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Modelling OR Practical Experience Jacques A. Ferland DIRO, Université de Montréal, Canada ferland@iro.umontreal.ca http://www.iro.umontreal.ca/~ferland/ X OPTIMA VI RED-M

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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

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

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Exemple 1: diet problem 3 types de 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 requirement of the diet

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Problem data 3 types de 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 requirement of the diet quantité g1 g2 g3 hebd. ENA 2 3 7 1250 ENB 1 1 0 250 ENC 5 3 0 900 END 0.6 0.25 1 232.5 $/kg 41 35 96

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Problem data 3 types de 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 requirement of the diet weekly g1 g2 g3 quantity ENA 2 3 7 1250 ENB 1 1 0 250 ENC 5 3 0 900 END 0.6 0.25 1 232.5 $/kg 41 35 96

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Problem variables 3 types de 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 requirement of the diet i) Activities or actions of the model Actions variables # kg de g1 x 1 # kg de g2 x 2 # kg de g3 x 3

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Objective function and constraints ii) Objective function Weekly cost of the diet = 41x 1 + 35x 2 + 96x 3 to minimise iii) Contraints ENA: 2x 1 + 3x 2 +7x 3 ≥ 1250 ENB: 1x 1 + 1x 2 ≥ 250 ENC: 5x 1 + 3x 2 ≥ 900 END: 0.6x 1 + 0.25x 2 + x 3 ≥ 232.5 Non negativity constraints: x 1 ≥ 0, x 2 ≥ 0, x 3 ≥ 0 weekly g1 g2 g3 quantity ENA 2 3 7 1250 ENB 1 1 0 250 ENC 5 3 0 900 END 0.6 0.25 1 232.5 $/kg 41 35 96

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Mathematical model ii) Objective function Weekly cost of the diet = 41x 1 + 35x 2 + 96x 3 to minimize iii) Contraints ENA: 2x 1 + 3x 2 +7x 3 ≥ 1250 ENB: 1x 1 + 1x 2 ≥ 250 ENC: 5x 1 + 3x 2 ≥ 900 END: 0.6x 1 + 0.25x 2 + x 3 ≥ 232.5 Non negativity constraints: x 1 ≥ 0, x 2 ≥ 0, x 3 ≥ 0 min z = 41x 1 + 35x 2 + 96x 3 s.t. 2x 1 + 3x 2 +7x 3 ≥ 1250 1x 1 + 1x 2 ≥ 250 5x 1 + 3x 2 ≥ 900 0.6x 1 + 0.25x 2 + x 3 ≥ 232.5 x 1 ≥ 0, x 2 ≥ 0, x 3 ≥ 0

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Linear Programming An Example. Problem The dairy "Fior di Latte" produces two types of cheese: cheese A and B. The dairy company must decide how many.

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