MATHEMATICS 1 CONTENTS Mathematical programming Linear - - PowerPoint PPT Presentation

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BUSINESS MATHEMATICS

Linear programming

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

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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

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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

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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.

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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 𝑏𝑗𝑘 ▪ the minimum amount of nutritional element 𝑘 is 𝑑

𝑘 ′

▪ the amount of food type 𝑗 in the diet 𝑦𝑗 ▪ the price of food type 𝑗 is 𝑞𝑗

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LINEAR PROGRAMMING Example (continued) Relationships ▪ cost: 𝐷 = σ𝑗=1

𝑛 𝑞𝑗𝑦𝑗

▪ nutritional value: 𝑑

𝑘 = σ𝑗=1 𝑛 𝑏𝑗𝑘𝑦𝑗

𝑘 = 1, … , 𝑜 Objective ▪ minimize cost 𝐷 Constraints ▪ staying healthy: 𝑑

𝑘 ≥ 𝑑 𝑘 ′

𝑘 = 1, … , 𝑜 ▪ “no nonsense”: 𝑦𝑗 ≥ 0 𝑗 = 1, … , 𝑛

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THE LP-PROBLEM Formulation of the problem minimize 𝐷 = ෍

𝑗=1 𝑛

𝑞𝑗𝑦𝑗 subject to 𝑑

𝑘 = ෍ 𝑗=1 𝑛

𝑏𝑘𝑗𝑦𝑗 ≥ 𝑑

𝑘 ′

𝑘 = 1, … , 𝑜 and 𝑦𝑗 ≥ 0 𝑗 = 1, … , 𝑛

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

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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

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GRAPHICAL SOLUTION Take: ▪ 𝑛 = 2:

▪ food 1 (meat); food 2 (potatoes) ▪ decision variables 𝑦1 (amount of meat); 𝑦2 (amount of potatoes)

▪ non-negativity constraints ▪ 𝑜 = 3:

▪ constraint 1 (calories); constraint 2 (vitamins); constraint 3 (proteins)

▪ iso-budget lines

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GRAPHICAL SOLUTION

𝑦1 𝑦2 constraint 1 constraint 2 constraint 3 𝐷1 𝐷2 𝐷3 𝐷𝑛𝑗𝑜

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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

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OLD EXAM QUESTION 10 December 2014, Q1e

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OLD EXAM QUESTION 24 March 2016, Q2c

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