Sample Midterm 5 Questions and Review

Agenda aPractice Exam to help you prepare for Tuesday aQuestions 1 Practice Exam: Problem 1 aAutopower Europe trying to ship motors from the ports to the plants. aWhat are the basic variables in this optimal solution? aSensitivity Report 2 Basic Variables aHow many should there be? `Answer: 6 `Why: There are 7 constraints (3 ports, 4 plants), but one is redundant aWhat must the 6 basic variables be? `The 6 with positive values `Careful, sometimes basic variables have value 0, but they always have Reduced Cost of 0. That’s how reduced cost is defined. 3 Question 2 aIs this the only optimal solution? aSensitivity Report 4 Unique Optimum? aNo. aHow do we know: The reduced cost of Amsterdam to Leipzig is zero. We can move flow onto this edge without increasing the objective. aCareful, if some of the basic variables had been zero, we may not have been able to move any flow onto this edge. 5 Question 3: aOne carrier would like to win business between Antwerp and Tilburg. At what price per unit would Autopower be willing to use this route? aSensitivity Report 6 Reduced Cost aCurrent Cost per unit is $110 aReduced Cost is $118 aEach unit we send costs $110 but decreases the objective value by $118. aThe carrier would have to pay us $8 for each unit we sent on this edge before it would be attractive! 7 Question 4 aIf we could shift a small amount of production from one plant to another, all else being equal, what plant would it be best to move production from and what plant should we move the production to? aSensitivity Report 8 Shadow Price aMove it from the plant with the highest shadow price and to the plant with the lowest shadow price. aSo, from Leipzig to Liege. aThe net effect should be to reduce the objective by 120 - 41 = $79 for each unit we move. 9 Modeling aDemands for Autopower's products outstrip its available supply of motors. In fact, demand in Liege has risen to 300. We estimate the cost of backordering (failing to meet current demand) at each plant to be: aPlant aLeipzig aLiege aNancy aTilburg Cost per motor backordered $50 $70 $30 $100 10 Backordering Autopower Transportation Model Unit Cost Leipzig From/To Amsterdam $ 120.0 Antwerp $ 61.0 The Hague $ 102.5 Backorder $ 50.0 Shipments From/To Amsterdam Antwerp The Hague Backorder Total Required Leipzig 300 100 400 400 Nancy Liege $ 130.0 $ 41.0 $ 40.0 $ 100.0 $ 90.0 $ 122.0 $ 30.0 $ 70.0 Nancy 700 200 900 900 Liege 300 300 300 Tilburg $ 59.5 $ 110.0 $ 42.0 $ 100.0 Tilburg Total 200 300 500 500 Total Cost Leipzig Nancy Liege Tilburg From/To $ $ 12,300 $ 11,900 Amsterdam $ Antwerp $ $ 28,000 $ $ The Hague $ 30,750 $ 18,000 $ $ 12,600 $ $ $ 5,000 $ Total $ 35,750 $ 46,000 $ 12,300 $ 24,500 500 700 800 100 - Available 500 700 800 100 Total $ 24,200 $ 28,000 $ 61,350 $ 5,000 $ 113,550 11 AMPL Model aset PORT; aset PLANT; aparam supply {PORT}; aparam demand {PLANT}; aparam cost {PORT, PLANT}; aparam BackOrderCost{PLANT}; 12 Model Cont’d var Trans {PORT, PLANT} >= 0; var BackOrder{PLANT} >= 0; minimize Total_Cost: sum{f in PORT, t in PLANT} cost[f,t]*Trans[f,t] + sum{t in PLANT} BackOrderCost[t]*BackOrder[t]; s.t. Supply {p in PORT}: sum{t in PLANT} Trans[p, t] <= supply[p]; s.t. MaxBackOrder: sum{t in PLANT} BackOrder[t] <= sum{t in PLANT}demand[t] - sum{p in PORT} supply[p]; s.t. Demand {t in PLANT}: sum{p in PORT} Trans[p,t] + BackOrder[t] >= demand[t]; 13 Modeling Time aSuppose the Singapore Electric Generator Company must send its completed generators to Australia for testing before they can be sold. This process takes an entire month so that, for example, generators made in January are not available for sale until February. Extend the Singapore Electric Generator Model accommodate this delay. 14 Modeling Time Singapore Electric Generator Production Unit Costs Dec Jan Feb Mar Apr. Production $ 29.00 $ 28.00 $ 27.00 $ 27.80 $ 29.00 Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty Production Limits Beginning Inventory Delivery Reqmts Ending Inventory 50 7 29 60 15 58 7 62 62 7 36 - 38 64 0 34 28 Production Cost $ 1,450.00 $ 812.00 $ 1,674.00 $ 1,056.40 $ Inventory Cost $ 3.30 $ 3.30 $ 1.05 $ 4.20 $ Total Cost $ 1,453.30 $ 815.30 $ 1,675.05 $ 1,060.60 $ May 0 66 28 59 Minimum 7 7 5.25 Total 5.25 $ 5,009.50 15 Short Answer aGive an example of an optimal, but not basic solution to a network flow problem. 16 Optimal, but not basic Netherlands 300-∆ 500 200+∆ Amsterdam The Hague * ∆ * 800 * Antwerp 700 700 500 Tilburg * Germany 200 * Liege 400-∆ 400 * Leipzig 200 200 * Nancy 900 Miles 0 50 100 17 Basic Variables aHow many basic variables will there be in a basic feasible solution to a Transportation Problem like Autopower's, but with 4 ports and 5 plants? a8 aWhy? 9 Constraints, but 1 is redundant. 18 Exam Structure aDistribute Word document and Excel Spreadsheets aReturn Word document and Excel spreadsheets, etc. aKeep time to 2 hrs. aAyla and Lincoln in class during class time. 19