A Little Problem James has to buy drinks for himself and 4 friends. He has £2.50 to spend. A can of cola costs 60p A can of orange costs 40p He can buy x cans of Cola and y cans of Orange; Explain why (i) x + y  5 (ii) 6x + 4y  25 Write down all the different combinations of each type of drink he can buy. 1 coke + 4 Oranges 6 Oranges 2 coke + 3 Oranges 5 Oranges

What % of each should they buy? A “Real-Life” problem Price Farm Chickens in Hereford supply Sun Valley Foods with live chickens. By law the chickens must be reared correctly according to strict dietry needs. Obviously, the firm must keep costs as low as possible but must obey the law. The law states that the chicks need the following: No less than 20% protein No less than 10 % fat Exactly 50% Carbohydrate No greater than 50% Fibre and Ash The farm can buy the following ingredients with the following contents and cost; Soya flour 50% prot, 20% fat, 20% carb, 10% F&A = £60 per ton Meat Meal 50% prot, 10% fat, 0% carb, 40% F&A = £50 per ton Enriched skimmed Milk 30% prot, 20% fat, 40% carb, 10% F&A = £70 per ton Sago Flour 100% carb = £40 per ton What % of each should they buy?

Linear Programming Modelling No! ……… Not that kind of modelling… Mathematical Modelling!!! A model is a way of building a simulation of a real life situation which we want to improve.  R.Jones 07/05/07

Optimisation Examples Linear Programming Optimisation Examples Battle Strategies Blending Ingredients Profit and Loss Management of people or products  R.Jones 07/05/07

We love Chicken Nuggets The Problem We love Chicken Nuggets Our company supplies chicken nuggets to Superbites. We need to ensure the chicks are fattened up before the kill, it must be more than 10%. We can feed them soya flour and powered milk. Soya flour has 50% protein, 20% fat, 20% carb, 10% F&A. Powdered milk has 30% protein, 10% fat, 0% carb, 40% F&A. The chicks must have more than 50% protein, 10% fat, exactly 50% Carbohydrate and no more than 50% F&A to be healthy. They must eat a blend of milk and soya our they will be sick. If Soya costs £60/ton and Milk costs £20/ton what percentage blend should we buy to spend the least?  R.Jones 07/05/07

The A, B , Cs of Optimisation Linear Programming The A, B , Cs of Optimisation A ADJUSTABLES (Things we can change) B BEST (What we are trying to improve) C CONSTRAINTS (The rules)  R.Jones 07/05/07

The A, B , Cs of Optimisation Linear Programming The A, B , Cs of Optimisation Adjustables : X = tonnes of Soya Flour y = tons of Powered Milk per day Best : Min Cost = 60 x + 20y Constraints : Protein 0.5x + 0.3y  0.5  5x + 3y  5 Fat 0.2x + 0.1y  0.1  2x + y  1 x  0 y  0 Carbohydrate 0.2x = 0.5  x = 2.5 Fibre & Ash 0.1x + 0.4y  0.5  x + 4y  5  R.Jones 07/05/07

Answer here (but decimal!!) Better here (why) Plot Objective Function line 0.5x+y = £50