Linear Programming allows us to tackle various real life problems, some of which we have already gone over in previous articles, such as the ** Transportation Problem**, the

**and the**

__Product Mix Problem__**. In the following article we will analyze a different classic application known as the**

__Diet Problem__**Production Inventory Problem**.

This problem is essentially used for establishing a policy for production within a time period that satisfies certain demand requirements, respecting the manufacturing limits and at a minimal cost. These types of models can be extended to use with several products, however, for this example we will only consider one single product:

We define the following linear optimization model: (Given: there is an initial inventory of 50 units => I0=50).

**1. Decision Variables:**

Xt: Units to produce in month t (t=1,…,6 with t=1 =>January; t=6=>June)

It: Units to be stored as inventory until the end of month t (t=1,…,6 with t=1 =>January; t=6=>June)

**2. Objective Function: **Minimize the production and inventory costs during the planning period, defined as: 60X1 + 60X2 + 55X3 + 55X4 + 50X5 + 50X6 + 15I1 + 15I2 + 20I3 + 20I4 + 20I5 + 20I6

**3. Constraints:**

Satisfy the demand requirements (known as the inventory balance constraint):

- X1 + 50 – I1 = 100 (January)
- X2 + I1 – I2 = 130 (February)
- X3 + I2 – I3 = 160 (March)
- X4 + I3 – I4 = 160 (April)
- X5 + I4 – I5 = 140 (May)
- X6 + I5 – I6 = 140 (June )

Respect the monthly production maximum capacity:

- X1 <= 120 X2 <= 120 X3 <= 150 X4 <= 150 X5 <= 150 X6 <= 150

Non-negativity conditions:

- Xt >= 0 It >= 0 For all t.

The following tutorial demonstrates how to implement this **Linear Programming** model using **Solver** in Excel:

The optimal solution is shown below with an optimal value of **$43,450**. One can see that a total of 780 units are produced between January and June, which together with the initial inventory of 50 units satisfies the monthly demand requirements.

*Would you like to receive the Excel file with Solver’s solution to this problem? **get free and immediate access to the file now: *Production and Inventory

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