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 Product Mix Problem and the Diet Problem. In the following article we will analyze a different classic application known as the 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
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
- Xt >= 0 It >= 0 For all t.
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