OPTIMAL COST

QUESTION

Question 1 (8 marks) 

Find the dual of the following LP:

minimize         -7x₁ + 6x₂ – 8x₃

subject to         2x₁ – 3x₂ + 4x₃ –  2x₄      ≤  -3

-x₁ + 2x₂           –  3x₄     ≥   4

x₂   + x₃ + 2x₄      =   1

x₁ –  2x₂  –  2x₃  +  x₄      ≤   0

x₁ ≤ 0, x₂ ≤ 0, x₃ ≥ 0, x₄ unrestricted-in-sign

Question 2 (12 marks)

Consider the following LP, (P):

maximize         2x₁ + 4x₂ + 3x₃  –   x₄

subject to         3x₁ +  x₂  +   x₃ –  4x₄     ≤  12

x₁  – 3x₂ +  2x₃  – 3x₄     ≤   7

2x₁ +  x₂  + 3x₃ +   x₄      =  10

x₁ ≥ 0, x₂ unrestricted-in-sign, x₃ ≥ 0, x₄ ≤ 0

Let y₁, y₂ and y₃ be the dual variables associated with the first, second and third constraints of (P). It has been suggested that (y₁, y₂, y₃) = (1, 0, 3) are optimal dual variable values for (P). Use duality and complementary slackness to confirm or refute this suggestion

Question 3 (18 marks)

Gates Enterprises manufactures an octal-core chip for a line of personal computers. The chips are manufactured in Seattle, Columbus and New York, and shipped to warehouses in Pittsburgh, Mobile, Denver, LA and Washington (DC), for further distribution. The following table shows the number of chips available at each manufacturing plant, the number of chips required by each warehouse, and the shipping costs (in dollars per chip shipped).

(a)  Find an initial basic feasible solution using the Northwest Corner Method.

(b)  Starting with your solution from part (a), find the amount that should be shipped from each plant to each warehouse to minimize the total shipping cost.

(c)   Find an initial basic feasible solution using the Minimum Cost Method. Is it optimal?

Question 4 (18 marks)  

Winston (4^th edition) Section 7.1 Question 2 (page 371). After you have formulated this Transportation Problem, use the Transportation Method to find its optimal solution, starting with the Northwest Corner initial basic feasible solution.

Question 5 (20 marks)

Consider the following TSP problem instance data provided, for the 14 locations A, B, … N; you may assume that Euclidean distance as a reasonable approximation of actual travel cost.

Apply each of the following TSP heuristics to this data, using lexicography to break ties. For each TSP solution, identify a specific modification of your solution that would improve the length of the TSP tour you found.

(a)  Double-ended Nearest Neighbour, starting at location H

(b)  Greedy

(c)  Convex Hull (Largest Angle) Insertion

(d)  Christofides’ Heuristic

Question 6 (24 marks)

Consider again the problem instance location data of Question 5. As shown in the diagram below, there are now two depots: Depot 1 and Depot 2. Depot 1 has 40 units of a homogeneous commodity available for delivery, while Depot 2 has 20 units available. Based at Depot 1 are two vehicles of capacity 20. Depot 2 has one vehicle of capacity 20 units based there. The demands of each of the customers are shown on the diagram. If there is only one number shown, it means that the customer’s demand may not be split; if there are two numbers shown, this indicates that the customers demand may be split, with the second number indicating the minimum amount of the commodity that the customer would accept per delivery. For example, customer J has a demand of 4 units, and it must receive that demand in one vehicle visit; customer C has 9 units of demand, and may receive its demand in more than one vehicle visit, as long as at least 3 units of demand are delivered during each vehicle’s visit.

A ‘good’ routing for this VRP is one in which each customer whose demand may be split is visited by at least one vehicle (all other customers must be visited by exactly one vehicle), the number of splits made is small, the routes are as balanced as possible (i.e., each route should be of approximately the same length and visit approximately the same number of customers), and the total distance travelled should be as short as possible. Use Euclidean distances.

Required: Determine a ‘good’ routing for the given problem instance data. Briefly outline the steps of the heuristic that you used to find your routing, and what principle(s) you used. (If you need to use more diagrams, you can find some at the end of this examination paper.)

 SOLUTION

The optimal total coat is 1020

X12= 10, x13=25 , x21= 45 , x23=5 , x32 = 10 , x34 = 30

cost incurred

plant             to             city                       cost incurred

1                                   1                                 0

10

                                      2

25

                                      3

4                              20

2                                    1                              45

2                              0

3                              5

4                                                    0

3                                    1                               0

2                             10

3                               0

4                                                  30

Cost is denoted by xij.

GH53

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