Linear programming Anders Ringgaard Kristensen Department of - PowerPoint PPT Presentation

Department of Veterinary and Animal Sciences Linear programming Anders Ringgaard Kristensen

Department of Veterinary and Animal Sciences Decision making in general When a decision is made concerning a unit, the following information is necessary: • The present state of the unit • The relation between factors and production • Immediate production • Future production • The farmer’s personal preferences • All constraints of legal, economic, physical or personal kind Slide 2

Department of Veterinary and Animal Sciences Linear programming Knowledge representation: • State of system: • Hidden in model formulation – often as constraints or as parameters • Factor/product relation: • Immediate production: • Linear function • Future production: • Static method • Farmer preferences: • Linear utility function • Constraints: • Linear constraints Slide 3

Department of Veterinary and Animal Sciences Example: Ration formulation A group of dairy cows is fed a ration consisting of x 1 kg silage and x 2 kg concentrates. The price of silage is p 1 and the price of concentrates is p 2 . The ration must satisfy some nutritional “demands”: • The energy content must be at least b 1 . The AAT 1 value must be at least b 2 • State of the cows The PBV 2 value must be at most b 3 • • The fill must be at most b 4 For both feeds, i , x i ≥ 0 Determine x 1 and x 2 so that the cost of the ration is minimized. 1 AAT = Amino Acids absorbed in the intestine 2 PBV = Protein balance in the rumen Slide 4

Department of Veterinary and Animal Sciences What are the states of the feeds? Energy (SFU/kg) of silage and concentrates: a 11 and a 12 , respectively. AAT (g/kg) of silage and concentrates: a 21 and a 22 , respectively. PBV (g/kg) of silage and concentrates: a 31 and a 32 , respectively. Fill (units/kg) of silage and concentrates: a 41 and a 42 , respectively. Slide 5

What are the states of the feeds? Silage Concentrates Energy a 11 a 12 AAT a 21 a 22 PBV a 31 a 32 Fill a 41 a 42

Department of Veterinary and Animal Sciences LP problem Objective function p 1 x 1 + p 2 x 2 = Min! (minimize costs) - subject to a 11 x 1 + a 12 x 2 ≥ b 1 (energy at least b 1 ) a 21 x 1 + a 22 x 2 ≥ b 2 (AAT at least b 2 ) a 31 x 1 + a 32 x 2 ≤ b 3 (PBV at most b 3 ) a 41 x 1 + a 42 x 2 ≤ b 4 (fill at most b 4 ) x 1 ≥ 0, x 2 ≥ 0 Constraints Slide 7

Department of Veterinary and Animal Sciences LP - problem in matrix notation px = Min! - subject to Ax ≤ b where p = ( p 1 , p 2 ), x = ( x 1 , x 2 )’, b = (- b 1 , - b 2 , b 3 , b 4 )’, and Slide 8

Department of Veterinary and Animal Sciences The linear relations: Two “activities” a 11 x 1 + a 12 x 2 ≥ b 1 ⇔ x 2 x 2 ≥ b 1 / a 12 – ( a 11 / a 12 ) x 1 b 1 / a 12 The permitted area is bounded by a straight line in the diagram x 1 Slope: a 11 / a 12 Slide 9

Graphical solution, convex area x 2 Energy, ≥ AAT, ≥ PBV, ≤ Fill, ≤ The per-mitted area forms a convex set x 1

Department of Veterinary and Animal Sciences The objective function p 1 x 1 + p 2 x 2 = c (costs) For fixed cost c’ we have x 2 = c’ / p 2 – ( p 1 / p 2 ) x 1 Combinations of x 1 and x 2 having same total cost form a straight line in the plane. Slide 11

Graphical solution, iso-cost line x 2 c’ / p 2 Combinations of x 1 and x 2 having cost c’ Slope: p 1 / p 2 We want to minimize cost c’ x 1

Graphical solution, minimization x 2 We want to minimize cost c’ Decrease c’ until the permitted area is reached. Decrease c’ until the lowest value of the permitted area is found. Optimal combination is x 1 ’, x 2 ’ x 2 ’ x 1 x 1 ’

Department of Veterinary and Animal Sciences What determines the optimum For a given set of constraints, the optimum is determined solely by the price relations. In our two-dimensional example it is simply the price ratio p 1 / p 2 Illustration Slide 14

Influence of prices, I x 2 Original setting x 2 ’ x 1 x 1 ’

Influence of prices, II x 2 Slight increase of p 1 Drastic change in terms of x 1 and x 2 but only sligtly in terms of costs c x 2 ’ x 1 x 1 ’

Influence of prices, III x 2 Large increase of p 2 Drastic change in terms of x 1 and x 2 and to some extent c x 2 ’ x 1 x 1 ’

Non-unique optimum x 2 Increase of p 2 All combinations of x 1 and x 2 along a border are optimal x 1

Department of Veterinary and Animal Sciences Properties of linear programming The permitted area is always a convex set The optimal solution is: • Either uniquely located in a corner • Or along a border line (accordingly also in two corners) It is therefore sufficient to search the corners of the convex set. A convex set A non-convex set Slide 19

Department of Veterinary and Animal Sciences Modeling tricks From maximization to minimization: • p 1 x 1 + p 2 x 2 = Max! ⇔ - p 1 x 1 - p 2 x 2 = Min! From greater than to less than: • a i 1 x 1 + a i 2 x 2 ≥ b i ⇔ - a i 1 x 1 - a i 2 x 2 ≤ - b i From “equal to” to “less than” • a i 1 x 1 + a i 2 x 2 = b i ⇔ • - a i 1 x 1 - a i 2 x 2 ≤ - b i • a i 1 x 1 + a i 2 x 2 ≤ b i Slide 20

Department of Veterinary and Animal Sciences More than two variables We cannot illustrate graphically anymore, but the same principles apply: • Convex (multi-dimensional) set • Optimum always in a corner Slide 21

Department of Veterinary and Animal Sciences The simplex algorithm The most commonly applied optimization algorithm for linear programming. The simplex algorithm: • Identifies a corner of the convex set forming the permitted area. • Searches along the edges to find a better corner. • Checks whether the present corner is optimal Implemented in modern standard spreadsheets Implemented in ration formulation programs Slide 22

Department of Veterinary and Animal Sciences Shadow prices What is the economic benefit of relaxing a constraint by one unit? • If it is not limiting it is zero! • Trivial method: • Change the constraint by one • Optimize again • How much did the result improve? • Software systems typically provides these values by default. Slide 23

Department of Veterinary and Animal Sciences Basic assumptions of LP Proportionality • No start-up costs • No decreasing returns to scale Additivity • No interactions between activities Divisibility • Any proportion of an activity allowed Certainty • All parameters (coefficients, prices, constraints) are known precisely. Slide 24

McCull and his sheep & steers McCull owns 200 ha of pasture where he can have ewes and/or Carrying capacity steers. Gross margins are $24 per ewe and Season Ewes Steers $60 per steer. Limited carrying capacity – refer to Spring 15 5 table Summer 24 4 Autumn 15 Winter 8

McCull: LP-problem Objective function: 24 x 1 + 60 x 2 = Max! Constraints: 1/15 x 1 + 1/5 x 2 ≤ 200 Spring grazing 1/24 x 1 + 1/4 x 2 ≤ 200 Summer grazing x 1 ≤ 3000 (i.e 15 x 200) Sheep in autumn x 1 ≤ 1600 (i.e 8 x 200) Sheep in winter x 1 ≥ 0, x 2 ≥ 0

Department of Veterinary and Animal Sciences McCull: Graphical solution 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 0 200 400 600 800 1000 Spring grazing Summer grazing Sheep in autumn Sheep in winter Iso-cost Slide 27

Department of Veterinary and Animal Sciences Properties of methods for decision support Herd constraints Optimization Linear programming Biological Functional variation limitations Slide 28 Uncertainty Dynamics