chapter 7 - Chapter 7 Transportation Assignment and...

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Chapter 7 Transportation, Assignment, and Transshipment Problems 2. a. Let x 11 : Amount shipped from Jefferson City to Des Moines x 12 : Amount shipped from Jefferson City to Kansas City x 23 : Amount shipped from Omaha to St. Louis Min 14 x 11 + 9 x 12 + 7 x 13 + 8 x 21 + 10 x 22 + 5 x 23 s.t. x 11 + x 12 + x 13 30 x 21 + x 22 + x 23 20 x 11 + x 21 = 25 x 12 + x 22 = 15 x 13 + x 23 = 10 x 11 , x 12 , x 13 , x 21 , x 22 , x 23 , 0 b. Optimal Solution: Amount Cost Jefferson City - Des Moines 5 70 Jefferson City - Kansas City 15 135 Jefferson City - St. Louis 10 70 Omaha - Des Moines 20 160 Total 435 4. a. P 2 W 2 1 P 3 W P 3 W 1 300 500 100 200 400 300 10 18 12 8 10 10 24 16 20 b. Let x ij = Amount shipped from plant i to warehouse j Min 20 x 11 + 16 x 12 + 24 x 13 + 10 x 21 + 10 x 22 + 8 x 23 + 12 x 31 + 18 x 32 + 10 x 33 s.t. x 11 + x 12 + x 13 300 x 21 + x 22 + x 23 500 x 31 + x 32 + x 33 100 7 - 1
Chapter 7 x 11 + x 21 + x 31 = 200 x 12 + x 22 + x 32 = 400 x 13 + x 23 + x 33 = 300 x ij 0 i = 1, 2, 3; j = 1, 2, 3 Optimal Solution: Amount Cost P 1 - W 2 300 4800 P 2 - W 1 100 1000 P 2 - W 2 100 1000 P 2 - W 3 300 2400 P 3 - W 1 100 1200 10,400 c. The only change necessary, if the data are profit values, is to change the objective to one of maximization. 7 - 2
Transportation, Assignment, and Transshipment Problems 6. a. Los Angeles Washington 5 4 3 Denver 2 Mobile 1 Pittsburg 1 Seattle 2 Columbus 3 New York 9000 4000 8000 3000 5000 4000 6000 3000 4 10 7 20 1 6 30 8 10 2 10 9 5 20 10 b. The linear programming formulation and optimal solution as printed by The Management Scientist are shown. The first two letters of the variable name identify the “from” node and the second two letters identify the “to” node. Also, The Management Scientist prints “<” for “ .” LINEAR PROGRAMMING PROBLEM MIN 10SEPI + 20SEMO + 5SEDE + 9SELA + 10SEWA + 2COPI + 10COMO + 8CODE + 30COLA + 6COWA + 1NYPI + 20NYMO + 7NYDE + 10NYLA + 4NYWA S.T. 1) SEPI + SEMO + SEDE + SELA + SEWA < 9000 2) COPI + COMO + CODE + COLA + COWA < 4000 3) NYPI + NYMO + NYDE + NYLA + NYWA < 8000 7 - 3
Chapter 7 4) SEPI + COPI + NYPI = 3000 5) SEMO + COMO + NYMO = 5000 6) SEDE + CODE + NYDE = 4000 7) SELA + COLA + NYLA = 6000 8) SEWA + COWA + NYWA = 3000 OPTIMAL SOLUTION Objective Function Value = 150000.000 Variable Value Reduced Costs -------------- --------------- ------------------ SEPI 0.000 10.000 SEMO 0.000 1.000 SEDE 4000.000 0.000 SELA 5000.000 0.000 SEWA 0.000 7.000 COPI 0.000 11.000 COMO 4000.000 0.000 CODE 0.000 12.000 COLA 0.000 30.000 COWA 0.000 12.000 NYPI 3000.000 0.000 NYMO 1000.000 0.000 NYDE 0.000 1.000 NYLA 1000.000 0.000 NYWA 3000.000 0.000 c. The new optimal solution actually shows a decrease of \$9000 in shipping cost. It is summarized. Optimal Solution Units Cost Seattle - Denver 4000 \$ 20,000 Seattle - Los Angeles 5000 45,000 Columbus - Mobile 5000 50,000 New York - Pittsburgh 4000 4,000 New York - Los Angeles 1000 10,000 New York - Washington 3000 12,000 Total: \$141,000 7 - 4
Transportation, Assignment, and Transshipment Problems