Network Flow Problems

Network Flow Problems • Transportation • Assignment • Transshipment • Production and Inventory Linear Programming

Network Flow Problems - Transportation • Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: • Northwood – 25 tons • Westwood – 45 tons • Eastwood – 10 tons • BBC has two plants, each of which can produce 50 tons per week. • BBC would like to minimize transportation costs. How should end of week shipments be made to fill the above orders given the following delivery cost per ton? Linear Programming

Network Representation - BBC Transportation Cost per Unit Destinations Plants (Origin Nodes) 1 Northwood 1 Plant 1 $24 25 50 $30 $40 2 Westwood 45 $30 2 Plant 2 $40 50 $42 3 Eastwood 10 Supply Distribution Routes - arcs Demand Linear Programming

Define Variables - BBC Let: xij = # of units shipped from Plant i to Destination j Linear Programming

General Form - BBC Min 24x11+30x12+40x13+30x21+40x22+42x23 s.t. x11 +x12 +x13 <= 50 x21 +x22+ x23 <= 50 x11 + x21 = 25 x12 + x22 = 45 x13 + x23 = 10 xij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand Linear Programming

Network Flow Problems • Transportation Problem Variations • Total supply not equal to total demand • Total supply greater than or equal to total demand • Total supply less than or equal to total demand • Maximization/ minimization • Change from max to min or vice versa • Route capacities or route minimums • Unacceptable routes Linear Programming

Network Flow Problems - Transportation • Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: • Northwood – 25 tons • Westwood – 45 tons • Eastwood – 10 tons • BBC has two plants, each of which can produce 50 tons per week. • BBC would like to minimize transportation costs. How should end of week shipments be made to fill the above orders given the following delivery cost per ton? • Suppose demand at Eastwood grows to 50 tons. Linear Programming

Network Representation - BBC Transportation Cost per Unit Destinations Plants (Origin Nodes) 1 Northwood 1 Plant 1 $24 25 50 $30 $40 2 Westwood 45 $30 2 Plant 2 $40 50 $42 3 Eastwood 50 10 Supply Distribution Routes - arcs Demand Linear Programming

General Form - BBC Min 24x11+30x12+40x13+30x21+40x22+42x23 s.t. x11 +x12 +x13 = 50 x21 +x22+ x23 = 50 x11 + x21 <= 25 x12 + x22 <= 45 x13 + x23 <= 50 xij >= 0 for i = 1, 2 and j = 1, 2, 3 Min 24x11+30x12+40x13+30x21+40x22+42x23 s.t. x11 +x12 +x13 <= 50 x21 +x22+ x23 <= 50 x11 + x21 = 25 x12 + x22 = 45 x13 + x23 = 10 xij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand Linear Programming

Network Flow Problems - Transportation • Building Brick Company (BBC) manufactures bricks. BBC has orders for 80 tons of bricks at three suburban locations as follows: • Northwood – 25 tons • Westwood – 45 tons • Eastwood – 10 tons • BBC has two plants, each of which can produce 50 tons per week. • BBC would like to maximize profit. How should end of week shipments be made to fill the above orders given the following profit per ton? Linear Programming

Network Representation - BBC Transportation Cost per Unit Profit per Unit Destinations Plants (Origin Nodes) 1 Northwood 1 Plant 1 $24 25 50 $30 $40 2 Westwood 45 $30 2 Plant 2 $40 50 $42 3 Eastwood 10 Supply Distribution Routes - arcs Demand Linear Programming

General Form - BBC Max Min 24x11+30x12+40x13+30x21+40x22+42x23 s.t. x11 +x12 +x13 <= 50 x21 +x22+ x23 <= 50 x11 + x21 = 25 x12 + x22 = 45 x13 + x23 = 10 xij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand Linear Programming

Network Flow Problems - Transportation • Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: • Northwood – 25 tons • Westwood – 45 tons • Eastwood – 10 tons • BBC has two plants, each of which can produce 50 tons per week. • BBC would like to minimize transportation costs. How should end of week shipments be made to fill the above orders given the following delivery cost per ton? • BBC has just been instructed to deliver at most 5 tons of bricks to Eastwood from Plant 2. Linear Programming

Network Representation - BBC Transportation Cost per Unit Destinations Plants (Origin Nodes) 1 Northwood 1 Plant 1 $24 25 50 $30 $40 2 Westwood 45 $30 2 Plant 2 $40 50 $42 3 Eastwood 10 At most 5 tons Delivered from Plant 2 Supply Distribution Routes - arcs Demand Linear Programming

General Form - BBC Min 24x11+30x12+40x13+30x21+40x22+42x23 s.t. x11 +x12 +x13 <= 50 x21 +x22+ x23 <= 50 x11 + x21 = 30 x12 + x22 = 45 x13 + x23 = 10 x23 <= 5 xij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand Route Max Linear Programming

Network Flow Problems - Transportation • Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: • Northwood – 25 tons • Westwood – 45 tons • Eastwood – 10 tons • BBC has two plants, each of which can produce 50 tons per week. • BBC would like to minimize transportation costs. How should end of week shipments be made to fill the above orders given the following delivery cost per ton? • BBC has just learned the route from Plant 2 to Eastwood is no longer acceptable. Linear Programming

Network Representation - BBC Transportation Cost per Unit Destinations Plants (Origin Nodes) 1 Northwood 1 Plant 1 $24 25 50 $30 $40 2 Westwood 45 $30 2 Plant 2 $40 50 $42 3 Eastwood 10 Route no longer acceptable Supply Distribution Routes - arcs Demand Linear Programming

General Form - BBC Min 24x11+30x12+40x13+30x21+40x22+42x23 s.t. x11 +x12 +x13 <= 50 x21 +x22+ x23 <= 50 x11 + x21 = 30 x12 + x22 = 45 x13 + x23 = 10 xij >= 0 for i = 1, 2 and j = 1, 2, 3 24x11+30x12+40x13+30x21+40x22 Plant 1 Supply x21 +x22 <= 50 Plant 2 Supply North Demand West Demand x13 = 10 East Demand Linear Programming

Network Flow Problems - Assignment • ABC Inc. General Contractor pays their subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects. • How should the contractors be assigned to minimize total distance (and total cost)? Linear Programming

Network Representation - ABC Transportation Distance Contractors (Origin Nodes) Electrical Jobs (Destination Nodes) 1 West 50 1 A 1 36 1 16 2 Fed 28 1 30 2 B 18 1 35 3 Goliath 32 1 20 3 C 25 25 1 4 Univ. 14 1 Possible Assignments - arcs Supply Demand Linear Programming

Define Variables - ABC Let: xij = 1 if contractors i is assigned to Project j and equals zero if not assigned Linear Programming

General Form - ABC Min 50x11+36x12+16x13+28x21+30x22+18x23+35x31+32x32+20x33+25x41+25x42+14x43 s.t. x11 +x12 +x13 <=1 x21 +x22 +x23 <=1 x31 +x32 +x33 <=1 x41 +x42 +x43 <=1 x11 +x21 +x31 +x41 =1 x12 +x22 +x32 +x42 =1 x13 +x23 +x33 +x43 =1 xij >= 0 for i = 1, 2, 3, 4 and j = 1, 2, 3 Linear Programming

Network Flow Problems • Assignment Problem Variations • Total number of agents (supply) not equal to total number of tasks (demand) • Total supply greater than or equal to total demand • Total supply less than or equal to total demand • Maximization/ minimization • Change from max to min or vice versa • Unacceptable assignments Linear Programming

Network Flow Problems - Transshipment • Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Rockwright) with customized shelving for its offices. Thomas and Washburn both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc. • Currently weekly demands by the users are: • 50 for Zrox, • 60 for Hewes, • 40 for Rockwright. • Both Arnold and Supershelf can supply at most 75 units to its customers. • Because of long standing contracts based on past orders, unit shipping costs from the manufacturers to the suppliers are: • The costs (per unit) to ship the shelving from the suppliers to the final destinations are: • Formulate a linear programming model which will minimize total shipping costs for all parties. Linear Programming

Network Representation - Transshipment Transportation Cost per Unit Transportation Cost per Unit Retail Outlets (Destinations Nodes) Plants (Origin Nodes) Warehouses (Transshipment Nodes) 5 Zrox 1 Arnold 3 Thomas $1 $5 50 75 Resembles Transportation Problem $5 $8 $8 6 Hewes Flow In 150 Flow Out 150 60 $3 $7 2 Super S. 4 Washburn $4 $4 75 $4 7 Rockwright 40 Supply Distribution Routes - arcs Demand Linear Programming

Define Variables - Transshipment Let: xij = # of units shipped from node i to node j Linear Programming

General Form - Transshipment Min 5x13+8x14+7x23+4x24+1x35+5x36+8x37+3x45+4x46+4x47 s.t. x13 +x14 <= 75 x23 +x24 <= 75 x35 +x36 +x37 = x13 +x23 x45 +x46 +x47 = x14 +x24 +x35 +x45 = 50 +x36 +x46 = 60 +x37 +x47 = 40 xij >= 0 for all i and j Flow In 150 Flow Out 150 Linear Programming

Network Flow Problems • Transshipment Problem Variations • Total supply not equal to total demand • Total supply greater than or equal to total demand • Total supply less than or equal to total demand • Maximization/ minimization • Change from max to min or vice versa • Route capacities or route minimums • Unacceptable routes Linear Programming

Network Flow Problems – Production & Inventory • A producer of building bricks has firm orders for the next four weeks. Because of the changing cost of fuel oil which is used to fire the brick kilns, the cost of producing bricks varies week to week and the maximum capacity varies each week due to varying demand for other products. They can carry inventory from week to week at the cost of $0.03 per brick (for handling and storage). There are no finished bricks on hand in Week 1 and no finished inventory is required in Week 4. The goal is to meet demand at minimum total cost. • Assume delivery requirements are for the end of the week, and assume carrying cost is for the end-of-the-week inventory. Linear Programming

Network Representation – Production and Inventory Production Nodes Production Costs Demand Nodes 1 Week 1 5 Week 1 $28 60 58 Inventory Costs $0.03 2 Week 2 6 Week 2 $27 62 36 $0.03 3 Week 3 7 Week 3 $26 64 52 $0.03 4 Week 4 8 Week 4 $29 66 70 Production - arcs Production Capacity Linear Programming Demand

Define Variables - Inventory Let: xij = # of units flowing from node i to node j Linear Programming

General Form - Production and Inventory Min 28x15+27x26+26x37+29x48+.03x56+.03x67+.03x78 s.t. x15 <= 60 x26 <= 62 x37 <= 64 x48 <= 66 x15 = 58+x56 x26 +x56 = 36+x67 x37 +x67 = 52+x78 x48 +x78 = 70 xij >= 0 for all i and j Linear Programming

Network Flow Problems