OR 6205 HW-S Due by EOD 12/2 Problem 1: A contractor, Susan Meyer, has to haul gravel to three building sites. She can purchase as much as 18 tons at a gravel pit in the north of the city and 14 tons at one in the south. She needs 10,5, and 10 tons at sites 1, 2, and 3, respectively. The purchase price per ton at each gravel pit and the hauling cost per ton are given in the table below Haullng Cost per Ton at Slle Flt 1 2 Price per Ton North 3100 $190 1300 South 180 110 40 420 Susan wishes to determine how much to haul from each pit to each site to minimize the total cost for purchasing and hauling gravel, a) Formulate a linear programming model for this problem. b) Now formulate this problem as a u'ansportation problem by constructing the appropriate parameter table, Compare the size of this table (and the corresponding transportation simplex tableau) used by the transportation simplex method with the size of the simplex tableaux from part (a) that would be needed by the simplex methodi Susan Meyer notices that she can supply sites 1 and 2 completely from the north pit and site 3 completely from the south pit Use the optimality test (but no iterations) of the transportation simplex method to check whether the corresponding BF solution is optimal. Starting with the northwest corner rule, interactively apply the transportation simplex method to solve the problem as formulated in part (b) As usual, let c5, denote the unit cost associated with source i and destination j as given in the parameter table constructed in part (b). For the optimal solution obtained in part (d ), suppose that the value of c,' for each basic variable xi] is xed at the value given in the parameter table, but that the Value of cu for each nonbasic Variable xIJ possibly can be altered through bargaining because the site manager wants to pick up the business Use sensitivity analysis to determine the allowable range for each of the latter c,\" and explain how this information is useful to the contractor O Show that the Vogel's approximation method nds the optimal solution obtained in (d) Problem 2: The coach of an age group swim team needs to assign swimmers to a ZOO-yard medley relay team to send to the Junior Olympics. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and the best times (in seconds) they have achieved in each of the strokes (for 50 yards) are stroke Chris Backstroke 323 Breaststroke 3 3.1 Buttery 23.5 Freestyle 26.4 The coach wishes to determine how to assign four swimmers to the four different strokes to minimize the sum of the corresponding best times. a) Formulate this problem as an assignment problem. b) Obtain an optimal solution