MATHEMATICS 1 CONTENTS Mathematical programming Linear - PowerPoint PPT Presentation

Linear programming BUSINESS MATHEMATICS 1

CONTENTS Mathematical programming Linear programming The LP-problem Old exam question Further study 2

MATHEMATICAL PROGRAMMING Mathematical programming ▪ refers to deciding on levels of decision variables in order to make an optimal decision Linear programming (LP) ▪ refers to a programming problem with a linear objective function Constraint (recall the Lagrangian) ▪ usually not all values of the decision variables are admissible 3

LINEAR PROGRAMMING Example: the army’s diet ▪ you must feed your soldiers ▪ you have a choice of food types ▪ potatoes, corn, meat, carrots, etc. ▪ each food type has nutritional characteristics ▪ calories, vitamins, proteins, etc. ▪ there is a minimum level of nutritional input (calories, vitamins, etc.) required to stay healthy and strong ▪ you want to select the cheapest diet (=composition of food types) that meets all constraints 4

LINEAR PROGRAMMING Example (continued) Decision variables: ▪ amount of potatoes, amount of corn, amount of meat, etc. Objective function: ▪ cost Constraints: ▪ minimum amount of calories, minimum amount of vitamins, etc. ▪ no negative amounts of potatoes, corn, etc. 5

LINEAR PROGRAMMING Example (continued) Symbols and variables: ▪ there are 𝑛 food types and 𝑜 nutritional elements ▪ food type 𝑗 contains nutritional element 𝑘 in an amount 𝑏 𝑗𝑘 ′ ▪ the minimum amount of nutritional element 𝑘 is 𝑑 𝑘 ▪ the amount of food type 𝑗 in the diet 𝑦 𝑗 ▪ the price of food type 𝑗 is 𝑞 𝑗 6

LINEAR PROGRAMMING Example (continued) Relationships 𝑛 𝑞 𝑗 𝑦 𝑗 ▪ cost: 𝐷 = σ 𝑗=1 𝑛 𝑏 𝑗𝑘 𝑦 𝑗 ▪ nutritional value: 𝑑 𝑘 = σ 𝑗=1 𝑘 = 1, … , 𝑜 Objective ▪ minimize cost 𝐷 Constraints ′ ▪ staying healthy: 𝑑 𝑘 ≥ 𝑑 𝑘 = 1, … , 𝑜 𝑘 ▪ “no nonsense”: 𝑦 𝑗 ≥ 0 𝑗 = 1, … , 𝑛 7

THE LP-PROBLEM Formulation of the problem 𝑛 minimize 𝐷 = ෍ 𝑞 𝑗 𝑦 𝑗 𝑗=1 𝑛 ′ subject to 𝑑 𝑘 = ෍ 𝑏 𝑘𝑗 𝑦 𝑗 ≥ 𝑑 𝑘 = 1, … , 𝑜 𝑘 𝑗=1 and 𝑦 𝑗 ≥ 0 𝑗 = 1, … , 𝑛 8

THE LP-PROBLEM In matrix notation minimize 𝐷 = 𝐪 ⋅ 𝐲 subject to 𝐝 = 𝐁𝐲 ≥ 𝐝 ′ ቐ and 𝐲 ≥ 𝟏 9

THE LP-PROBLEM Similar to previous constrained maximization/minimization problem But with: ▪ linear objective function ▪ many constraints ▪ inequality constraints ▪ non-negativity constraints Constraints define a feasible/admissable solution space Optimal solution is always on a vertex 10

GRAPHICAL SOLUTION Take: ▪ 𝑛 = 2 : ▪ food 1 (meat); food 2 (potatoes) ▪ decision variables 𝑦 1 (amount of meat); 𝑦 2 (amount of potatoes) ▪ non-negativity constraints ▪ 𝑜 = 3 : ▪ constraint 1 (calories); constraint 2 (vitamins); constraint 3 (proteins) ▪ iso-budget lines 11

GRAPHICAL SOLUTION constraint 3 𝑦 2 constraint 1 constraint 2 𝐷 3 𝐷 𝑛𝑗𝑜 𝑦 1 𝐷 1 𝐷 2 12

NON-GRAPHICAL SOLUTION When 𝑛 > 2 Dedicated algorithm ▪ Simplex method ▪ in Excel Solver Use “Simplex LP” for linear programming problems Use “GRG Nonlinear” or “Evolutionary” for other problems 13

OLD EXAM QUESTION 10 December 2014, Q1e 14

OLD EXAM QUESTION 24 March 2016, Q2c 15

FURTHER STUDY Sydsæter et al. 5/E 17.1 Tutorial exercises week 6 Linear programming1 Linear programming2 16