# Food Manufacture I

### Food Manufacture I Example

Both this model and Food Manufacture II are examples of a blending problem. In blending optimization problems, multiple raw materials are combined in a way the meets the stated constraints for the lowest cost. These problems are common in numerous industries including, for example, the oil industry (blending different types of crude oil at a refinery) and agriculture (manufacturing feed that meets the different nutritional requirements of different animals). In this particular example, we’ll model and solve a production planning problem where we must create a final product from a number of ingredients, each of which has different costs, restrictions and features. The aim is to create an optimal multi-period production plan that maximizes profit. More details can be found on the Problem Description and Model Formulation tabs below. In Food Manufacture II, we’ll extend this example with additional constraints that change the problem type from a linear program (LP) to a mixed integer program (MIP), which makes it harder to solve. Note: you can download the model, implemented in Python, here. More information on this type of model can be found in the fifth edition of Model Building in Mathematical Programming, by H. Paul Williams.

Reference: H. Paul Williams, Model Building in Mathematical Programming, fifth edition (Page 253-254, 296-298, 349-350)

## Problem Description

A manufacturer needs to refine several raw oils and blend them together to produce a given food product that can be sold. The raw oils needed can be divided into two categories:

Category Oil
Vegetable oils: VEG 1
VEG 2
Non-vegetable oils: OIL 1
OIL2
OIL 3

The manufacturer can choose to buy raw oils for the current month and/or buy them on the futures market for delivery in a subsequent month. Prices for immediate deliver and in the futures market are given below in $/ton: Month VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 January 110 120 130 110 115 February 130 130 110 90 115 March 110 140 130 100 95 April 120 110 120 120 125 May 100 120 150 110 105 June 90 100 140 80 135 There are a number of additional factors that must be taken into account. These include: 1. The final food product sells for$150 per ton.
2. Each category of oil (vegetable and non-vegetable) needs to be refined on a different production line.
3. There is limited refinement capacity such that in any given month a maximum of 200 tons of vegetable oils and 250 tons of non-vegetable oils can be refined.
4. Also, there is no waste in the refinement process, so the sum of the raw oils refined will equal the amount of refined oils available.
5. The cost of refining the oils may be ignored.