How to build a deterministic optimization model for the following fc

I am not sure how to model this. I have like this table that has dates and orders



dates order

1/16/2021 12 units

1/21/201 13 unit

1/27/2021 21

1/29/2021 14

2/23/2021 15

2/24/2022 27

2/25/2022 39

I am trying to create an optimization model. Basically I have a FC which fulfill customer order. We have orders listed in the above following dates The orders are basically on the column orders. We have a starting inventory of 50.

Basically the FC orders from a DC with unlimited inventory to satisfy the order. We have a lead time of 3 days between shipping the order and fulfilling it.

We have the decision variable $$x_1$$,$$x_2$$,...,$$x_{365}$$ this is how much the FC orders from the distribution center. Where 1=January 1 and 365=Dec 31 2021 Our time frame is 1 year.

But I am not sure how to model this. I know the time between ordering something and getting it to the FC is 3 days. We have order cost of 50 dollar per unit. We have holding cost of 5 dollar per unit We have lost sales cost of 30 dollar per unit. We have min order quantity of 25 and batch size of multiple of 5.

We have our FC inventory depletes as we fulfill the orders. We have the FC can hold at most 100 units. The FC will run out of inventory at some time and have to order from the DC

But I am not sure how to model this as linear optomization problem.

I think I would have 365 decision variables.

We want to minimize all the different cost and know on which days of the year to place orders.

For each day, you will want to have variables representing how many units you ship to customers, how many units of sales you lose, how much inventory you have at the end of the day, and how many five you batches you order from the DC. Most of those variables can be continuous (they will naturally take integer values), but the DC order size variables will need to be integers with domain 0, 5, 6, ... (so that you cannot order 1 to 4 five-unit batches in any day). The rest of the model should fall into place given those variables.

• I am wondering I think if $I_1$ is the amount of inventory I have at the end of day 1 and have $I_1=50-x_1$ where $x_1$ is the amount customer orders for day 1. How could I model the lost sales? Commented Nov 13, 2022 at 19:16
• Lost sales is demand minus shipments. Closing inventory is opening inventory (inventory from the day before) plus arriving orders from the DC minus shipments.
– prubin
Commented Nov 13, 2022 at 19:30
• So would demand be a parameter because I know the demand from my spread sheet. So its not a variable. Commented Nov 13, 2022 at 19:36
• Correct, demand would be a parameter.
– prubin
Commented Nov 13, 2022 at 20:26
• Not if you allow lost sales.
– prubin
Commented Nov 13, 2022 at 21:42