MATH143 Lec 3 Quantitative Methods Charles W Jackson Andrew Nunekpeku MATH143 – Fall 2011 1/30/2018

Agenda for today Announcement – Return Polya’s “How to Solve It” to Library since this is for Freshman maths Translation of Word Problems into LP mathematical statements Another Example, solved in class Translation to LP math Graphical Technique Identifying Corner Points and Evaluating There Discovering the Optimum Program MATH143 – Fall 2011 1/30/2018

Linear Programing Allocate some limited resources One sentence description: Allocate some limited resources Between competing activities In a best possible way As we found in A1, you need all three parts to have a meaningful answer. MATH143 – Fall 2011 1/30/2018

Word Problem Translation Step One: What are the Decision Variables? These are the minimum info required to describe what you want done during the period. What are the competing activities? Step Two: What are the Constraints? Which linear combinations of activities is limited? May be more or less than number of decision variables. Beware of implied non-negativity constraints. MATH143 – Fall 2011 1/30/2018

Word Problem Translation Step Three: What is the Objective Function? Describe mathematically how you will decide which of two plans (levels of the activities) you would prefer. Often, this can be cast in the notion of maximizing profit for profit-making enterprises, but other situations might not be based on that metric. MATH143 – Fall 2011 1/30/2018

Berekuso Farm We are seeing both corn and pineapple on the hillsides near here... 1/30/2018

Word Problem Take this“word problem” (business situation): A Berekuso seller owns a 100-hectare farm on which she could plant corn and pineapple. Every hectare planted with corn requires 50 liters of water per day, as well as 20 kg of fertilizer. Every hectare planted with pineapple requires 75 liters of water per day and 15 kg of fertilizer. From experience, she figures it will take 2 days of labour to harvest each hectare of corn, while pineapple requires 2.5 days per hectare. MATH143 – Fall 2011 1/30/2018

Word Problem Continuing the “word problem”: The Berekuso farmer can sell as many bushels of corn as she harvests at GHS 3 each, pineapples sell at GHS 1. Again from experience, she expects to be able to harvest 90 bushels corn per hectare, or 300 pineapples per hectare. Her borehole can only deliver 6000 liters per day, while fertilizer is available in unlimited amounts at GHS10 per 50 kilo bag. Labour is plentiful at GHS 5 per day. What should she plan so that she will make the most profit this season? MATH143 – Fall 2011 1/30/2018

Word Problem Identify the Objective (what is “best”) Identify the Decision Variables (what defines the “program”) Identify the Constraints (what resources are limited?) Maximize the Profit for the season. The number of hectares of corn (x) and the number of hectares of pineapples (y) Water and land are limited, otherwise it says she can get as much fertilizer and labour as she want to buy. MATH143 – Fall 2011 1/30/2018

In Algebraic Form Maximize: 90∙3x+300∙1y-20∙0.2x-15∙0.2y-2∙5x-2.5∙5y (Total Profit = revenues – fertilizer - labour) Subject to: 50x + 75y ≤ 6000 (Water) x + y ≤ 100 (Size of farm) Also assume non-negativity on activities: x ≥ 0 (non-negative corn Ha) y ≥ 0 (non-negative pineapple Ha) (Notice that the marginal costs deduct from the profit computation, but are not in themselves a constraint, since we are told we can get as much fertilizer and labour as we pay for.) MATH143 – Fall 2011 1/30/2018

2-D Solutions - Graphical Method Add in constraints one at a time. The constraints form straight lines. The inequalities include the line and the points on one side of each line. As we add constraints, the feasible area contracts. Lastly, draw constant-profit lines, find solution at a corner or along an edge. MATH143 – Fall 2011 1/30/2018

MATH143 – Fall 2011 1/30/2018

Thanks, Math is OK! Footer Text 1/30/2018