# Introduction to Linear Programming (LP) Mathematical Programming (MP) Concept

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1 Introduction to Linear Programming (LP) Mathematical Programming Concept LP Concept Standard Form Assumptions Consequences of Assumptions Solution Approach Solution Methods Typical Formulations Massachusetts Institute of Technology Intro. to Linear Programming Slide 1 of 20 Mathematical Programming (MP) Concept Definition MP includes a range of powerful computer-based optimization methods Approach to Optimization MP methods exploit peculiar features of the structure of a problem to get solutions efficiently Different MP methods exploit different structures or features Understanding which features apply to a particular problem -- and thus which method can apply -- is important Massachusetts Institute of Technology Intro. to Linear Programming Slide 2 of 20 1

2 MP: Two important Categories Those Valid for Convex Feasible Regions Linear Programming, etc These use a local search routine to reach global optimum Keep going up/down until reach top/bottom Very efficient -- but not valid for non-convex regions Those Valid for Non-Convex Feasible Regions Dynamic programming, etc These "enumerate" solutions to discover optimum They prune, that is, eliminate, possible solutions because these can be shown to be dominated Computations limit applicability to special situations, but good for options analysis Massachusetts Institute of Technology Intro. to Linear Programming Slide 3 of 20 LP Overview Special form of mathematical programming Equations must be linear Uses simple solution procedures Linear algebra Very powerful Extremely large problems 100,000 variables 1000's of constraints Note difference! Useful design information by Sensitivity Analysis Answers to "what if" questions Massachusetts Institute of Technology Intro. to Linear Programming Slide 4 of 20 2

3 Standard Form of LP - Three Parts Objective function (OF) maximize or minimize Y = Σ c i x i Y = c 1 x 1 + c 2 x c n x n X i known as decision variables Constraints subject to: a 11 x 1 + a 12 x < = > b 1 a 21 x 1 + a 22 x < = > b 2 a 31 x 1 + a 32 x < = > b 3 Non-Negativity x i > 0 for all i Massachusetts Institute of Technology Intro. to Linear Programming Slide 5 of 20 Standard Form of LP -- Summary Optimize subject to: Y = c X A X (< or = or >) b X > 0 Massachusetts Institute of Technology Intro. to Linear Programming Slide 6 of 20 3

4 Three LP Assumptions (1) Linearity ; Additivity ; Non-Negativity Linearity of Objective Function and Constraints Essential Condition is: f(kx) = k f(x) for example: f(x) = 3 + 4X 1 + 2X 2 is NOT linear in the LP sense Implies Constant returns to scale (only first order terms) No "fixed charges" (no constants) Massachusetts Institute of Technology Intro. to Linear Programming Slide 7 of 20 Additivity: Three LP Assumptions (2) f(x 1, X 2,..., X n ) = f(x 1 ) + f(x 2 ) f(x n ) no interactive effects among X i terms assumes that individual segments of the problem operate as well independently as together Non-Negativity X i > 0 no fundamental difficulties except in particular situations Massachusetts Institute of Technology Intro. to Linear Programming Slide 8 of 20 4

5 Consequences of Assumptions Convexity of feasible region (if it exists!) Convex feasible region, with linear OF, implies: Optimum will be on an edge of the feasible region Since edges are also linear Optimum is at a corner point (can be several in special cases) Corner points are a small, finite set, defined by solution of linear equations Bottom Line: Assumptions imply the existence of an efficient solution strategy Massachusetts Institute of Technology Intro. to Linear Programming Slide 9 of 20 Example for Assumptions minimize: Z = 3X 1 + 5X 2 s.t. X 1 + X 2 > 5 3X 1 + 2X 2 < 18 6X 1 + 5X 2 < 42-7X 1 + 8X 2 < 0 0 < X 1 < 4 0 < X 2 < 6 Solution: X 1 * = 4 X 2 * = 1 Massachusetts Institute of Technology Intro. to Linear Programming Slide 10 of 20 5

6 Example Solution from Excel Solver X(SUB1) X(SUB2) OBJ FCN EXPRESSION CONSTRAINT LIMIT X1 +X2 5 >= 5 3 X1 + 2 X2 14 <= 18 6 X1 +5 X2 29 <= 42 (-7 X1) + 8 X2-20 <= 0 X1 4 >= 0 X1 4 <= 4 X2 1 >= 0 X2 1 <= 6 Massachusetts Institute of Technology Intro. to Linear Programming Slide 11 of 20 General Solution Approach Find a corner point An "initial feasible solution" Proceed to improved corner points Stop when no further improvements are possible For large problems, a variety of more sophisticated approaches are used! Massachusetts Institute of Technology Intro. to Linear Programming Slide 12 of 20 6

7 Solution Calculations To find a corner point it is necessary to solve system of constraint equations from linear algebra, this requires working with matrix of constraint equations, specifically, manipulating the determinants Amount of effort set by number of constraints So number of constraints defines amount of effort This is why LP can handle many more decision variables than constraints Massachusetts Institute of Technology Intro. to Linear Programming Slide 13 of 20 Solution Methods Simplex -- The textbook method For step 2, select improved corners Always goes to best corner Searches until no further improvement possible Inefficient for real problems Not used in practice Many practical methods - often proprietary Step 2 takes many forms Each best for different cases Very great efficiency possible A real art! Massachusetts Institute of Technology Intro. to Linear Programming Slide 14 of 20 7

8 Some LP Software Several LP methods available for simple problems Solver an Excel tool What's Best (free trials available) GAMS (big problems) They use standard equations and need no special organization They provide all kinds of sensitivity information The difficult part is setting up the LP equations so that they make sense! Massachusetts Institute of Technology Intro. to Linear Programming Slide 15 of 20 Discussion of Set-up for Solver The slide showing a solution has extra information to make slide easy to understand The left-hand column (with the equations) and the center columns (with the less or greater than symbols) are not necessary See help menu for Solver in Excel Massachusetts Institute of Technology Intro. to Linear Programming Slide 16 of 20 8

9 Typical Formulations: "Transportation" Problem Objective = Minimize cost of moving a single commodity from sources "i" to uses "j" = Σ ij C ij X ij Subject to: Amount shipped < Amount available Amount delivered > Uses Σ j X ij < S i Σ i X ij > S j Note: Matrix of constraint coefficients are all 0's and 1's ==> Particularly efficient solutions Massachusetts Institute of Technology Intro. to Linear Programming Slide 17 of 20 Typical Formulations: "Blending" or "Diet" Problems Objective = Minimize cost of materials = Σ i C i X i Subject to: Limits on availability X i < Amounts given Maxima or minima on impurities, trace elements, nutritional requirements, etc... Σ i a ij X i < = > b j Example: Minimize cost of steel alloy when only so much scrap is available, subject to limitations on carbon content, trace elements, etc. Massachusetts Institute of Technology Intro. to Linear Programming Slide 18 of 20 9

10 Typical Formulations: Activity Problems Objective = Minimize cost of production = Σ i C i X i where X i represent activities, that is, specific ways or fixed ratios of using resources Subject to: Limits on resources Σ i a ij X i < = > b j Example: Minimize cost of delivering cargo where each activity represents the use of a different size of ship, each with its own implications for the use of crew, fuel, etc. Massachusetts Institute of Technology Intro. to Linear Programming Slide 19 of 20 Summary LP can handle very large problems Basic mathematics simple: User can focus on definition of the problem Realistic problems require special techniques to deal with non-linearities, integral variables, etc. Can be very sophisticated This presentation is only an introduction! Massachusetts Institute of Technology Intro. to Linear Programming Slide 20 of 20 10

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