Problem 2. If all the capacities in a network are integer-valued, show that the maxi- mum flow returned by the Max-Flow Min-Cut Algorithm is integer-valued.
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- Scott and Associates, Inc., is an accounting firm that has three new clients. Project leaders will be assigned to the three clients. Based on the different backgrounds and experiences of the leaders, the various leader-client assignments differ in terms of projected completion times. The possible assignments and the estimated completion times in days are as follows. Client Project Leader Jackson Elis Smith 22 24 24 (a) Develop a network representation of this problem. (Submit a file with a maximum size of 1 MB.) This answer has not been graded yet Min (b) Formulate the problem as a linear program. (Express your answers in the form x, where x, represents the completion time from leader / to client J.) st. Jackson Elis Smith Client 1 Client 2 Client 3 x20 for all, J. Solve Relationship Assignment Time (days) "11 *12 *13 *21 10 16 32 *22 *31 *32 711 What is the total time required (in days)? daysQ1) Four workers are to be assigned for 4 different jobs on the basis of the worker's relative skill levels on the various machines. Four Jobs are available, so assign one job to worker. In order to maximize profitability, we wish to minimize the total cost of the assignment. Use the cost matrix given in the following, and the Hungarian assignment algorithm, to determine the optimal assignment of workers to machines, and give the cost of the optimal assignment. J1 J2 J3 J4 W1 8 26 17 11 W2 13 28 26 W3 38 19 18 15 W4 19 26 24 10 4.Five jobs are to be assigned to 5 machines to minimize the total time required to process the jobs on machines. The times in hours for processing each job on each machine are given in the matrix below. By using assignment algorithm make the assignment for minimizing the time of processing. Jobs U V W X Y A 2 4 3 5 4 B 7 4 6 8 4 C 2 9 8 10 4 D 8 6 12 7 4 E 2 8 5 8 8
- To test the effectiveness of a job training program on the subsequent wages of workers, we specify the model log(wage) = Bo + Bitrain + Bzeduc + B;exper + u, where train is a binary variable equal to unity if a worker participated in the program. Think of the error term u as containing unobserved worker ability. If less able workers have a greater chance of being selected for the program, and you use an OLS analysis, what can you say about the likely bias in the OLS estimator of B,? (Hint: Refer back to Chapter 3.)An account on server A is more expensive than an account on server B. However, server A is faster. To see if it’soptimal to go with the faster but more expensive server, a manager needs to know how much faster it is. A certaincomputer algorithm is executed 30 times on server A and 20 times on server B with the following results:T-Value = -3.12 P-Value = 0.002 DF = 48Is server A really faster? How strong is the evidence? Formulate the suitable hypothesis and alternative and conclude at5% level of significance.Needs Complete solution with 100 % accuracy.
- 3.9. Is the reduced echelon form of a matrix unique? Justify your conclusion. Namely, suppose that by performing some row operations (not necessarily fol- lowing any algorithm) we end up with a reduced echelon matrix. Do we always end up with the same matrix, or can we get different ones? Note that we are only allowed to perform row operations, the "column operations"' are forbidden.7. Use the algorithm introduced in section 2.2 to find the inverse, if it exists, of the following matrix. -2 -3 1 -3 1 -4 1 -3 -28. Show the adjacency matrix for this graph, where vertex i is represented by the i-th row and i-th column. Then show and explain how to find the total number of walks of length 3 in the graph that either start or end at vertex 5 by using the adjacency matrix. (A walk is a path which may repeat edges of the graph. Be careful not to double count walks that start AND end at vertex 5! Draw this graph above your solution. 1 6 5 4
- 5. Use the Gaussian Elimination Algorithm to solve the following linear systems, if possible, and deter- mine whether row interchanges are necessary: - + 3x =2, 3x,-3x+ y =-1. Handwriting 2-1.5x+3x= 1, + 2r, = 3, 4x,-4.5x+5x, 1. = 3.Explain why it is problematic to include a constraintsuch as the following in an LP model for a blendingproblem:Total octane in gasoline 1 blendBarrels of gasoline 1 blended daily$ 102. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1
