Get Assessment Solutions on MAT2200 – Operations Research 1 Assignment 4 – Semester 2

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QUESTION  1                                  (20 marks)

Use the branch and bound method to solve the following Integer Programming problem:

max z = 2.7x₁ + 3.9x₂

s.t.

7x₁ + 14x₂ ≤ 385

5x₁ + 14x₂ ≤ 338

12x₁ + 16x₂ ≤ 536

x₁, x₂ ≥ 0 and are integers

QUESTION 2                                    (30 marks)

Water from a proposed desalination plant is to be pumped through an existing network of pipes leading from the plant to a storage dam.  The map in Figure 1 below represents the network of pipes where the nodes (2 – 13) represent pumping stations and node (1) is the desalination plant and node (14) is the storage dam. The numbers on the arcs denote the maximum flow for each pipe in Megalitres per day. Moreover, flow in each pipe is restricted to one way but can be in either direction (see Figure 1 below).

Figure 1: Map of water pipeline network showing maximum flow restriction in each pipeline section.

a).  What is the maximum volume of water (in Megalitres) that can be pumped per day from the desalination plant to the storage dam? (15 marks)

b).  Indicate clearly on a diagram the direction in which the water should be pumped in the network in order to maximize the volume of water pumped per (5 marks)

c).  To satisfy the manufacturers specifications each pumping station must be connected to a network of higher quality pipes. The costs for upgrading each section of pipeline to the higher quality pipeline (in millions of dollars) are shown in Figure 2. The local council proposes to only initially upgrade some of the pipes so that the plant, dam and each pumping station, (1) through (14), is linked to every other location by sections consisting of only upgraded

Which pipes should be upgraded if costs are to be minimized?                                                     (10 marks)

Figure 2: Map of water pipeline network showing the cost of upgrading each section.

QUESTION  3                                  (30 marks)

A manager of theme park needs to check the paths between the park’s attractions are clean and tidy before opening for the day. To do this the manager will choose a route such that she travels along each path once and once only to see and check if it is suitably clean. She does not mind if she has to pass through an attraction more than once as long she only travels along each path once and that the route begins and ends at the main gate located at L.

Figure 3 shows the main areas of the theme park with the main gate designated as L and the attractions denoted by A, B, C, . . . , K, M. The paths between each of the attractions and with the main gate are denoted as a, b, c . . . w.

Figure 3:  A map of the theme park with the gate L and attractions at A, B, C,. . . , K, M.

• Determine the route (if possible) that would satisfy manager’s requirements.

Draw this route. What type of path is this route?                                                             (15 marks)

• A visitor to the theme park wants to see all the attractions at A, B, C, . . . , K, and  M  by beginning at the main gate L and finishing at L. However the visitor gets easily bored and so only wants to visit each attraction once and once only.

Determine the route (if possible) that would satisfy visitor’s requirements.

Draw this route. What type of path is this route?                                                            (15 marks)

QUESTION 4                                    (20 marks)

Figure 4 below shows the travel times between 14 customer locations, and the travel times between some of these locations and the warehouse (W). What is the shortest travel time from the warehouse to each of the customers? Present your answer in a table with three columns showing:

Customer number,

The shortest time, and,

The associated route

Hint: Find the shortest routes by starting from node W.

Figure 4: The travel times between customer locations and between the warehouse and customer locations.

Assignment Criteria

Question 1

1. Branch and Bound Method

Branch and bound method is used – answer not obtained by rounding, guessing, or the integer enumeration method.

Fathoming tree is shown clearly indicating hierarchy of LP problems (or stated in the solution).

All required LP sub-problems are investigated with the reason for each branch clearly stated.

Additional constraints are correct for the additional two sub-problems when branching-off a non-integer variable.

Each branch of the tree is correctly fathomed.

2. LP Solution for each problem

If software is used then it is named and both the input and the output are given.

If completed by hand (simplex or graphical method) then all working is shown and is correct.

Correct answer for each sub-problem is given.

3. Final Solution

The solution to the integer programming problem is correct.

Decision variables and objective function value clearly stated.

Solution is both feasible and optimal.

Question 2

A).  Appropriate algorithm is used to obtain the

Full working with labels are shown (or stated in solution) and are correct.

The flows through each node are feasible and balanced.

All appropriate flows are considered.

Cut method is used correctly to determine optimality.

The maximum volume of water flow per day is correct and is stated clearly.

B).  Direction of the flow in each pipeline is clearly shown on the network or clearly stated in

Direction of the flow is correct and is feasible.

C).  Appropriate algorithm is used to find the

All working is shown and is correct.

Pumping stations are attached by all required upgraded pipelines.

The minimum cost is stated and is correct.

The upgraded sections of pipelines are correct and are stated clearly.

Question 3

A).  Manager’s path

Obeys all of the manager’s requirements (if possible) i.e. starts and ends at the appropriate location, visits all required items of interest, etc.

Route is given clearly using location and path code names and is correct.

A diagram of the route is given.

Type of path is given and is correct.

B).  Visitor’s path

Obeys all of visitor’s requirements (if possible) i.e. starts and ends at the appropriate location, visits all required items of interest, etc.

Route is given clearly using location and path code names and is correct.

A diagram of the route is given.

Type of path is given and is correct.

Question 4

Appropriate algorithm is used to obtain the solution.

Full working with labels is shown and is correct.

The shortest travel times from the warehouse to each customer location are given and are correct.

All shortest routes from the warehouse to each customer location are given and are correct.

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