Minimizing cost of transportation and storage of items

Question

I am looking for an optimization algorithm to minimize the cost of transportation and storage cost in warehouse.

Let's assume the following table gives us the weekly forecast of Demand.

+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
|              | Week 1  | Week 2  | Week 3  | Week 4  | Week 5  | Week 6  | Week 7  | Week 8  | Week 9  | Week 10 | Week 11 | Week 12 |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| Demand       |  2,300  |  1,800  |  1,100  |  2,300  |  2,000  |  1,600  |  2,200  |  2,000  |  2,900  |  1,900  |  2,000  |   1,000 |
| Buffer       |    600  |    500  |    300  |    600  |    500  |    400  |    600  |    500  |    800  |    500  |    500  |     300 |
| Total Demand |  2,900  |  2,300  |  1,400  |  2,900  |  2,500  |  2,000  |  2,800  |  2,500  |  3,700  |  2,400  |  2,500  |   1,300 |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+

The transportation cost follows the below mentioned price buckets given the ordered amount of units.

+---------------+----------------------+
|     Units     | Delivery Cost / Unit |
+---------------+----------------------+
| Less than 500 |                   50 |
| 500 - 1000    |                   40 |
| 1000 - 1500   |                   35 |
| 1500 - 2000   |                   30 |
| 2000 - 2500   |                   25 |
| 2500 - 3000   |                   20 |
| 3000 above    |                   15 |
+---------------+----------------------+

There is fixed 30$ storage cost to maintain inventory in warehouse. Time for delivery of stock from order point is 5 days. So we need to order 5 days prior if a certain quantity is required today.

I am looking for an algorithm that would help minimize the total cost while meeting the demand for each week. Also the order point should be provided.

Primarily, I am looking for readings or tutorial that would help understand how to solve such a problem. It would be great if someone can help with this particular example.

Answer 1

I won't write a python solution as i am not familiar with any python modeling language but i can describe the approach i took in the past to solve problems like this.

I would solve this problem using a technique of finite horizon optimal control where we have an $x_t$ a state vector for each time point, a control signal $u_t$, a prediction $p_t$.

The core of idea is that you define a transition function which defines $x_t = f_t(x_{t-1},x_{t-2}, ... ,x_{0},u_{t-1},u_{t-2}, ... ,u_{0}, p_t, p_{t+1}, ... , p_{t+h})$ depending on past state, past controls and future predictions of demand and those as constraint of your model the time interval your model runs over. Futhermore you define constraints involving the predictions $p_t$, $x_t$, $u_t$ that ensure you always have enough stock. You define your objective in terms of $x_t$ and $u_t$.

The idea is that on day $\tau$ you run a the model with updated predictions, the 5 past $u_{\tau-1}, u_{\tau-2}, u_{\tau-3}, u_{\tau-4}, u_{\tau-5}$ as you ordered and they will arrive, $x_{\tau-1}, x_{\tau-2}, x_{\tau-3}, x_{\tau-4}, x_{\tau-5}$ as the historic stock levels and $p_t, ..., p_{t+h}$ as predict the demand on these days, you leave $x_{\tau}, ... x_{\tau + h}$ as free variables, constraint by transition function and leave $u_\tau, ... , u_{\tau + h}$ as free variables where $h$ is the length of time horizon. The choice of $h$ is a trade off between model performance and amount of computation. After running the model you get an $u_\tau$ and you order that many. The next day you rerun the model with updated predictions and one day further.

In your case the transition constraint would look like $x_{t} = x_{t-1} + u_{t-5} -p_{t-1}$. The stock level of the next day is the stock level of the previous day plus what you order 5 days ago minus that you expect to sell. You in addition can pose a contraint how much buffer you want atleast for each $x_t$.

The objective would be a sum of a piece wise function applied to how much was order plus storage costs. There is a lot of research how you can express piecewise function as mixed integer problems but some of them are about continous functions which yours is not.

It think you could express this problem completly in the language of Mixed Integer Linear Programming. I hope this gave you an idea how to approach it.

Answer 2

I would approach this as a mixed integer linear programming (MILP) problem. There are a number of MILP solvers, some open source, some commercial (with some of the commercial solvers providing free licenses for educational use). Many of them either have a Python API or can be used with PuLP (mentioned in comment to the selected answer).

You might want to search the web for references to integer programming models that involve "all units discounts", which describes your delivery costs. All units discounts frequently occur in the context of order costs rather than shipping costs, but the method of modeling them would be easily transferable.

Answer 3

Depending on what your decision variables are you are looking at an integer, mixed integer, or mixed integer linear programming problem. It seems to me that you want to solve it using a Python model. If that is the case, you can look at Pyomo or PuLP systems if you're looking for open-source solvers.