1Chapter 1 Solutions121Supplement to the 2nd edition ofOptimization inOperations Research,by Ronald L. Rardin, PearsonHigher Education, Hoboken NJ,○c 2017.2As of June 4, 20151-1.(a)The only unsettled quantity isdecision variables.(b)Given quantities orparameters ared,pandb.(c)Minimize themaximum error, i.e. objective min(d/s)2(d)We must have an integer number of sensorsand not exceed the available budget, i.e.constraintsps≤b,snonnegative and integer.1-2.(a)Feasible because 3.5(4)14≤, andoptimal because any largerswould not befeasible.(b)Infeasible and thus not optimalbecause 3.5(6)/≤14.(c)Feasible because3.5(2)≤14, but not optimal because feasiblesolutions= 4 yields a better objective value.1-3.(a)The only quantities to bedetermined arex1andx2, the numbers of lotson the 2 lines.(b)Given quantities orparameters aret1,t2,c1,c2,bandT.(c)Minimize total production cost or objectiveminc1x1+c2x2.(d)t1x1+t2x2≤T(atmostThours of production),x1+x2=b(produceblots),x1, x2≥0and integer(numbers nonnegative integers).1-4.(a)Infeasible and thus not optimalbecause10(0) + 20(3)/≤40.(b)Feasiblebecause10(2) + 20(1)≤40 and 2 + 1 = 3.Also optimal because no more or lessexpensivex2can be used ifb= 3 lots are torun.(c)Feasible because10(3) + 20(0)≤40and 3 + 0 = 3, but not optimal becausex1= 2,x2= 1yields a lower cost.1-5.(a)Exact numerical optimizationbecause it is the maximum feasible choice forthe given set of parameter values.(b)Descriptive modeling because we have merelyevaluated the consequences of a given choiceof decision variables and parameters.(c)Closed-form optimization because an optimalsolution is specified for each choice of decisionvariables.(d)Heuristic optimization becausea good feasible solution is identified for thegiven choice of parameter values, but anon-usual layout might yield superior results.1-6.(a)Provides optimum for all choices ofinput parameters, not just one.(b)Providesa provably best solution, not just a goodfeasible one.(c)Systematically searches for agood feasible solution, rather than justevaluating the consequences of one.1-7.Higher tractability usually means loss ofvalidity, so results from the model might notbe useful in the application.1-8.(a)(3 for the first)·(3 for the second)·. . .·(3 for thenth) =3ncombinations.(b)One run per second is 3,600 per hour, 86,400per day, 31,536,000 per year. The310= 59,049requires 59,049/3,600 = 16.4hours;315= 14,348,907requires 166.1 days;320≈3.49×109requires 110.6 years; and330≈2.06×1014requires 6.5 million years.(c)Practical computation would be limited toa few days which could accommodate no morethan 10−11 decision variables.1-9.(a)Random variable because short termrainfall is unpredictable.(b)Deterministicquantity because annual rainfall averages arefairly stable.(c)Deterministic quantity be-cause history can be known with certainty.(d)Random variable because future stock marketbehavior is highly uncertain.(e)Deterministicquantity because the seating capacity is fairlyfixed.(f )Random variable because night tonight arrivals are usually variable.(g)Ran-dom variable because breakdowns make the ef-fective production rate uncertain.(h)Deter-ministic quantity because a reliable robot has apredictable rate of production.(i)Determin-istic quantity because short term demand forsuch an expensive product would be fairly wellknown for the next few days.(j)Random vari-able because long term demand for a productis usually uncertain.Preview Mode
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