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by: Clifford Mertz


Clifford Mertz
Texas A&M
GPA 3.72


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Class Notes
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This 10 page Class Notes was uploaded by Clifford Mertz on Wednesday October 21, 2015. The Class Notes belongs to AGEC 622 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 69 views. For similar materials see /class/225922/agec-622-texas-a-m-university in Agricultural & Resource Econ at Texas A&M University.

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Date Created: 10/21/15
Basics of Linear Programming LP Problems Notes for AGEC 622 Bruce McCarl Regents Professor of Agricultural Economics Texas AampM University Basic LP Problem A LP problem is a linear form of mathematical programing Max chl 02X2 c3X3 chn s1 61le a12X2 a13 X3 611an S 91 a21X1 6122X2 a23X2 612an 2 b2 a21X1 a32X2 a33X3 613an 2 93 alel am2 X2 am3X3 amn X S bm X1 X2 X3 X1 2 0 This formulation may also be expressed in matrix notation Max CX Subject to AX g b X Z O Basic LP Application Joe s Van Shop Suppose Joe converts plain vans into custom vans and produces two grades ne and fancy Joe makes money from this Both types require a 25000 plain van After conversion fancy vans sell for 37000 Joe must use 10000 in parts and there is a pro t margin of 2000 per van A ne van uses 6000 in parts and sells for 32700 yielding a pro t of 1700 Joe Wishes to maximize his pro ts Assuming that pro ts are constant per van this can be written mathematically as Maximize Z 2000 X 1700 X fancy ne Basic LP Application Joe s Van Shop Joe is limited to the number of vans he can convert The shop can work on no more than 12 vans in a week so X X 512 fancy ne Labor is also limiting Joe has seven employees who work eight hours per day ve days a week Therefore he has at most 280 hours of labor available in a week Joe has found that a fancy van will take 25 labor hours to make while a ne van takes 20 25X 20X lt280 fancy fine Joe can also only convert nonnegative numbers of vans So X gt0 fancy 7 X ne Basic LP Application Joe s Van Shop So Joe s LP problem is Maximize 2000 Xfamy 1700Xfine st Xf X lt 12 ancy fine 25 xf 20 x lt 280 ancy fine fancy Xfine Z 0 Such a problem can be solved a number of ways The answer is Zpro ts 22800 X 8 fancy ne This solution uses all of our labor and capacity We also get shadow prices Capacity value 500 per van Labor value 60 per hour Basic LP Application Joe s Van Shop Speci c properties of the solution How much pro t is produced is the objective value Zpro ts 22800 How much of each good is made These are the decision variables Xfancy 8 7 X ne 4 How much the resources are worth in their current application are found in the shadow prices or Lagrange multipliers since we use all of our labor and capacity Capacity value 500 per van Labor value 60 per hour A shadow price is an important LP concept and is an estimate of how much the objective function will change with a one unit change in the right hand side of an equation Assumptions of LP Attributes of model objective function appropriateness decision variable appropriateness constraint appropriateness Math in model proportionality additivity divisibility certainty Assumptions Attributes of model Objective Function Appropriateness The objective function is the proper and sole criteria for choosing among the feasible values of the decision variables Decision Variable Appropriateness The decision variables are all fully manipulatable Within the feasible region and are under the control of the decision maker All appropriate decision variables have been included in the model Assumptions Attributes of model Constraint Appropriateness The constraints fully identify the bounds placed on the decision variables by resource availability technology the external environment etc Any selection of decision variables Which simultaneously satis es all the constraints is admissible The resources used andor supplied Within any single constraint are homogeneous items that can be used or supplied by any decision variable appearing in that constraint Constraints do not improperly eliminate admissible values of the decision variables The constraints are inviolate No considerations involving model variables other than those included in the model can lead to the relaxation of the constraints Assumptions Math in model Proportionality The contribution per unit of a variable to any equation is constant There are no economies of scale Additivity Contributions of variables to an equation are additive The objective function and resource use equals the sum of contributions of each variable times levels This rules out interaction or multiplicative terms in the objective function or the constraints Divisibility All decision variables can take on any nonnegative value including fractional ones In other words the decision variables are continuous Certainty All parameters are known constants The optimum solution is predicated on perfect knowledge of all parameter values


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