a Allocate the units to the cells by Northwest Corner Stone method in the following table. Plants 1 Warehouses B $8 A $4 C Supply $8 56 2 $16 $24 $16 82 3 Demand $8 $16 72 $24 102 41 77 215 b What is the total cost at this stage? c Evaluate the open cells for optimality. d What is the maximum saving you can generate by moving one unit on the loop and how many units should you move? e Make the move in the following table. Plants 1 Warehouses B $8 A $4 C Supply $8 56 2 $16 $24 $16 82 3 $8 $16 Demand 72 $24 102 41 77 215 f What is the total savings you can generate? g Evaluate the open cells for optimality. h What is the maximum saving you can generate by moving one unit on the loop and how many units should you move? i Make the move in the following table. Plants 1 Warehouses B $8 A $4 C Supply $8 56 2 $16 $24 $16 82 3 Demand $8 $16 72 $24 102 41 77 215 j Evaluate the open cells for optimality. k What is the maximum saving you can generate by moving one unit on the loop and how many units should you move? l Make the move in the following table. Plants 1 Warehouses B $8 A $4 C Supply $8 56 2 $16 $24 $16 82 3 Demand $8 $16 72 $24 102 41 77 215 m Evaluate the open cells for optimality. The following tables are spare ones. Plants 1 Warehouses B $8 A $4 C Supply $8 56 2 $16 $24 $16 82 3 $8 $16 Demand Plants 1 72 $24 102 Warehouses B $8 A $4 41 C 77 215 Supply $8 56 2 $16 $24 $16 82 3 Demand $8 $16 72 $24 102 41 77 215 a. Allocate the units to the cells by Northwest Corner Stone method in the following table. Warehouses Plants 1 A B $4 C $8 Supply $8 56 2 $16 56 $24 16 3 $8 $16 66 $16 82 $24 36 41 77 Demand 72 102 41 215 Please go to the powerpoint slides file for this problem for a step by step details. b. What is the total cost at this stage? $224 + $256 + $1,584 + $576 + $984 =$3,624 c. Evaluate the open cells for optimality. A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C3+B3-B2+A2-A1 =+8-24+16-24+16-4 =-12 C2: +C2-C3+B3-B2 =+16-24+16-24 =-16 d. What is the maximum saving you can generate by moving one unit on the loop and how many units should you move? $16 on +C2-B2+B3-C3, you can move 41 units. e. Make the move in the following table. Warehouses Plants 1 A B $4 C $8 Supply $8 56 2 $16 56 $24 $16 16 3 $8 25 $16 41 82 $24 77 77 Demand 72 102 41 215 f. What is the total savings you can generate? $16 x 41 = $656; The total costs = $3,624 - $656 = $2,968 g. Evaluate the open cells for optimality. Warehouses Plants 1 A B $4 C $8 Supply $8 56 2 $16 56 $24 $16 16 3 $8 25 $16 41 $24 77 Demand A3: B1: C1: C3: +A3-A2+B2-B3 +B1-B2+A2-A1 +C1-C2+A2-A1 +C3-B3+B2-C2 72 82 77 102 =+8-16+24-16 =+8-24+16-4 =+8-16+16-4 =+24-16+24-16 41 =0 =-4 =4 =16 215 h. What is the maximum saving you can generate by moving one unit on the loop and how many units should you move? $4 on +B1-A1+A2-B2 and you can move 25 units. i. Make the move in the following table. Warehouses Plants 1 A B $4 $8 31 2 $16 $8 25 $24 56 $16 41 3 $8 Supply C 41 $16 $24 77 Demand 72 82 102 77 41 215 j. Evaluate the open cells for optimality. Warehouses Plants 1 A B $4 $8 31 2 $16 $8 25 $24 56 $16 41 3 $8 Supply C 41 $16 82 $24 77 Demand 72 102 =+8-4+8-16 =+24-16+4-8 =+8-16+16-4 =+24-16+8-4+16-16 77 41 215 A3: B2: C1: C3: +A3-A1+B1-B3 +B2-A2+A1-B1 +C1-C2+A2-A1 +C3-B3+B1-A1+A2-C2 =-4 =4 =4 =12 k. What is the maximum saving you can generate by moving one unit on the loop and how many units should you move? $4 on +A3-B3+B1-A1 and you can move 31 units. l. Make the move in the following table. Warehouses Plants 1 A B $4 C $8 Supply $8 56 2 $16 $24 56 $16 41 3 $8 $16 Demand m. 41 82 $24 31 46 72 102 77 41 215 Evaluate the open cells for optimality. Warehouses Plants 1 A B $4 C $8 Supply $8 56 2 $16 $24 56 $16 41 3 Demand $8 41 $16 82 $24 31 46 72 102 77 41 215 A1: +A1-B1+B3-A3 =+4-8+16-8 =4 B2: +B2-B3+A3-A2 =+24-16+8-16 =0 C1: +C1-C2+A2-A3+B3-B1 =+8-16+16-8+16-8 =8 C3: +C3-A3+A2-C2 =+24-8+16-16 =16 The evaluations of the open cells show no more savings. Stop here. Rules for Northwest Corner Stone Method: First a. Start by allocating units in the upper left hand corner. b. End by allocating units in the lower right hand corner. c. In making each allocation decision, consider whether the supply or demand is smaller and use this amount. d. Always proceed to exhaust supply and/or demand before moving downward or to the right in the rows and columns of the matrix. Second a. The open cell being evaluated should be treated as an increasing cost, since products are being moved into it. This is designated by a plus sign. b. Paths should be constructed from the open cell to an occupied cell in either a vertical or horizontal direction. Diagonal paths are not permitted. c. Right-angle turns should be made, only occupied cell being used, until the path leads back to the open cell. d. A completed path must involve an equal number of increased costs (+) and decreased costs (-). e. The plus and minus signs must balance each other, so that the supply and demand restraints are not exceeded. Using Northwest Corner method. The following costs apply to shipping one unit from each factory to each warehouse. Work out a solution which gives you the lowest total shipping costs. You should: a. Allocate all the units from their sources to destinations, starting from northwest corner. What are the total costs? b. Evaluate the open cells by moving one unit to the open cells (show your work). On which route you can generate largest savings per unit move? By how much? c. Choose the cell that the maximum saving can be generated. d. What is the total number of units that can be moved in to this cell? e. What is the total savings that can be generated? f. Continue to evaluate the open cells after the first move. The optimal solution is reached when there is no negative value in the open cells. What are the optimal costs? Warehouses B Plants 1 $4 $8 $8 Supply 56 2 $16 $24 $16 82 3 $8 $16 $24 77 Demand A 72 102 C 41 215 Plants 1 $4 2 $16 3 $8 Demand Warehouses B A 56 16 $8 $8 Supply 56 $24 $16 82 $24 77 $16 72 66 36 102 C 41 41 215 b. What is the total cost at this stage? =$4*56+$16*16+$24*66+$16*36+$24*41=$3,624 c. Evaluate all open cells Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand A3: 16 $16 72 +A3-A2+B2-B3 66 36 102 41 41 =+8-16+24-16 215 =0 Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand 16 $16 72 66 36 102 41 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand 16 $16 72 66 36 102 41 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C3+B3-B2+A2-A1 =+8-24+16-24+16-4 =-12 Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand 16 $16 72 66 36 102 41 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C3+B3-B2+A2-A1 =+8-24+16-24+16-4 =-12 C3: +C2-C3+B3-B2 =+16-24+16-24 =-16 Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand 16 $16 72 66 36 102 41 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C3+B3-B2+A2-A1 =+8-24+16-24+16-4 =-12 C3: +C2-C3+B3-B2 =+16-24+16-24 =-16 Any saving? Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand 16 $16 72 66 36 102 41 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C3+B3-B2+A2-A1 =+8-24+16-24+16-4 =-12 C3: +C2-C3+B3-B2 =+16-24+16-24 =-16 C3 has $16 saving, the largest. Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand 16 $16 72 66 36 102 41 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C3+B3-B2+A2-A1 =+8-24+16-24+16-4 =-12 C3: +C2-C3+B3-B2 =+16-24+16-24 =-16 C3 has $16 saving, the largest. Then how many to move? Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand 16 $16 72 66 36 102 41 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C3+B3-B2+A2-A1 =+8-24+16-24+16-4 =-12 C3: +C2-C3+B3-B2 =+16-24+16-24 =-16 C3 has $16 saving, the largest. Move 41 units since 41 is smaller than 66. total saving is $16*41=$656. Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 $24 77 56 2 $16 3 $8 Demand 16 $16 72 66 36 102 41 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B1: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C3+B3-B2+A2-A1 =+8-24+16-24+16-4 =-12 C3: +C2-C3+B3-B2 =+16-24+16-24 =-16 C3 has $16 saving, the largest. Move 41 units since 41 is smaller than 66. total saving is $16*41=$656. make move in the following slide Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 56 2 $16 3 $8 Demand 16 $16 72 25 77 102 Evaluate all open cells 41 $24 77 41 215 Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 56 2 $16 3 $8 Demand A3: 16 $16 72 +A3-A2+B2-B3 25 77 102 41 $24 77 41 =+8-16+24-16 215 =0 Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 56 2 $16 3 $8 Demand 16 $16 72 25 77 102 41 $24 77 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B2: +B1-B2+A2-A1 =+8-24+16-4 =-4 Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 56 2 $16 3 $8 Demand 16 $16 72 25 77 102 41 $24 77 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B2: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 56 2 $16 3 $8 Demand 16 $16 72 25 41 $24 77 102 77 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B2: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B2-C2 =+24-16+24-16 =16 Any saving? Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 56 2 $16 3 $8 Demand 16 $16 72 25 41 $24 77 102 77 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B2: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B2-C2 =+24-16+24-16 =16 B2 has $4 saving. Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 56 2 $16 3 $8 Demand 16 $16 72 25 77 102 41 $24 77 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B2: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B2-C2 =+24-16+24-16 =16 B2 has $4 saving. Move 25 units since 25 is smaller than 41. total saving is $4*25=$100. Plants 1 Warehouses B A $4 C $8 $8 Supply 56 $24 $16 82 56 2 $16 3 $8 Demand 16 $16 72 25 77 102 41 $24 77 41 215 A3: +A3-A2+B2-B3 =+8-16+24-16 =0 B2: +B1-B2+A2-A1 =+8-24+16-4 =-4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B2-C2 =+24-16+24-16 =16 B2 has $4 saving. Move 25 units since 25 is smaller than 41. total saving is $4*25=$100. make move in the following slide Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 72 C 25 77 102 41 77 41 215 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 72 A3: +A3-A1+B1-B3 C 25 77 102 41 77 41 =+8-4+8-16 215 =-4 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 72 C 25 77 102 41 77 41 215 A3: +A3-A1+B1-B3 =+8-4+8-16 =-4 B2: +B2-A2+A1-B1 =+24-16+4-8 =4 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 72 C 25 77 102 41 77 41 215 A3: +A3-A1+B1-B3 =+8-4+8-16 =-4 B2: +B2-A2+A1-B1 =+24-16+4-8 =4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 C 25 72 77 102 41 77 41 215 A3: +A3-A1+B1-B3 =+8-4+8-16 =-4 B2: +B2-A2+A1-B1 =+24-16+4-8 =4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B1-A1+A2-C2 =+24-16+8-4+16-16 =12 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 C 25 72 77 102 41 77 41 215 A3: +A3-A1+B1-B3 =+8-4+8-16 =-4 B2: +B2-A2+A1-B1 =+24-16+4-8 =4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B1-A1+A2-C2 =+24-16+8-4+16-16 =12 Any saving? Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 C 25 72 77 102 41 77 41 215 A3: +A3-A1+B1-B3 =+8-4+8-16 =-4 B2: +B2-A2+A1-B1 =+24-16+4-8 =4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B1-A1+A2-C2 =+24-16+8-4+16-16 =12 A3 has $4 saving. Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 C 25 72 77 102 41 77 41 215 A3: +A3-A1+B1-B3 =+8-4+8-16 =-4 B2: +B2-A2+A1-B1 =+24-16+4-8 =4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B1-A1+A2-C2 =+24-16+8-4+16-16 =12 A3 has $4 saving. How many to move? Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 72 C 25 77 102 41 77 41 215 A3: +A3-A1+B1-B3 =+8-4+8-16 =-4 B2: +B2-A2+A1-B1 =+24-16+4-8 =4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B1-A1+A2-C2 =+24-16+8-4+16-16 =12 A3 has $4 saving. Move 31 units since 31 is smaller than 77. total saving is $4*31=$124. Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 31 2 $16 3 $8 Demand 41 72 C 25 77 102 41 77 41 215 A3: +A3-A1+B1-B3 =+8-4+8-16 =-4 B2: +B2-A2+A1-B1 =+24-16+4-8 =4 C1: +C1-C2+A2-A1 =+8-16+16-4 =4 C3: +C3-B3+B1-A1+A2-C2 =+24-16+8-4+16-16 =12 A3 has $4 saving. Move 31 units since 31 is smaller than 77. total saving is $4*31=$124. make move in the following slide Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 C 56 2 $16 3 $8 Demand 41 31 72 46 102 41 77 41 215 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 C 56 2 $16 3 $8 Demand 41 31 72 A1: +A1-B1+B3-A3 46 102 =+4-8+16-8 41 77 41 215 =4 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 C 56 2 $16 3 $8 Demand 41 31 72 46 102 41 77 41 215 A1: +A1-B1+B3-A3 =+4-8+16-8 =4 B2: +B2-B3+A3-A2 =+24-16+8-16 =0 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 C 56 2 $16 3 $8 Demand 41 31 72 46 102 41 77 41 215 A1: +A1-B1+B3-A3 =+4-8+16-8 =4 B2: +B2-B3+A3-A2 =+24-16+8-16 =0 C1: +C1-C2+A2-A3+B3-B1 =+8-16+16-8+16-8 =8 Plants 1 Warehouses B A $4 $8 Supply 56 $24 $16 82 $16 $24 $8 C 56 2 $16 3 $8 Demand 41 31 72 46 102 41 77 41 215 A1: +A1-B1+B3-A3 =+4-8+16-8 =4 B2: +B2-B3+A3-A2 =+24-16+8-16 =0 C1: +C1-C2+A2-A3+B3-B1 =+8-16+16-8+16-8 =8 C3: +C3-A3+A2-C2 =+24-8+16-16 =16 The evaluations of the open cells show no more savings. Stop here