8.2-21 a, b, c, d, e. Make sure that each unit cost contains both a purchasing and a hauling component. For example, the total cost of supplying a ton of gravel from the north pit to building site 1 is $400. In question (e) the adjustment to add to the reduced cost of variable xij is D(cij - ui - vj) = cij, since dual variables ui and vj are constant (basic solution remains unchanged). After you add the adjustment to the corresponding reduced cost, apply the optimality condition to obtain an inequality for cij. Solve the inequality to obtain the range of cij and cij so that the current solution remains optimal.

Note that to invoke an improvement over the current optimal solution the unit cost has to decrease below the allowable range.

Add two more questions to this problem:

(f) Show that the Vogel's approximation method finds the optimal solution obtained in (d);

(g) Formulate the transportation problem using LINGO'S modeling language: Define sets, such as sources, destinations and routes (origin-destination pairs) and solve. Verify the optimal solution by comparing it with the one obtained in (d). Generate the ranges report and verify the answer obtained in (e).

Solve the prototype example of the Generalized Transportation problem with the IOR Tutorial by selecting as Area the "Transportation Problem" and as Procedure "Enter or Revise a Transportation Problem" and "Solve Automatically by the Transportation Simplex Method." Enter costs, supplies and demands in IOR Tutorial as they appear in the balanced transportation tableau (with 6 sources and 7 destinations) in Lesson 2, solve the transportation problem by the "automatic" procedure and verify the solution.

8.2-21. 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. Pit Hauling Cost per Ton at Site 2 3 $100 $190 $160 $180 $110 $140 North South Price per Ton $300 5420 Susan wishes to determine how much to haul from each pit to cach site to minimize the total cost for purchasing and hauling gravel. (a) Formulate a linear programming model for this problem. Using the Big M method, construct the initial simplex tableau ready to apply the simplex method (but do not actually solve). (b) Now formulate this problem as a transportation 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 method. D (C) Susan Meyer notices that she can supply sites 1 and 2 com- pletely 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 corre- sponding BF solution is optimal. D. (d) Starting with the northwest comer rule, interactively apply the transportation simplex method to solve the problem as formulated in part (b). (e) As usual, let ; 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), sup- pose that the value of Cij for each basic variable Xij is fixed at the value given in the parameter table, but that the value of c, for each nonbasic variable X.; 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 Cij, and explain how this information is useful to the contractor. 8.2-21. 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. Pit Hauling Cost per Ton at Site 2 3 $100 $190 $160 $180 $110 $140 North South Price per Ton $300 5420 Susan wishes to determine how much to haul from each pit to cach site to minimize the total cost for purchasing and hauling gravel. (a) Formulate a linear programming model for this problem. Using the Big M method, construct the initial simplex tableau ready to apply the simplex method (but do not actually solve). (b) Now formulate this problem as a transportation 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 method. D (C) Susan Meyer notices that she can supply sites 1 and 2 com- pletely 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 corre- sponding BF solution is optimal. D. (d) Starting with the northwest comer rule, interactively apply the transportation simplex method to solve the problem as formulated in part (b). (e) As usual, let ; 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), sup- pose that the value of Cij for each basic variable Xij is fixed at the value given in the parameter table, but that the value of c, for each nonbasic variable X.; 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 Cij, and explain how this information is useful to the contractor