I am trying to optimize for a given date of deposit and withdrawal of a given commodity in a storage considering that the storage max capacity is 50 and the rate of injection/withdrawal is 5/day. Assume that it costs 0.01 per month to store the commodity. My commodity prices list is equally spaced in monthly dates.
At first I want to try to optimize for a single injection and withdrawn and next extend for multiples injections and withdrawns for the given timeframe i am analyzing.

So I created few constraint functions

i. withdrawal date > deposit date

ii. total volume injected – total volume withdrawn < 50

iii. volume injected = volume withdrawn

iv. withdrawn <= 5 per day

v. deposit <= 5 per day

The profit function is given by
i. volume withdrawn * commodity_price[date_withdrawn] – volume injected * commodity_price[date injected] – cost*(date withdrawn – date injected)

The code is as follows:

    ##Volume in mm BBTUs
    max_volume = 50
    ##Volume in mm BBTUs per day
    rate = 5
    ##Cost in million dollars per day
    costs = 0.01

    commodity_prices = hw_nat_gas['Complete'].to_list()


    from scipy.optimize import minimize


    def constraint1(dates):
        deposit_date, withdrawal_date, volume_deposit, volume_withdrawal = dates
        return withdrawal_date - deposit_date

    # Define constraint function for volume constraint
    def constraint_volume(dates):
        deposit_date, withdrawal_date, volume_deposit, volume_withdrawal = dates  
        total_volume = volume_deposit - volume_withdrawal 
        return max_volume - total_volume 

    def constraint_volume2(dates):
        deposit_date, withdrawal_date, volume_deposit, volume_withdrawal = dates  
        return volume_deposit - volume_withdrawal

    def profit_function(dates):
        deposit_date, withdrawal_date, volume_deposit, volume_withdrawal = map(int, dates)
        profit = volume_withdrawal*commodity_prices[withdrawal_date] -     volume_deposit*commodity_prices[deposit_date] - costs*(deposit_date-withdrawal_date)
        return -profit

    def withdrawal(dates):
        deposit_date, withdrawal_date, volume_deposit, volume_withdrawal = map(int, dates)
        return rate - volume_withdrawal

    def deposit(dates):
        deposit_date, withdrawal_date, volume_deposit, volume_withdrawal = map(int, dates)
        return rate - volume_deposit

    constraints = [{'type': 'ineq', 'fun': constraint1},
                   {'type': 'ineq', 'fun': constraint_volume},
                   {'type': 'ineq', 'fun': withdrawal},
                   {'type': 'ineq', 'fun': deposit},
                   {'type': 'eq', 'fun' : constraint_volume2}]

    bounds = [(0, 58), (0, 58), (0,5), (0,5)]

    initial_guess = [7, 12, 0, 0]

    result = minimize(profit_function, initial_guess, bounds=bounds, constraints=constraints)

At first what I am getting is that there is no variation in the dates of deposit and withdrawal, it simply optimizes for the initial guess, the value i am inserting in the scipy.minimize function. Second, it is not varying as well the volumes for injection.

If I instead of using initial guess of [7, 12, 0, 0] use [7, 12, 5, 5] the value of the profit increases, but i wanted the function to perform this analysis by itself, by varying the parameters and obtaining the combination of values that get the max profit.

Any suggestions?