Mathematical+Programs+-+Objective+Function

= Objective Functions - Case Study: Home Delivery Bags  =

**Author:** Joseph Warfel, 2011

Purpose
The purpose of this lesson is to build on the previous two lessons, in which the students learned about decision variables and constraints (respectively). Together with the objective function, these are the three components of a mathematical program. The resulting mathematical program for home delivery bags is relatively small, which is good, because it is the first one they will use.

Overview
This set of lessons is an extended discussion of the Home Delivery (HD) Program at Lakeview Pantry, a food pantry in Chicago. The HDP consists of filling two grocery bags with food and delivering it to clients who are unable to come to the pantry to pick up food. Each client in the program receives a bag once a month. The decision about which items to put in the bag can be described as a mathematical program, the objective function of which changes depending on which type of bag is being produced – for a diabetic bag, we should seek to minimize sugar; for a cardiac bag, we should seek to minimize sodium; for a regular bag, there are several possible objectives.

In the previous lessons, the students will have studied the meaning and formulation of constraints, and how they define a feasible set of solutions to the problem. In this lesson, I will introduce the objective function as a way to differentiate solutions – as a way to determine which solutions are better than others. (Finding the best, or “optimal,” solution is a task I expect students to carry out in Excel, and therefore it is outside the scope of this lesson.)

Student Outcomes
SWBAT - Identify the objective of a decision-maker in a problem. - Formulate an objective that is a linear combination. - Rank feasible solutions by objective value. Muchin does not use Illinois state standards.

Time
Length of time needed for the activity, in minutes and/or class periods

Level
Grade level and class name

Materials and Tools
Technology including software and files Suggested resources or equipment needed Supplementary documents or handouts

[|Food bank vehicles.pdf] PowerpointPresentation.pdf

Preparation
Any suggestions that teachers need to do in advance of the lesson, including setup.

Prerequisites
Activities, or alternative activities, that should have been done before

Background
The students will need to understand summation notation (capital sigma notation), sets, and subscripts, as well as how to calculate a linear combination. All of these are topics that we covered at least two lessons before and which were practiced in the previous lesson.

The students also need to be able to determine whether a given solution is feasible for the constraints of the problem. This was covered in the previous lesson.

The students also need to know some general knowledge about food and health conditions – specifically, that diabetics need to reduce their consumption of certain types of foods, in particular those with added sugar, and that people with cardiac conditions need to reduce their consumption of sodium.

Teaching Notes
This should be directed towards the teachers. Spell out the steps the teacher would take to facilitate this lesson plan. List alternatives here as well.

Assessment
Lakeview Pantry has two types of bags that require an objective function. I will teach one of the bags (the diabetic bag) as a lesson, and then let the students carry out the same task for the other (the cardiac bag). If there is only time to do this, then the cardiac bag exercise will be the assessment; however, if there is more time, we could discuss the options of different types of objectives for the regular bags to assess only the first two objectives listed above.

Additional Information
Any other explanations necessary of data, technology development, resources. = =