Linear Programming Session 1 PDF

INTRODUCTION TO OPTIMIZATION: LINEAR PROGRAMMING  Optimization Problems Primarily concerned with either allocation of limited resources in “best” possible manner; Determining the level at which some activity must be performed so that it is “best” for the organization; and so on…...

INTRODUCTION TO OPTIMIZATION: LINEAR PROGRAMMING  Optimization Problems Primarily concerned with either allocation of limited resources in “best” possible manner; Determining the level at which some activity must be performed so that it is “best” for the organization; and so on… Essentially, making decisions such that a given objective is achieved in the best possible manner while satisfying conditions imposed by internal or external environment of the organization  An Example: Product Mix Problem Gia’s woodcarving manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Gia’s variable labour and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Gia’s variable labour and overhead costs by $10. Each soldiers requires 2 hours of finishing and 1 hour of carpentry labour and each train requires 1 hour of finishing and 1 hour of carpentry labour. Each week Gia can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 soldiers are bought each week. If Gia wants to maximize her weekly profit, how many soldiers and trains should she produce every week? Product Mix Problem: Summary Data Soldier Train Available Selling Price $27 $21 Raw Material Cost $10 $9 Variable Labour and Overhead Cost $14 $10 Finishing Hours 2 1 100 Carpentry Hours 1 1 80 Demand 40 unlimited Product Mix Problem: Summary Data Soldier Train Available Selling Price $27 $21 Raw Material Cost $10 $9 Variable Labour and Overhead Cost $14 $10 Contribution to Profit $3 $2 Finishing Hours 2 1 100 Carpentry Hours 1 1 80 Demand 40 unlimited  Product Mix Problem Decision to be made – Number of soldiers to be produced per week – Number of trains to be produced per week Objective – To maximize weekly profit To ensure viability – Number of hours used for manufacturing in finishing and carpentry department must NOT be more than the number of hours available – Number of soldiers produced must NOT be more than the demand Product Mix Problem: Optimization Model Decision to be made Variables Decision – Number of soldiers to be produced per week – Number of trains to be produced per week Objective Objective Function – To maximize weekly profit To ensure viability – Number of hours used for manufacturing in finishing and carpentry department must NOT be more than the Constraints number of hours available – Number of soldiers produced must NOT be more than the demand Linear Programming Problem (LPP) Decision Variables – X1: #soldiers to be produced per week – X2: #trains to be produced per week Linear in decision variables! Objective Function – Z = maximize 3X1+ 2X2 Constraints 2X1+1X2