretty technical and 
understands optimization, so don’t be afraid to include quantitative material. Your boss is also 
busy, so be sure to include some visualizations to get the important points across. For the 
purpose of your report, you can assume that your boss is interested in the data posted with the 
project. Write this as if you were going to give it to your boss as a technical report that may be 
shared with the CEO. That means, make it look GOOD! 
9) HINT: In order to help understand how to formulate the quadratic part of the problem, 
try doing this matrix multiplication: 
(?! ?" ?#) $? 
0 
0 
0 
0 
0 
0 
0 
0' $
?
! 
?
" 
?
#' 
What do you get? This just tells you how to do the quadratic part, there is still a linear 
part you’ll need to figure out.