Quiz 1B : Optimization of the formulation of animal feed

Length : around 5 minutes

« An animal feed company produces concentrates for cattle using various types of ingredients : protein crops (peas), cereal (maize) and cattle cake (soya-bean).

The concentrates obtained must have a minimum guaranteed content of TNM (Total Nitrogenous Matter) and fat as well as maximum mineral matter and crude fibre contents

Minimum contents	TNM (g/kg)	Fat (g/kg)
Concentrate	200	25

Maximum contents	Mineral matter (g/kg)	Crude fibre (g/kg)
Concentrate	70	65

The content of each of these components is known for every ingredient.

Contents	TNM (g/kg)	Fat (g/kg)	Mineral Matter (g/kg)	Crude fibre (g/kg)
Maize	81	37	12	22
Peas	207	10	30	52
Soya-bean cakes	453	19	64	60

The cost of maize is 148€/T, that of peas 195€/T and soya-bean cakes 355€/T. »

Source : P. Haurez, Utilisation des protéagineux dans l’alimentation des bovins viande, synthèse réalisée par l’Institut de l’Elevage, 2002.

It is possible to develop a linear programming model which helps determine the optimal formulation of a kilo of concentrate so as to respect the guaranteed contents while minimizing production costs.

Consider the linear programme and answer the questions.

Let C be the cost of concentrates (euros)
M the amount of maize used in a kg of concentrate (in kg)
P the amount of peas used in a kg of concentrate (in kg)
T the amount of soya-bean cakes used in a kg of concentrate (in kg)

How can the sentence “the goal of the company is to minimize its production costs” be written in algebraic form ?

Max C = (148M+195P+355T)/1000

Min C = (P+M+S)/1000

Min C = (148M+195P+355T)/1000

The mineral equation says that the total mineral content per concentrate unit must at most be equal to the guaranteed content and is written as follows :

12M+30P+64T ≤ 70

12M+30P+64T ≥ 70

12M+30P+64T = 70

The TNM equation says that the TNM total content per concentrate unit must at least be equal to the guaranteed minimum content and is written as follows

81M+207P+453T ≥ 200

M+T+P = 200

200*(P+M+T) ≥ 81+207+453

The fat equation says that the fat total content per concentrate unit must at least be equal to the guaranteed minimum content and is written as follows :

25*(M+P+T) ≥ 37+10+19

P+M+T ≥ 25

37M+10P+19T ≥ 25

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« An animal feed company produces concentrates for cattle using various types of ingredients : protein crops (peas), cereal (maize) and cattle cake (soya-bean).

The concentrates obtained must have a minimum guaranteed content of TNM (Total Nitrogenous Matter) and fat as well as maximum mineral matter and crude fibre contents

Minimum contents	TNM (g/kg)	Fat (g/kg)
Concentrate	200	25

Maximum contents	Mineral matter (g/kg)	Crude fibre (g/kg)
Concentrate	70	65

The content of each of these components is known for every ingredient.

Contents	TNM (g/kg)	Fat (g/kg)	Mineral Matter (g/kg)	Crude fibre (g/kg)
Maize	81	37	12	22
Peas	207	10	30	52
Soya-bean cakes	453	19	64	60

The cost of maize is 148€/T, that of peas 195€/T and soya-bean cakes 355€/T. »

Source : P. Haurez, Utilisation des protéagineux dans l’alimentation des bovins viande, synthèse réalisée par l’Institut de l’Elevage, 2002.

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In mathematical notation, this problem can be written as follows :

Minimizer   C = 148M + 195P + 355T

with       M + P + T = 1
              81M + 207P + 453T ≥ 200               (TNM constraint)
              37M + 10P + 19T ≥ 25                   (fat constraint)
              12M + 30P + 64T ≤ 70                     (mineral constraint)
              22M + 52P + 60T ≤ 65                     (fibre constraint)

M ≥ 0 ; P ≥ 0 ;T ≥ 0

where         C           cost of concentrates (euros)
                 M           the quantity of maize which will be used in the concentrates
                 P           the quantity of peas which will be used in the concentrates
               T           the quantity of soya-bean cakes which will be used in the concentrates

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Question 1 of 6

1. Question
« An animal feed company produces concentrates for cattle using various types of ingredients : protein crops (peas), cereal (maize) and cattle cake (soya-bean).

The concentrates obtained must have a minimum guaranteed content of TNM (Total Nitrogenous Matter) and fat as well as maximum mineral matter and crude fibre contents

Minimum contents TNM (g/kg) Fat (g/kg)

Concentrate 200 25

Maximum contents Mineral matter (g/kg) Crude fibre (g/kg)

Concentrate 70 65

The content of each of these components is known for every ingredient.

Contents TNM (g/kg) Fat (g/kg) Mineral Matter (g/kg) Crude fibre (g/kg)

Maize 81 37 12 22

Peas 207 10 30 52

Soya-bean cakes 453 19 64 60

The cost of maize is 148€/T, that of peas 195€/T and soya-bean cakes 355€/T. »

Source : P. Haurez, Utilisation des protéagineux dans l’alimentation des bovins viande, synthèse réalisée par l’Institut de l’Elevage, 2002.

It is possible to develop a linear programming model which helps determine the optimal formulation of a kilo of concentrate so as to respect the guaranteed contents while minimizing production costs.

Let C be the cost of concentrates (euros)
       M the amount of maize used in a kg of concentrate (in kg)
       P the amount of peas used in a kg of concentrate (in kg)
       T the amount of soya-bean cakes used in a kg of concentrate (in kg)

How can the sentence “the goal of the company is to minimize its production costs” be written in algebraic form ?
- Max C = (148M+195P+355T)/1000
- Min C = (P+M+S)/1000
- Min C = (148M+195P+355T)/1000
Correct

Knowing that “the cost of maize is 148€/T, that of peas is 195€/T and that of soya bean cakes is 355€/T”, if C represents the cost per kilo, M the quantity of maize used in the concentrates, P the quantity of peas used in the concentrates and T the quantity of cakes used in the concentrates, then C = (148M+195P+355T)/1000. Furthermore, considering that our objective is to MINIMIZE the cost, the equation is written as follows : Min C = (148M+195P+355T)/1000.

Incorrect

Knowing that “the cost of maize is 148€/T, that of peas is 195€/T and that of soya bean cakes is 355€/T”, if C represents the cost per kilo, M the quantity of maize used in the concentrates, P the quantity of peas used in the concentrates and T the quantity of cakes used in the concentrates, then C = (148M+195P+355T)/1000. Furthermore, considering that our objective is to MINIMIZE the cost, the equation is written as follows : Min C = (148M+195P+355T)/1000.
Question 2 of 6

2. Question
The mineral equation says that the total mineral content per concentrate unit must at most be equal to the guaranteed content and is written as follows :
- 12M+30P+64T ≥ 70
- 12M+30P+64T = 70
- 12M+30P+64T ≤ 70
Correct

Mineral contents must not be too high. They must be LOWER THAN OR EQUAL TO the maximum guaranteed contents. As a consequence : 12M+30P+64T ≤ 70.

Incorrect

Mineral contents must not be too high. They must be LOWER THAN OR EQUAL TO the maximum guaranteed contents. As a consequence : 12M+30P+64T ≤ 70.
Question 3 of 6

3. Question
The TNM equation says that the TNM total content per concentrate unit must at least be equal to the guaranteed minimum content and is written as follows :
- 200*(P+M+T) ≥ 81+207+453
- 81M+207P+453T ≥ 200
- M+T+P = 200
Correct

In order to obtain the total contents, the quantities of raw materials must be multiplied by their contents. That is to say, 81*M+207*P+453*S. These total contents must be higher than or equal to the minimum guaranteed contents.

Incorrect

In order to obtain the total contents, the quantities of raw materials must be multiplied by their contents. That is to say, 81*M+207*P+453*S. These total contents must be higher than or equal to the minimum guaranteed contents.
Question 4 of 6

4. Question
The fat equation says that the fat total content per concentrate unit must at least be equal to the guaranteed minimum content and is written as follows :
- P+M+T ≥ 25
- 25*(M+P+T) ≥ 37+10+19
- 37M+10P+19T ≥ 25
Correct

The quantity of each raw material must be multiplied by its fat contents. These total contents must be higher than or equal to the minimum guaranteed contents.

Incorrect

The quantity of each raw material must be multiplied by its fat contents. These total contents must be higher than or equal to the minimum guaranteed contents.
Question 5 of 6

5. Question
The fibre equation says that the fibre total content per concentrate unit must be less than or equal to the guaranteed maximum content and is written as follows :
- 22M+52P+60T ≥ 65
- P+M+T = 65
- 22M+52P+60T ≤ 65
Correct

The quantity of each raw material must be multiplied by its fibre contents. These total contents must be lower than or equal to the minimum guaranteed contents.

Incorrect

The quantity of each raw material must be multiplied by its fibre contents. These total contents must be lower than or equal to the minimum guaranteed contents.
Question 6 of 6

6. Question
How can “the total quantity of ingredients must amount to a kilo” be written ?
- P = 1 ; M = 1 ; S = 1
- P+M+S = 1
Correct

The TOTAL quantity must be equal to a kilo ! If M is the quantity of maize used in the concentrates, P the quanitty of peas used in the concentrates and T the quantity of cakes used in the concentrates, then M+P+T = 1

Incorrect

The TOTAL quantity must be equal to a kilo ! If M is the quantity of maize used in the concentrates, P the quanitty of peas used in the concentrates and T the quantity of cakes used in the concentrates, then M+P+T = 1

Length : around 5 minutes

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