🕋 Eid Mubarak, Selamat Hari Raya Haji ☪️

I have written about Linear Programming – Allocation model in my previous post Linear Programming and Linear Programming with LibreOffice Calc Solver. This post would talk about Linear Programming – Covering models.

First question would be – what’s the difference between Allocation model and Covering model. There is no difference in the optimization function. The difference exists in the constraints. All our constraints talk about maximum in allocation model. All those constraints had ≤ symbol. Covering models talk about minimization, usually cost.

### Example

I’ll use the data set given in https://paginas.fe.up.pt/~mac/ensino/docs/OR/otherDocs/PowellAllocationCoveringBlendingConstraints.pdf

Dahlby Outfitters wishes to introduce packaged trail mix as a new product. The ingredients for the trail mix are seeds, raisins, flakes, and two kinds of nuts. Each ingredient contains certain amounts of vitamins, minerals, protein, and calories.

The marketing department has specified that the product be designed so that a certain minimum nutritional profile is met. The decision problem is to determine the optimal product composition—that is, to minimize the product cost by choosing the amount for each of the ingredients in the mix. The data shown below summarize the parameters of the problem:

Component |
Grams per pound |
Nutritional Requirement |
||||

Seeds |
Raisins |
Flakes |
Pecans |
Walnuts |
||

Vitamins | 10 | 20 | 10 | 30 | 20 | 20 |

Minerals | 5 | 7 | 4 | 9 | 2 | 10 |

Protein | 1 | 4 | 10 | 2 | 1 | 15 |

Calories | 500 | 450 | 160 | 300 | 500 | 600 |

Cost/pound | 4 | 5 | 3 | 7 | 6 |

Lets denote the product names as S, R, F, P and W. Our objective function would be like this.

Total Cost = 4S+5R+3F+7P+6W

Rewriting the above statement as –

Zmin = 4S+5R+3F+7P+6W

subject to constraints –

Vitamin content |
10S + 20R + 10F + 30P + 20W | greater than or eq | 20 |

Mineral content |
5S + 7R + 4F + 9P + 2W | greater than or eq | 10 |

Protein content |
1S + 4R + 10F + 2P + 1W | greater than or eq | 15 |

Calorie content |
500S + 450R + 160F + 300P + 500W | greater than or eq | 600 |

Rewriting the above constraints as linear equations as given below,

10S + 20R + 10F + 30P + 20W ≥ 20

5S + 7R + 4F + 9P + 2W ≥ 10

1S + 4R + 10F + 2P + 1W ≥ 15

500S + 450R + 160F + 300P + 500W ≥ 600

### Spreadsheet method (LibreOffice Calc)

Prepare the data set. G9 is highlighted in yellow colour. This would be our minimizing figure.

The data have given the cost of each product already. So, our aim is to find how much amount of each product shall be produced. This would be the decision variable. We need to find out. The cells of the decision variables are also highlighted in yellow colour.

Let’s write the constraints now. Our aim is to find how much vitamin, mineral etc to be added in our product. Those cells are highlighted in yellow colour.

Let’s open the Solver now. Following is my selection.

- Target cell is where we find the minimum cost.
- As we are talking about minimum, we choose ‘optimize result to’ as ‘Minimum’
- By changing cells = Decision variable cells
- Limiting Constraints are highlighted with => operator.

Following is the result.

The answer I get in Calc is not equal to what I see in the reference PDF. However, let’s take it as the decision at the moment –

We would take 24.6, 10, 15, 600 for vitamins, minerals, protein and calories.

Linear programming suggests us to avoid pecans and walnuts.

0.5 x seeds, 0.3 x Raisins and 1.3 x Flakes are sufficient.

With this, we would be able to provide 24.6 vitamins, 10 minerals, 15 protein and 600 calories.

With this I’m closing the statistics post. I’d be starting the next part of this series soon, which is R programming.