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Combinatorics and Optimization
CO 250
Christine Dupont

Assignment 1 Due: Wednesday January 23 at the BEGINNING of class 1. This question will test your ability to understand AMPL code. (a) The following is the .mod ▯le for a particular AMPL model. Write out the general structure of the linear program using standard mathematical notation. # A business assigns workers to a specific shift during the day. # In a work cycle, workers get one day off. set day; # declares a set of days set shift; # declares a set of shifts param minworkers {shift, day} >=0; # declares a 2D-array of non-negative elements # rows are indexed by shifts and columns are indexed by days # minworkers[s,d] = min no. of workers the business needs # during shift s on day d var x {shift,day} >=0; # declares a 2D-array of non-negative variables # where x[s,d] denotes the no. of workers # that will work on shift s but get day d off minimize numworkers: sum {d in day, s in shift} x[s,d]; subject to shiftreq {s in shift, d in day}: sum {dd in day: dd != d} x[s,dd] >= minworkers[s,d]; (b) The following is the .dat ▯le for the same model in part (a). Write out the com- plete linear program using these values. (Write out all variables and constraints.) set day := Mon, Tue, Wed, Thu; set shift := Morning, Afternoon; param minworkers: Mon Tue Wed Thu := Morning 20 14 16 22 Afternoon 23 12 11 21 ; (c) Solve the model from part (a) with the data from part (b) using AMPL. What is the number of workers that the business should hire? How should the workers that are hired be assigned their shifts? 2. Example of a Blending Problem - Optimizing an Oil Re▯nery An oil re▯nery obtains crude oil from 3 di▯erent sources. The re▯nery distills up to 100,000 barrels of crude oil per day in its fractionater. The fractionater heats up the crude oil until various products called DN (distillation naphtha), DHO (distillation heating oil), and DGO (distillation gas oil), are given o▯ in vapour form and collected at various levels. Other non-usable byproducts are also produced and discarded. It costs $0.75 to run the fractionater on a barrel of crude oil. The table below illustrates the outputs of the fractionater for each type of crude. Fractionater Yield (barrels/barrel) from outputs Crude 1 Crude 2 Crude 3 DN 0.17 0.12 0.11 DHO 0.29 0.33 0.31 DGO 0.35 0.37 0.38 Price ($/barrel) 55.50 57.00 56.25 Availability (barrels/day) 65,000 80,000 45,000 DHO and DGO can be sold as heating oil and diesel fuel respectively. They can also be processed further in a catalytic cracker to obtain CN (catalytic naptha). The cracker can only process one of DHO or DGO at a time and can run for 20 hours per day. It cost $1.05 to process any barrel in the cracker. The output of the cracker is in the table below: CN output (barrels/barrel) Time (seconds/barrel) DHO 0.33 11 DGO 0.7 19 The re▯nery blends the outputs of the fractionater and cracker into various products to be sold, listed in the table below: Product recipe Selling Price Daily Demand Gas 1 45% DN, 55% CN $111 50,000 Gas 2 40% DN, 60% CN $108 80,000 Heating Oil 100% DHO $109 60,000 Diesel Fuel 100% DGO
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