MATH 171 University of Southern California Linear Programming MATLAB Report

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MATH 171

University of Southern California

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The last question requires using MATLAB.

If graphs are asked need to be drew, you can use Desmos.com and then printed it out to annotate the corner points and stuff. (It's fine if you have other preferred ways to draw the graphs)

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Math 171A: Linear Programming Instructor: Philip E. Gill © 2021 (Not to be Reposted) Winter Quarter 2021 Homework Assignment #1 Due Friday January 15, 2021 I know that you are all aware of the importance of doing the homework assignments. This is the best way to keep up with the class and do well in the midterm and final examinations. Unfortunately, the graders will not have the time to grade every exercise. Instead, they will grade two or three exercises (including at least one Matlab exercise) and give a fixed score for every other exercise attempted. The starred exercises require the use of Matlab. Remember that it is necessary to do all the Matlab assignments to obtain credit for the class. Exercise 1.1. In the JuiceCo mixture problem considered in class, x represents gallons of cranapple, y gallons of appleberry. (a) How many gallons of each juice mixture are represented by each corner point? Evaluate the profit formula 3x + 4y at each corner point of the feasible region. (b) Which corner point represents the maximum profit for each of the following profit formulas: (i) profit = 3x + 4y (the definition used in class); (ii) profit = 2x + 5y; (iii) profit = 5x + 3y. Exercise 1.2. Convert the JuiceCo problem into “all-inequality form” min cTx, subject to Ax ≥ b and define the quantities A, b, and c. Exercise 1.3. Consider the following constraints in two variables: (1) x1 − 2x2 ≤ −1; (2) x1 − x2 ≥ −3; (3) 21 ≤ x2 ≤ 4; (4) 2x1 − 2x2 ≤ 6; (5) x1 + x2 ≤ 6; (6) x1 ≥ 0; and (7) x2 ≥ 0. (a) Define the matrix A and vector b that express these constraints in the form Ax ≥ b. (b) Draw the feasible region defined by the 8 constraints. Outline the feasible region on your graph and label each of the corner points. (c) Draw the level curve along which the linear function −2x1 − x2 is equal to −8. (d) Draw the level curve along which the linear function −x1 − x2 is equal to −6. Exercise 1.4. Suppose that your diet consists of a selection of the following items from a well-known fast food restaurant (we give each food a nickname to assist in referring to it): QP: MD: BM: FF: MC: Quarter Pounder McLean Deluxe Big Mac Filet-O-Fish McGrilled Chicken FR: Fries, small SM: Sausage McMuffin 1M: 1% Lowfat Milk OJ: Orange Juice 2 Mathematics 171A You are interested in providing your diet with appropriate amounts of the following seven “nutrients”: Prot: Protein VitA: Vitamin A VitC: Vitamin C Calc: Calcium Iron: Iron Cals: Calories Carb: Carbohydrates Using the internet you have found how much of each nutrient is in one serving of each food, and the total of each nutrient that you require. You also found the price per serving of each food. The relevant costs, requirements, and nutritional values are: Cost Prot VitA VitC Calc Iron Cals Carb QP 1.84 28 15 6 30 20 510 34 MD 2.19 24 15 10 20 20 370 35 BM 1.84 25 6 2 25 20 500 42 FF 1.44 14 2 0 15 10 370 38 MC 2.29 31 8 15 15 8 400 42 FR 0.77 3 0 15 0 2 220 26 SM 1.29 15 4 0 20 15 345 27 1M 0.60 9 10 4 30 0 110 12 OJ 0.72 1 2 120 2 2 80 20 Req’d 55 100 100 100 100 2000 350 Formulate a linear program with a solution that defines the least expensive combination of the foods providing a day’s nutritional requirements. Write the problem in the form min cTx, subject to Ax ≥ b. Exercise 1.5.∗ You have formulated the diet problem above as a linear program of the form min cTx subject to Ax ≥ b. (a) In class we defined a “corner point” as a feasible point that lies at the intersection of n hyperplanes. Give an upper limit on the number of corner points for the diet problem. (Don’t just guess a number, give an estimate based on the row and column dimensions of the constraint matrix.) (b) Find three corner points and compute the value of the objective function at each one. Exercise 1.6. Consider the inequality constraint aTx ≤ b where a 6= 0. Show that the constraint normal points “into” the infeasible half-space.
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Assignment

MATH 171 A: Linear Programming
Type University name here

Date: January 15, 2021
Submitted to: Prof. Philip E. Gill
Submitted by:

Exercise 1.1.
(a) Solution. The optimization problem for the JuiceCo mixture in mathematical form was
formulated in the class. The constraints for JuiceCo problem are given below:
3𝑥 + 2𝑦 ≤ 200
𝑥 + 2𝑦 ≤ 100
𝑥≥0
𝑦≥0
Here, x represents gallons of craneapple and y represents gallons of appleberry.
The graph for the above problem can be plotted in Matlab or on desmos with a little effort.
The graph drawn on desmos is shown below:

Figure 1.1: Graph for the optimization model with corner points

The gallons of each juice for corner points can be tabulated as follows:
Corner point
(0,0)
(0,50)
(50,25)
(200/3,0)

Gallons of cranapple
0
0
50
200/3

Gallons of appleberry
0
50
25
0

It is given that the profit is 𝑃 = 3𝑥 + 4𝑦
At (0,0), the profit is 𝑃 = 3 × 0 + 4 × 0 = 0
At (0,50), the profit is 𝑃 = 3 × 0 + 4 × 50 = 200
At (50,25), the profit is 𝑃 = 3 × 50 + 4 × 25 = 250
At (200/3,0), the profit is 𝑃 = 3 ×

200
3

+ 4 × 0 = 200

It’s observed that the profit is maximum at x =50 and y = 25 and the maximum profit is 250
(b)
(i) 𝑝𝑟𝑜𝑓𝑖𝑡 = 3𝑥 + 4𝑦
The maximum profit is for the corner point (50, 25) and the maximum profit is 250
(ii) 𝑝𝑟𝑜𝑓𝑖𝑡 = 2𝑥 + 5𝑦
Table for the profit can be given as follows
Corner points

Profit

(0, 0)

2×0+5×0=0

(0, 50)

2 × 0 + 5 × 50 = 250

(50, 25)

2 × 50 + 5 × 25 = 225

(200/3, 0)


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