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# Chapter 06 - CHAPTER 6 NETWORK OPTIMIZATION PROBLEMS...

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1 CHAPTER 6 NETWORK OPTIMIZATION PROBLEMS SOLUTION TO SOLVED PROBLEMS 6 .S 1 Distribution at Heart Beats Heart Beats is a manufacturer of medical equipment. The company’s primary product is a device used to monitor the heart during medical procedures. This device is produced in two factories and shipped to two warehouses. The product is then shipped on demand to four third-­‐party wholesalers. All shipping is done by truck. The product distribution network is shown below. The annual production capacity at Factories 1 and 2 is 400 and 250, respectively. The annual demand at Wholesalers 1, 2, 3, and 4 is 200, 100, 150, and 200, respectively. The cost of shipping one unit in each shipping lane is shown on the arcs. Due to limited truck capacity, at most 250 units can be shipped from Factory 1 to Warehouse 1 each year. Formulate and solve a network optimization model in a spreadsheet to determine how to distribute the product at the lowest possible annual cost. This is a minimum-­‐cost flow problem. To set up a spreadsheet model, first list all of the arcs as shown in B4:C11, along with their capacity (F4) and unit cost (G4:G11). Only the arc from F1 to WH1 is capacitated. Then list all of the nodes as shown in I4:I11 along with each node’s supply or demand (L4:L11).

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2 The changing cells are the amount of flow to send through each arc. These are shown in Flow (D4:D11) below, with an arbitrary value of 10 entered for each. The flow through the arc from F1 to WH1 must be less than the capacity of 250, as indicated by the constraint D4 <= F4. 3 4 5 6 7 8 9 10 11 B C D E F G H I J K L From To Capacity Unit Cost Nodes Supply/Demand F1 WH1 250 \$40 F1 400 F1 WH2 \$35 F2 250 F2 WH2 \$25 WH1 0 WH1 WS1 \$60 WH2 0 WH1 WS2 \$35 WS1 -200 WH2 WS2 \$55 WS2 -100 WH2 WS3 \$50 WS3 -150 WH2 WS4 \$65 WS4 -200 3 4 5 6 7 8 9 10 11 B C D E F G From To Flow Capacity Unit Cost F1 WH1 10 <= 250 \$40 F1 WH2 10 \$35 F2 WH2 10 \$25 WH1 WS1 10 \$60 WH1 WS2 10 \$35 WH2 WS2 10 \$55 WH2 WS3 10 \$50 WH2 WS4 10 \$65
3 For each node, calculate the net flow as a function of the changing cells. This can be done using the SUMIF function. In each case, the first SUMIF function calculates the flow leaving the node and the second one calculates the flow entering the node. For example, consider the F1 node (I4). SUMIF(From, Nodes, Flow) sums each individual entry in Flow (the changing cells in D4:D11) if that entry is in a row where the entry in From (B4:B11) is the same as in that row of Nodes (i.e., F1). Since I4 = F1 and the only rows that have F1 in From (B4:B11) are rows 4 and 5, the sum in the ship column is only over these same rows, so this sum is D4+D5. The goal is to minimize the total cost of shipping the product from the factories to the wholesalers. The cost is the SUMPRODUCT of the Unit Costs with the Flow, or Total Cost = SUMPRODUCT(UnitCost, Flow). This formula is entered into TotalCost (D13).

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