Write a Python program that solve each of the following problems by dynamic programming. You expect the price of a commodity to behave as follows over the next $12$ months:

Month $123456789101112$

Price $($$$/$unit$)10878605697130141100705990133$

Your inventory capacity is $25$ units and you have $3$ units in inventory at the start of month $1.$ In any given month, you can buy up to $10$ units or sell up to $15$ units. The prices per unit sold or bought are the same within each month. However, in each month that you buy or sell any units, you must pay a $$flat$$ $$120$ transaction charge. For example, if you were to buy $3$ units in month $2,$ your cash flow for that month would be $(3$ units$)\backslash$times $($$$78$ per unit$)$ $$120=$ $$354.$ Conversely, if you were to sell $2$ units in month $8,$ your cash flow for that month would be $+(2$ units$)\backslash$times $($$$100$ per unit$)$ $$120=$ $$80.$ The only way to avoid the $$120$ charge for a month is to neither buy nor sell. Your holding cost is $$11$ per unit per month, assessed on the inventory at the end of each month. There is no residual value or cost associated with having any units of the commodity left over after month $12.$

You want to identify the trading pattern that maximizes your total profit over the $12-$month time span.

To aid your thinking, in this problem:

The time periods are the $12$ months

The state is how many units of the commodity you have in stock at the start of the month

The decision in each time period is how many units to buy or sell $($including $0$ for doing neither$)$

The interpretation of the value function is $$ft$($i$)$ is the largest profit you can obtain in months t through $12$ if you have i units of the commodity in stock at the start of month t$.$

Hint: you may find it helpful to use the convention that a positive value of the inner loop decision variable d $($whose optimal choice you will store in x$[$t$,$i$])$ denotes buying d units, a negative value of d denotes selling $|$d$|$ units and a zero value denotes neither buying nor selling. For example,

If you have $3$ units in stock at the beginning, you may consider selling up to $3$ units, doing nothing, or buying up to $10,$ so the decision loop variable d should have the range $3,2,1,0,1,2,3,4,5,6,7,8,9,10.$

If you have $20$ units in stock, you may consider selling up to $15$ units, doing nothing, or buying up to $5=2520,$ so the decision loop variable d should have the range $15,14,13,...,2,1,0,1,2,3,4,5.$