SIE340: Deterministic Operations Research Midterm Exam - University Of Arizona

SIE340: Deterministic Operations Research

Midterm Exam

(1 hour and 45 minutes)

Name (PRINT): __________________________________ Instruction: • This exam is closed book and closed notes; • A simple calculator with basic functions (only perform “+,-,*,/” operations) can be used; • Including this cover page, this exam has 10 pages with 6 problems; • Please read the problems carefully and show your steps to obtain final solutions.

Good luck and Bear Down!

© University of Arizona (You should not distribute this exam to any one or post online.)

SIE340 Midterm Exam 1

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1. True or False (10 points, 2 points each question) Please write in the line “T” if the statement is true, or “F” if the statement is false. a) ____ A feasible solution is a solution that satisfies all the constraints. b) ____ A linear function can include the product of two variables, e.g. 𝑓 (𝑥$ , 𝑥& ) = 𝑥$ 𝑥& .

c) ____ A constraint is nonbinding if the left-hand side and right-hand side of the constraint are equal when the optimal values of the variables are plugged in the constraint.

d) ____ The set in Figure 1 (below) is convex.

Figure 1. The set under study is represented by the gray area.

e) ____ For a maximization problem, the optimal objective values for an integer programming problem is less than or equal to the optimal objective value of its LP (linear programming) relaxation.

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2. LP model and graphical method (30 points: 15 points modeling, 15 points solving) Following the 5 steps introduced in class, model the following problem as a linear program and solve it by the graphical method (a template is provided in Figure 2, next page).

LightCo produces two types of lamps (type I and type II) that require metal parts and electrical components. Each lamp type I requires 1 unit of metal part, and 2 units of electrical components, and it can be sold for $1. Each lamp type II requires 3 units of metal parts and 2 units of electrical components, and it can be sold for $2. LightCo has 200 units of metal parts and 300 units of electrical components every day. There is no limit for the demand of lamp I, while the customer is not willing to buy more than 60 lamps of type II every day. The management team wants to maximize the profits of LightCo. 2.1)

Please model this problem as an LP (decision variables must be clearly explained).

SIE340 Midterm Exam 1

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2.2)

Please use graphical method to solve the model (clearly indicate the optimal solution).

𝑥&

𝑥$ Figure 2. Template for graphical method

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3. Network models by LP. (15 points modeling+5 points question) DuckCo produces canned food. It has 3 processing plants that can be used to satisfy the demand of 4 cities. Plant A can produce up to 335 thousand cans per day, plant B can produce up to 270 thousand cans per day, and plant C can produce up to 600 thousand cans per day.

City 1 requires 222 thousand cans per day, city 2 requires 355 thousand cans per day, city 3 requires 100 thousand cans per day, and city 4 requires 528 thousand cans per day. The costs of processing and sending thousand cans from a plant to a city are given in Table 1. The management team wants to minimize the shipping cost.

City

1

2

3

4

A

$5

$7

$2

$8

B

$2

$5

$3

$7

C

$5

$3

$7

$5

Plant

Table 1. Cost of shipping a thousand cans of canned food.

3.1) Following the 5 steps introduced in class, model the above problem as an LP. Do not solve.

3.2) (Please answer after modeling the problem) What can you say about the problem if city 4 requires 600 thousand cans per day instead of 528 thousand?

SIE340 Midterm Exam 1

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[for answering Problem 3]

Answer to Question 3.2):

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4. Integer Programming Models. (15 points modeling+5 points question) My cell phone has three different SIM cards, each of them associated with a different company: SIM card 1 is associated with company A, SIM card 2 is associated with company B, and SIM card 3 is associated with company C (SIM cards association cannot be changed).

Company A charges me $16 fixed a month, and $0.25 per minute of call. Company B charges me $25 fixed a month, and $0.21 per minute of call. Company C charges me $18 fixed a month, and $0.22 per minute of call. I need to make calls for at least 200 minutes every month. If I don’t pay the fixed charge of a company, I cannot make calls with this company. I can use any combination of companies to make all the calls I require. I need some help to minimize my cell phone bill.

4.1) Following the 5 steps introduced in class, model the above problem. Do not solve.

4.2) (Please add the following requirement after modeling, you do not need to write the whole model again) Instead of using any combination of companies to make calls, you can use at most two companies to make calls.

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[for answering Problem 4]

Answer to Question 4.2):

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5. Dijkstra’s Algorithm. (15 points) Please use Dijkstra’s Algorithm to find a shortest path from node 1 to node 6. Please show the steps how you can obtain the shortest path. (The number next to each arc is the length of that arc.)

Answer these questions and perform iterations: Please fill the set 𝑉 and set 𝐴: 𝑁 = (𝑉, 𝐴), 𝑉 = { 𝐴={

},

}

Iterations:

The shortest path from node 1 to node 6 is: -- -- -- , and the length for this path is _______.

SIE340 Midterm Exam 1

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6. Standard form of an LP. (5 points) Please write the standard form of the following linear programming problem: (maximum objective, only equality constraints): min 𝑧 = 𝑥$ − 2𝑥& + 3𝑥7 𝑠. 𝑡. − 𝑥$ − 5𝑥& + 6𝑥7 ≤ 2 −2𝑥$ + 𝑥& + 3𝑥7 ≥ 5 𝑥$ , 𝑥& , 𝑥7 ≥ 0

SIE340 Midterm Exam 1

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