LP Formulations For Network Problems 1. Transportation Model 2. Transshipment Model 3. Shortest Path Problem
1. Transportation Model -- Model Description: Set of suppliers (factories) with limited capacities Set of buyers (retailers) requesting for products Route between each supplier and buyer Find most efficient shipment plan
-- A Graphical Representation Capacities Suppliers Customers Demands
-- Example: Powerco has three electric power plants that supply the power needs of four cities. Each power plant can supply the amounts shown in Table 1 (in millions of kilowatt-hours of electricity). The peak power demand (again in millions of kwh) at each city is given in Table 2. Finally, the cost (in dollars) of sending a million kwh from each plant to each city is given in Table 3. Table 1: Capacity of each plant Plant Plant 1 Plant 2 Plant 3 Supply 35 50 40
Table 2: Demand of each city City City 1 City 2 City 3 City 4 Demand 45 20 30 30
Table 3: Shipping cost between each plant and city City 1 City 2 City 3 City 4 Plant 1 8 6 10 9 Plant 2 9 12 13 7 Plant 3 14 9 16 5
a. Find the lowest-cost shipment plan for Powerco to meet the demand of the four cities. b. If the connection between Plant 1 and City 2 is not available, resolve the problem. c. If the supply of Plant 1 increases to 40, do we need to change our setup? d. If the demand of City 4 increases to 35, resolve the problem.
-- Excel Data Organization: Unit cost and total cost Shipment plan 2. Transshipment Model (An Extension of Transportation Model) -- Model Description: Set of suppliers (factories) with limited capacities Set of buyers (retailers) requesting for products Set of warehouses in between suppliers and buyers Route between each supplier, warehouse, and buyer Find most efficient shipment plan
-- A Graphical Representation Capacities Suppliers Warehouse Customers Demands
-- An Example: Foodco produces food at three plants (nodes 1, 2, and 3). Food can be shipped directly to customers (node 6 and 7) or it can be shipped to warehouses (node 4 and 5) and then to the customers. The cost of producing food at each plant is the same, so Foodco is concerned with minimizing the total shipping cost incurred in meeting customer demands. The production capacity of each plant (in tons per year) and the demand of each customer are listed in Table 1 and 2. The cost of shipping a ton of food (in thousands of dollars) between each pair of points is given in Table 3, where a dash indicates that Foodco cannot ship from a node to itself. At most 200 tons of food can be shipped between any two nodes. Foodco wants to determine a minimum cost shipping schedule. Table 1: Capacity of each plant Plant Plant 1 (node 1) Plant 2 (node 2) Plant 3 (node 3) Supply 200 300 100
Table 2: Customer demands Customer Customer 1 (node 6) Customer 2 (node 7) Demand 400 180
Table 3: Unit shipping cost To Node 1 2 3 4 5 6 7 1 -- 5 3 2 1 2 4 From 2 9 -- 9 1 1 8 9 Node 3 0.4 8 -- 1 0.5 10 8 4 1 1 0.5 -- 1.2 5 1 5 2 1 0.6 0.8 -- 2 7 6 2 9 1 0.6 0.7 -- 3 7 7 3 6 1 0.3 7 --
-- Solution Procedure: 1. Calculate gap = sum(supplies) - sum(demands). 2. If gap > 0, add a dummy customer with demand = gap, flow-in capacity = gap, and flow-out capacity = 0. 3. If gap < 0, add a dummy supplier with supply = gap, flow-in capacity = 0 and flow- out capacity = gap. 4. The unit shipment costs from and to the dummy are set to 0 in both cases. 5. In the constraint, set net out-flow of each node = given out-flow (supply, -demand, or 0), where net out-flow = out-flow - in-flow.
-- Two Excel Tricks Used in the Example: 1. Define a name for a range: Mark the range, and enter the name in the name cell at the up-left corner of the sheet. 2. Convert a row vector into a column vector: Mark the range of the column vector, enter the formula transpose(range of the row vector) in the first cell of the marked column vector. Press Control, Shift, and then Enter. 3. Shortest path problem and applications
-- Description of the problem Given: A network with nodes and arcs Length of each arc Find: Shortest path from origin to destination
-- Idea of the solution procedure: Treat the length of each arc as unit shipment cost Try to send 1 unit of flow from the origin to the destination with minimum cost
-- A simple example Find the shortest path from node 2 to node 6 in the following network.
5 8 9 6 6 2 6
4 3 6
7 3 4
-- Applications to equipment replacement: Tradeoff between replacement cost and aging cost.
-- Example (best policy for trade-in car) Jane Driver has just purchased (at time 1) a new car for $12,000. The cost of maintaining a car during a year depends on the age of the car at the beginning of the year, as given in the following table. To avoid high maintenance costs associated with an older car, Jane can trade in a car and purchase a new car. The prices she receives on a trade-in are also given in the table. To simplify the computations, we assume that at any time it costs Jane $12,000 to purchase a new car. Jane’s goal is to minimize the net costs (purchasing costs plus maintenance costs minus money received in trade-ins) incurred during the next five years. Age of Car (years) Annual Maintenance Cost Trade-in Price 0 $2000 N/A 1 $4000 $7000 2 $5000 $6000 3 $9000 $2000 4 $12000 $1000 5 N/A $0
-- Network representation of car trade-in problem
Beginning of End of year 5 year 1
-- Calculating cost for each option Cij: the total net cost incurred by owning and operating a car from the beginning of year i to the beginning of year j if a new car is purchased at the beginning of year i and this car is traded in at the beginning of year j.
Cij =
-- Example: C12 = C13 = C14 = ... C23 = C24 = ...
-- Excel data organization: Input data Calculating cost of each option and total cost Trading plan Homework Problems for Network Related Problems:
Problem 1. General Ford produces cars at LA and Detroit and has a warehouse in Atlanta. The company supplies cars to customers in Houston and Tampa. The costs of shipping a car between various points are listed in the following table, where a dash means that a shipment is not allowed. LA can produce up to 1100 cars, and Detroit can produce up to 2900 cars. Houston must receive 2400 cars, and Tampa must receive 1500 cars. Determine how to minimize the cost of meeting demands at Houston and Tampa.
To LA Detroit Atlanta Houston Tampa LA - $140 $100 $90 $225 From Detroit $145 - $111 $110 $119 Atlanta $105 $115 - $113 $78 Houston $89 $109 $121 - - Tampa $210 $117 $82 - - Total Shipping Cost: $420,500
Problem 2. Solve the project management example in my lecture (house construction) by formulating the problem as a shortest (longest) path problem. Find out the minimum project duration and all critical activities. Use the maximum time (b) for each task.
26 days
Problem 3. Solve the car replacement problem by Excel.
Total Cost: $31,000
Problem 4. (Extra Credit, 0.5% bonus toward your final grade) A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box are given in the following table. The variable cost (in dollars) of producing each box is equal to the volume of the box. A fixed cost of $1000 is incurred to produce any of a particular box. If the company desires, demand for a box can be satisfied by a box of a larger size. Determine how to minimize the cost of meeting the demand for boxes.
Box 1 2 3 4 5 6 7 Size 33 30 26 24 19 18 17 Demand 400 300 500 700 200 400 200
Hint: Compare this problem with the car-trade example and try to change the problem to a shortest path problem. For example, one possible solution of the problem could be producing 700 boxes of size 33, 1200 boxes of size 26, and 800 boxes of size 19. This feasible solution can be represented a path in the network representation of this problem.
$72,100