1. 1. Introduction

2. Introduction Modeling
Use a set of mathematical relations to represent mathematically a real life situation
Compromise between a closer image of the reality and the difficulty of solving the model

3. Mathematical model Three items to identify
The set of actions (activities) of the decision maker (variables)
The objective of the problem specified in terms of a mathematical fonction (objective fonction)
The context of the problem specified in terms of mathematical relations (contraint functions)

4. Solving the problem Three items to identify
The set of actions (activities) of the decision maker (variables)
The objective of the problem specified in terms of a mathematical fonction (objective fonction)
The context of the problem specified in terms of mathematical relations (contraint functions)
Solving method
Use a procedure (algorithm) to determine
the values of the variables indicating how the different activities are used
to optimise the objective fonction (to reach the objective)
and satisfying the contraints

5. Two specific properties
Linear model
Two specific properties
Additivity of the values of the variables: the total effect of any set of actions (variables) is equal to the sum of the individual effect of each action (variable) in the set.
There is no cross action of the variables
The variables are always non negative

6. Exemple 1: diet problem
3 types of grains are available to feed an herb: g1, g2, g3
Each kg of grain includes 4 nutritional elements: ENA, ENB, ENC, END
The weekly quantity required for each nutritional element is specified
The price per kg of each grain is also specified.
Problem: Determine the quantity (in kg) of each grain to specify the minimum cost diet for the herb satisfying the nutritional requirements of the diet

7. Problem data
3 types of grains are available to feed the herb: g1, g2, g3
Each kg of grain includes 4 nutritional elements: ENA, ENB, ENC, END
The weekly quantity required for each nutritional element is specified
The price per kg of each grain is also specified.
Problem: Determine the quantity (in kg) of each grain to specify the minimum cost diet for the herb satisfying the nutritional requirements of the diet
quantité
g g g hebd.
ENA
ENB
ENC
END
$/kg

8. Problem data
3 types of grains are available to feed the herb: g1, g2, g3
Each kg of grain includes 4 nutritional elements: ENA, ENB, ENC, END
The weekly quantity required for each nutritional element is specified
The price per kg of each grain is also specified.
Problem: Determine the quantity (in kg) of each grain to specify the minimum cost diet for the herb satisfying the nutritional requirements of the diet
weekly
g g g quantity
ENA
ENB
ENC
END
$/kg

9. Problem variables
3 types of grains are available to feed the herb: g1, g2, g3
Each kg of grain includes 4 nutritional elements: ENA, ENB, ENC, END
The weekly quantity required for each nutritional element is specified
The price per kg of each grain is also specified.
Problem: Determine the quantity (in kg) of each grain to specify the minimum cost diet for the herb satisfying the nutritional requirements of the diet
i) Activities or actions of the model
Actions variables
# kg de g x1
# kg de g x2
# kg de g x3

10. Objective function and constraints
ii) Objective function
Weekly cost of the diet =
41x1 + 35x2 + 96x3
to minimise
iii) Contraints
ENA: x x x3 ≥ 1250
ENB: 1x x ≥ 250
ENC: 5x x ≥ 900
END: x x2 + x3 ≥ 232.5
Non negativity constraints:
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
weekly
g g g quantity
ENA
ENB
ENC
END
$/kg

11. Mathematical model ii) Objective function Weekly cost of the diet =
41x1 + 35x2 + 96x3
to minimize
iii) Contraints
ENA: x x x3 ≥ 1250
ENB: 1x x ≥ 250
ENC: 5x x ≥ 900
END: x x2 + x3 ≥ 232.5
Non negativity constraints:
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
min z = 41x1 + 35x2 + 96x3
s.t.
2x x x3 ≥ 1250
1x x ≥ 250
5x x ≥ 900
0.6x x x3 ≥ 232.5
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

12. Exemple 2: Restaurant owner problem
Seafoods available:
30 sea-urchins
24 shrimps
18 oysters
Two types of seafood plates to be offered:
$8 : including 5 sea-urchins, 2 shrimps et 1 oyster
$6 : including 3 sea-urchins, 3 shrimps et 3 oysters
Problem: determine the number of each type of plates to be offered by the
owner in order to maximize his revenue according to the
seafoods available

13. Problem variables i) Activities or actions Actions variables
Seafoods available:
30 sea-urchins
24 shrimps
18 oysters
Two types of seafood plates to be offered:
$8 : including 5 sea-urchins, 2 shrimps et 1 oyster
$6 : including 3 sea-urchins, 3 shrimps et 3 oysters
Problem: determine the number of each type of plates to be offered by the owner in order to maximize his revenue according to the seafoods
available
i) Activities or actions
Actions variables
# plates $ x
# plates $ y

14. Objective function and contraints
Seafoods available:
30 sea-urchins
24 shrimps
18 oysters
Two types of seafood plates to be offered:
$8 : including 5 sea-urchins, 2 shrimps et 1 oyster
$6 : including 3 sea-urchins, 3 shrimps et 3 oysters
Problem: determine the number of each type of plates to be offered by the owner in order to maximize his revenue according to the seafoods
available
i) Activities or actions
ii) Objective function
owner’s revenue =
8x + 6y
to maximise
iii) Contraints
sea-urchins: 5x + 3y ≤ 30
shrimbs: 2x + 3y ≤ 24
oysters: 1x + 3y ≤ 18
Non negative variables:
x,y ≥ 0

15. Mathematical model max 8x + 6y s.t. 5x + 3y ≤ 30 2x + 3y ≤ 24
i) Activities or actions
ii) Objective function
owner’s revenue =
8x + 6y
to maximise
iii) Contraints
sea-urchins: 5x + 3y ≤ 30
shrimbs: 2x + 3y ≤ 24
oysters: 1x + 3y ≤ 18
Non negative variables:
x,y ≥ 0

16. Exemple 3: Knapsack problem

17. Exemple 4: Assignment problem

18. Exemple 4: Assignment problem

20. Exemple 5: Multicommodity Distribution System Design

21. Multicommodity Distribution System Design