25-09-2014 Decision making in general When a decision is made - - PDF document

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Linear programming

Anders Ringgaard Kristensen

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

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

Example: Ration formulation

A group of dairy cows is fed a ration consisting of x1 kg silage and x2 kg concentrates. The price of silage is p1 and the price of concentrates is p2. The ration must satisfy some nutritional “demands”:

• The energy content must be at least b1.
• The AAT1 value must be at least b2
• The PBV2 value must be at most b3
• The fill must be at most b4

For both feeds, i, xi ≥ 0 Determine x1 and x2 so that the cost of the ration is minimized.

1AAT = Amino Acids absorbed in the intestine 2PBV = Protein balance in the rumen

State of the cows

What are the states of the feeds?

Energy (SFU/kg) of silage and concentrates: a11 and a12, respectively. AAT (g/kg) of silage and concentrates: a21 and a22, respectively. PBV (g/kg) of silage and concentrates: a31 and a32, respectively. Fill (units/kg) of silage and concentrates: a41 and a42, respectively.

What are the states of the feeds?

Silage Concentrates Energy a11 a12 AAT a21 a22 PBV a31 a32 Fill a41 a42

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LP problem

p1x1 + p2x2 = Min! (minimize costs)

• subject to

a11x1 + a12x2 ≥ b1 (energy at least b1) a21 x1 + a22x2 ≥ b2 (AAT at least b2) a31 x1 + a32x2 ≤ b3 (energy at most b3) a41 x1 + a42x2 ≤ b4 (fill at most b4) x1 ≥ 0, x2 ≥ 0 Objective function Constraints

LP - problem in matrix notation

px = Min!

• subject to

Ax ≤ b where p = (p1 , p2), x = (x1, x2)’, b = (-b1, -b2, b3, b4)’, and

The linear relations: Two “activities”

The permitted area is bounded by a straight line in the diagram

a11x1 + a12x2 ≥ b1 ⇔ x2 ≥ b1/a12 – (a11/a12)x1 x1

b1/a12

x2

Slope: a11/a12

Graphical solution, convex area

Energy, ≥ AAT, ≥ PBV, ≤ Fill, ≤ The per-mitted area forms a convex set

x1 x2

The objective function

Combinations of x1 and x2 having same total cost form a straight line in the plane.

p1x1 + p2x2 = c (costs) For fixed cost c’ we have x2 = c’/p2 – (p1/p2)x1

Graphical solution, iso-cost line

x1 x2 Combinations of x1 and x2 having cost c’ c’/p2 Slope: p1/p2

We want to minimize cost c’

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Graphical solution, minimization

x1 x2

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 x1’, x2’

x2’ x1’

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 p1/p2 Illustration

Influence of prices, I

x1 x2

Original setting

x2’ x1’

Influence of prices, II

x1 x2

Slight increase of p1 Drastic change in terms of x1 and x2 but only sligtly in terms of costs c

x2’ x1’

Influence of prices, III

x1 x2

Large increase of p2 Drastic change in terms of x1 and x2 and to some extent c

x2’ x1’

Non-unique optimum

x1 x2

Increase of p2 All combinations of x1 and x2 along a border are optimal

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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 non-convex set A convex set

Modeling tricks

From maximization to minimization:

• p1x1 + p2x2 = Max! ⇔ -p1x1 - p2x2 = Min!

From greater than to less than:

• ai1x1 + ai2x2 ≥ bi ⇔ -ai1x1 - ai2x2 ≤ -bi

From “equal to” to “less than”

• ai1x1 + ai2x2 = bi ⇔
• -ai1x1 - ai2x2 ≤ -bi
• ai1x1 + ai2x2 ≤ bi

More than two variables

We cannot illustrate graphically anymore, but the same principles apply:

• Convex (multi-dimensional) set
• Optimum always in a corner

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

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.

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.

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McCull and his sheep & steers

McCull owns 200 ha of pasture where he can have ewes and/or steers. Gross margins are $24 per ewe and $60 per steer. Limited carrying capacity – refer to table

Season Carrying capacity Ewes Steers Spring 15 5 Summer 24 4 Autumn 15 Winter 8

McCull: LP-problem

Objective function: 24x1 + 60x2 = Max! Constraints: Spring grazing 1/15 x1 + 1/5 x2 ≤ 200 Summer grazing 1/24 x1 + 1/4 x2 ≤ 200 Sheep in autumn x1 ≤ 3000 (i.e 15 x 200) Sheep in winter x1 ≤ 1600 (i.e 8 x 200) x1 ≥ 0, x2 ≥ 0

McCull: Graphical solution

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 200 400 600 800 1000 Spring grazing Summer grazing Sheep in autumn Sheep in winter Iso-cost

Herd constraints Optimization Biological variation Uncertainty Functional limitations Dynamics

Linear programming

Properties of methods for decision support