3.17 Following is the connectivity matrix of a graph. Use the shortest path la algorithm to find the shortest route from node 1 to node 6. The symbol ode the absence of a path. 5 00 6 00 00 00 1 00 8 00 00 2 00 00 3 00 00 00 2 00 8. 8. 8. 3. 8. 8. LO 8. 8. 8.
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- We have 60 meters of fence and want to fence a triangular shaped area. Please formulate an NLP (do not try to solve) that will enable us to maximize the fenced area (Hint: The area of a triangle with sides of length a, b, and c is ( s (s – a) (s – b) (s – c))1/2, where s is half the parameter of the triangle).The area of a triangle with sides of length a, b, and c iss(s a)(s b)(s c), where s is half the perimeter ofthe triangle. We have 60 ft of fence and want to fence atriangular-shaped area. Formulate an NLP that will enableus to maximize the fenced area.We illustrated how a machine replacement problem can be modeled as a shortest path problem. This is probably not the approach most people would think of when they first see a machine replacement problem. In fact, most people would probably never think in terms of a network. How would you model the problem? Does your approach result in an LP model?
- A person starting in Columbus must visit Great Falls, Odessa, and Brownsville, and then return home to Columbus in one car trip. The road mileage between the cities is shown. Columbus Great Falls Odessa Brownsville Columbus --- 102 79 56 Great Falls 102 --- 47 69 Odessa 79 47 --- 72 Brownsville 56 69 72 --- a)Draw a weighted graph that represents this problem in the space below. Use the first letter of the city when labeling each b) Find the weight (distance) of the Hamiltonian circuit formed using the nearest neighbor algorithm. Give the vertices in the circuit in the order they are visited in the circuit as well as the total weight (distance) of the circuit.Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 2), (0, 3), (2, 2), (2, 0), (1, 1), (2, 1), (3, 3)}. (a) Draw the directed graph of R. (b) Is R reflexive? Explain. (c) Is R symmetric? Explain. (d) Is R transitive? ExplainThe distance between two cities in the United States can be approximated by the following formula, where lat1 and long1 are the latitude and longitude of city 1 and lat2 and long2 are the latitude and longitude of city 2. 69 (lat1 − lat2)2 + (long1 − long2)2 Ted's daughter is getting married, and he is inviting relatives from 15 different locations in the United States. The file Wedding gives the longitude, latitude, and number of relatives in each of the 15 locations. Ted would like to find a wedding location that minimizes the demand-weighted distance, where demand is the number of relatives at each location. Assuming that the wedding can occur anywhere, find the latitude and longitude of the optimal location. (Hint: Notice that all longitude values given for this problem are negative. Make sure that you do not check the option for Make Unconstrained Variables Non-Negative in Solver. Round your answers to three decimal places.) latitude of the optimal wedding location:…
- (5) Consider the following linear programming model with 4 regular constraints:Maximize 3X + 5Y (a) Draw your graph in the space below:subject to: 4X + 4Y ≤ 48 (constraint #1) 4X + 3Y ≤ 50 (constraint #2) 2X + 1Y ≤ 20 (constraint #3) X ≥ 2 (constraint #4) X, Y ≥ 0 (non-negativity constraints)(a) Which of the constraints is redundant? Constraint #______.Justify by drawing a graph similar to Figure 7.14 on p.263.(b) Is point (9,3) a feasible solution? _____. Explain your answer (by analyzing each of the constraints).Constraint #1: _______________________________________________________________Constraint #2: _______________________________________________________________Constraint #3: _______________________________________________________________Constraint #4: _______________________________________________________________(c) Which of the following points yields the best solution? Underline the best solution: (7,5), (9,2), (6,6).Justify your answer (using the data from the above LP…Find solution using BigM (penalty) method.Maximize Z = x1 + 2x2 + 3x3 - x4subject to the constraintsx1 + 2x2 + 3x3 = 152x1 + x2 + 5x3 = 20x1 + 2x2 + x3 + x4 = 10and x1, x2, x3, x4 ≥ 0A health Centre will be built to serve 7 communities. The geographical location of the communities and their population is shown in the table below. Communities A, X =2.5 (km), Y=4.5(km), Population( 000's)=2 Communities B, X =2.5 (km), Y=2.5(km), Population( 000's)=5 Communities C, X =5.5 (km), Y=4.5(km), Population( 000's)=10 Communities D, X =5 (km), Y=2(km), Population( 000's)=7 Communities E, X =8 (km), Y=5(km), Population( 000's)=10 Communities F, X =7 (km), Y=2(km), Population( 000's)=20 Communities G, X =9 (km), Y=2.5(km), Population( 000's)=14 There is a possibility of building the Centre in two communities, community C and F. Based on the Demand-Distance criterion what is your recommendation for the site of the Centre. The distances to be measured based on Rectilinear method. Based on…
- In this problem, we explain why the MST algorithmworks. DefineS minimum spanning treeCt nodes connected after iteration t of MSTalgorithm has been completedCt nodes not connected after iteration t of MSTalgorithm has been completedAt set of arcs in minimum spanning tree after titerations of MST algorithm have beencompleted Suppose the MST algorithm does not yield a minimumspanning tree. Then, for some t, it must be the case that allarcs in At1 are in S, but the arc chosen at iteration t (call itat) of the MST algorithm is not in S. Then S must containsome arc at that leads from a node in Ct1 to a node in Ct1). Show that by replacing arc at with arc at, we can obtain ashorter spanning tree than S. This contradiction proves that allarcs chosen by the MST algorithm must be in S. Thus, theMST algorithm does indeed find a minimum spanning tree.Draw a graph that identifies the feasible region for the following set of constraints. 0.25 A + 0.25 B ≥ 30 0.5 A + 5 B ≥ 200 0.75 A + 1.5 B ≤ 150 A, B ≥ 0Example 1. The Big M Method solve LP with mixed constraints Minimize: Z = 4x1 + 2x2 + x3 Subject to: 2x1 + 3x2 + 4x3 ≤ 14 3x1 + x2 + 5x3 ≥ 4 x1 + 4x2 + 3x3 ≥ 6 x1, x2, x3≥ 0