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FinalExam - OEM Deterministic Methods in Operations...

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OEM: Deterministic Methods in Operations Research Final Exam Please turn off all cell phones. There are five questions. In all questions, please show your work (at least briefly!). Any ansWers without at I.east some explanation will be counted as wrong. Name:
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Question 1 (5 + 5 + 5 + 10 = 25 points). Consider the following "transportation problem." We have 3 supply points, where supply points 1, 2, and 3 have 40,50, and 70 units of supply, respectively. There are 3 demand points, where demand points 1, 2, and 3 have 35, 45, and 70 units of demand, respectively. We may push flow only between supply and demand points (not vice versa, not from supply to supply, and not from demand to demand). We wish to minimize the costs to satisfy all demands. The per unit flow cost from supply node i to demand node j is given by the following table: J , I Cost values Demand 1 Demand 2 Demand 3 Supply 1 $5 $3 $7 Supply 2 $2 $8 $3 Supply 3 $9 $6 $2 Thus, the optimization model is given as follows: Minimize subject to ~ 3X12 +~ + 2X21 +~ + 3X23 +~+ 6X32 + 2X33 Xu + X12 + Xl3 $ 40 (Constraint "supply 1") X21 + X22 + X23 $ 50 (Constraint "supply 2") X31 + X32 + X33 $ 70 (Constraint "supply 3/1) Xu + X21 + X31 = 35 (Constraint "demand 1/1) X12 + X22 + X32 = 45 (Constraint /ldemand 2/1) Xl3 + X23 + X33 = 70 (Constraint "demand 3/1) x~O The optimal solution sends: 40 units from supply 1 to demand 2 ')(\L 35 units from supply 2 to demand 1 X1..-1 5 units from supply 2 to demand 3 )( L.,~ 5 units from supply 3 to demand 2 'j.}L / ( ) 65 units from supply 3 to demand 3 )( -;(~O) +- ~75 i-- }(5")+ G(S) + c(6))=- /(JJ+10+ 15+>0 1-~o~ 565 ~ The shadow prices are: Supply 1: -4 Demand 1: 2 Supply 2: 0 Demand 2: 7 Supply 3: -1 Demand 3: 3 The ranges are: Supply 1: [30,45] Supply 2: [40,00] Supply 3: [60, 75] Demand 1: [0,45] Demand 2: [40,55] Demand 3: [65,80]
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(All parts are independent.) @J.
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