Model: Iron Ore Production with Exogenous Stochastic Price (IRONORE)

Description:

Iron ore is traded on the spot market, facing an exogenous, stochastic price. There is enormous demand for iron, and so for the purposes of a small or medium-sized iron ore mine, we assume that any quantity of ore can be instantaneously sold at current market rates.

Let there be \(T\) time periods (days), holding cost of \(h\) per unit, production cost of \(c\) per unit, maximum production per day of \(m\) units, and maximum holding capacity of \(K\) units. Let the iron ore market price for the day be \(P_t\).

Let \(x_1\) be the price at which to begin production, \(x_2\) be the inventory level at which to cease production, \(x_3\) be the price at which to cease production, and \(x_4\) be the price at which to sell all current stock.

The daily order of operations in the simulation is as follows:

  1. Sample the market price, \(P_t\). Let current stock be \(s_t\).

  2. If production is already underway,

  1. if \(P_t\) ≤ \(x_3\) or \(s_t\) ≥ \(x_2\), cease production.

  2. else, produce \(min(m, K − s_t)\) at cost \(c\) per unit.

  1. If production is not currently underway, and if \(P_t\) ≥ \(x_1\) and \(s_t\) < \(x_2\), begin production.

  2. If \(P_t\) ≥ \(x_4\), sell all stock (after production) at price \(P_t\).

  3. Charge a holding cost of \(h\) per unit (after production and sales).

Sources of Randomness:

1. Let \(P_t\) be a meanreverting random walk, such that \(P_t = \mbox{trunc}(P_t - 1 + N_t (\mu,\sigma))\), where \(N_t\) is a normal random variable with standard deviation \(\sigma\) and mean \(\mu_t = \mbox{sgn}(\mu_0 − P_t−1) * (| \mu_0 − P_t − 1 |)^{\frac{1}{4}}\). Here \(\mbox{trunc}(x)\) truncates the price to lie between a specified minimum and maximum price.

Model Factors:

  • mean_price: Mean iron ore price per unit.

    • Default: 100.0

  • max_price: Maximum iron ore price per unit.

    • Default: 200.0

  • min_price: Minimum iron ore price per unit.

    • Default: 0.0

  • capacity: Maximum holding capacity.

    • Default: 10000

  • st_dev: Standard deviation of random walk steps for price.

    • Default: 7.5

  • holding_cost: Holding cost per unit per period.

    • Default: 1.0

  • prod_cost: Production cost per unit.

    • Default: 100.0

  • max_prod_perday: Maximum units produced per day.

    • Default: 100

  • price_prod: Price level to start production.

    • Default: 80.0

  • inven_stop: Inventory level to cease production.

    • Default: 7000

  • price_stop: Price level to stop production.

    • Default: 40

  • price_sell: Price level to sell all stock.

    • Default: 100

  • n_days: Number of days to simulate.

    • Default: 365

Respones:

  • total_profit: The total profit over the time period

  • frac_producing: The fraction of days spent producing iron ore

  • mean_stock: The average stocks over the time period

References:

N/A

Optimization Problem: Maximize Profit (IRONORE-1)

Decision Variables:

  • price_prod

  • inven_stop

  • price_stop

  • price_sell

Objectives:

Maximize total_profit over the \(T\) time periods.

Constraints:

All decision variables should be non-negative. Logically, we should also have price_stop <= price_prod <= price_sell, but this is not enforced.

Problem Factors:

  • budget: Max # of replications for a solver to take

    • Default: 1000

Fixed Model Factors:

  • N/A

Starting Solution:

  • initial_solution: \(x_1 = 80\), \(x_2 = 7000\), \(x_3 = 40\), \(x_4=100\)

Random Solutions:

  • \(x_1\): Sample an lognormal random variate with 2.5- and 97.5-percentiles of 10 and 200.

  • \(x_2\): Sample an lognormal random variate with 2.5- and 97.5-percentiles of 1000 and 10000.

  • \(x_3\): Sample an lognormal random variate with 2.5- and 97.5-percentiles of 10 and 200.

  • \(x_4\): Sample an lognormal random variate with 2.5- and 97.5-percentiles of 10 and 200.

Optimal Solution:

Unknown

Optimal Objective Function Value:

Unknown