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Chapter 7 Transportation, Assignment, and Transshipment Problems 12. a. b. Min 10x11 + 16x12 + 32x13 + 14x21 + 22x22 + 40x23 + 22x31 + 24x32 + s.t. x11 + x12 + x13 x21 + x22 + x23 x31 + x32 + x11 + x21 + x31 x12 + x22 + x32 x13 + x23 + 34x33 x33 x33 = = = 1 1 1 1 1 1 xij 0 for all i, j Solution x12 = 1, x21 = 1, x33 = 1 13. a. Total completion time = 64 Optimal assignment: Jackson to 1, Smith to 3, and Burton to 2. Time requirement is 62 days. b. Considering Burton has saved 2 days. c. Ellis. 7-1 Chapter 7 14. a. b. Max 100x11 + 95x12 + 85x21 + 80x22 + 90x31 + 75x32 s.t. x11 + x12 x21 + x22 x31 + x32 x11 + x21 + x31 x12 + x22 + x32 = = 1 1 1 1 1 xij 0, i = 1,2,3; j = 1,2 Optimal Solution: x12 = 1, x31 = 1 Value: 185 Assign Bostock to the Southwest territory and Miller to the Northwest territory. 16. a. 7-2 Transportation, Assignment, and Transshipment Problems 1 1 10 Shoe 6 12 8 Location 1 1 15 1 2 18 Toy 5 11 Location 2 17 10 3 1 1 13 Auto Parts 16 Location 14 1 3 12 4 1 Housewares 13 10 14 1 5 Video Location 4 16 6 12 7-3 1 Chapter 7 b. 1 if department i is assigned location j Let xij 0 otherwise Max + + 10x11 + 17x31 + 14x51 + 6x12 + 10x32 + 16x52 + 12x13 + 8x14 + 15x21 + 18x22 13x33 + 16x34 + 14x41 + 12x42 6x53 + 12x54 x11 x12 + x13 + + 5x23 + 11x24 13x43 + 10x44 s.t. x11 + x31 + x51 + + x52 x21 + x22 + x14 x12 x13 x32 + + + + x14 x33 + x34 x53 x31 + + x54 + x23 + x24 x32 + x33 + x21 + x22 + x23 + x24 x41 + x42 + x43 + x41 + x42 + x34 + x51 + x43 + x44 x52 + x53 + x44 x54 = = = = 1 1 1 1 1 1 1 1 1 xij 0 for all i, j Optimal Solution: Toy: Auto Parts: Housewares: Video: Profit: 61 18. a. Location 2 Location 4 Location 3 Location 1 This is the variation of the assignment problem in which multiple assignments are possible. Each distribution center may be assigned up to 3 customer zones. The linear programming model of this problem has 40 variables (one for each combination of distribution center and customer zone). It has 13 constraints. There are 5 supply ( 3) constraints and 8 demand (= 1) constraints. The problem can also be solved using the Transportation module of The Management Scientist. The optimal solution is given below. Plano: Flagstaff: Springfield: Boulder: 19. Assignments Kansas City, Dallas Los Angeles Chicago, Columbus, Atlanta Newark, Denver Total Cost - Cost ($1000s) 34 15 70 97 $216 b. the Nashville distribution center is not used. c. All the distribution centers are used. Columbus is switched from Springfield to Nashville. Total cost increases by $11,000 to $227,000. A linear programming formulation and the optimal solution are given. For the variables, we let the first letter of the sales representatives name be the first subscript and the sales territory be the second subscript. 7-4 Transportation, Assignment, and Transshipment Problems Max 44xWA + 80xWB + 52xWC + 60xWD + 60xBA + 56xBB + 40xBC + 72xBD + 36xFA + 60xFB + 48xFC + 48xFD + 52xHA + 76xHB + 36xHC + 40xHD s.t. xWA + xWB + xWC + xWD xBA + xWA + xBB + xBA xWB xBC + xBD xFA + xFB + xFA + xBB + xWC + + xWD xHA + xHB + xHC + xHD xHA + + + xFD xFB xBC + xFC + xHB xFC xBD + + xHC xFD + xHD 1 1 1 1 = 1 = 1 = 1 = 1 xij 0 for all i, j Optimal Solution Sales Washington - B Benson - D Fredricks - C Hodson - A 80 72 48 52 252 Total 20. A linear programming formulation of this problem can be developed as follows. Let the first letter of each variable name represent the professor and the second two the course. Note that a DPH variable is not created because the assignment is unacceptable. Max 2.8AUN + 2.2AMB + 3.3AMS + 3.0APH + 3.2BUN + ·· · + 2.5DMS s.t. AUN + AUN + AMB BUN + + AMS + BMB + CUN + BUN AMB + + CUN + BMB + AMS + All Variables 0 Optimal Solution: A to MS course B to Ph.D. course C to MBA course D to Undergraduate course Max Total Rating Rating 3.3 3.6 3.2 3.2 13.3 7-5 APH BMS CMB DUN DUN CMB BMS APH + + + BPH CMS + CPH DMB + DMS + + + DMB CMS + DMS BPH + CPH = = = = 1 1 1 1 1 1 1 1 Chapter 7 21. a. Min 150x11 + 210x12 + 170x21 + 270x13 + 230x22 + 180x31 + 220x23 + 230x32 + 160x41 + 225x33 + 240x42 + 230x43 s.t. x11 + x11 x12 x21 + + +x21 x12 x13 x22 x31 + + +x31 +x22 x13 x23 x32 x41 +x41 +x32 +x23 + + x33 x42 + +x42 +x33 +x43 x43 = = = 1 1 1 1 1 1 1 xij for all i, j Optimal Solution: x12 = 1, x23 = 1, x41 = 1 Total hours required: 590 Note: statistician 3 is not assigned. b. The solution will not change, but the total hours required will increase by 5. This is the extra time required for statistician 4 to complete the job for client A. c. The solution will not change, but the total time required will decrease by 20 hours. d. The solution will not change; statistician 3 will not be assigned. Note that this occurs because increasing the time for statistician 3 makes statistician 3 an even less attractive candidate for assignment. 7-6