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Unformatted text preview: Space IOE 202: lecture 5 outline Announcements Last time... Restricting the variables to be integer Other modeling possibilities with integer variables IOE 202: Operations Modeling, Fall 2009 Page 1 Space Last time Linear programming models:
Shipping model Blending model IOE 202: Operations Modeling, Fall 2009 Page 2 Space Postal employee scheduling A post oﬃce requires diﬀerent numbers of employees on diﬀerent days of the week. The number of employees required is as follows: Day Min. required Mon 17 Tue 13 Wed 15 Thu 19 Fri 14 Sat 16 Sun 11 Union rules state that each employee must work 5 consecutive days and then receive 2 days oﬀ. For example, an employee might work Wednesday through Sunday, and be oﬀ Monday and Tuesday. The post oﬃce wants to minimize the number of employees it needs to hire while meeting the daily requirements. Decisions to make: The employee schedule in this problem is completely determined by how many employees start their shifts on each of the seven days of the week. IOE 202: Operations Modeling, Fall 2009 Page 3 Space Modeling Postal Employee schedule Schedules: Start Day Mon Tue Wed Thu Fri Sat Sun Mon Tue Wed Thu Fri Sat Sun Variables: x1  number of employees starting on Mondays x2  number of employees starting on Tuesdays ··· x7  number of employees starting on Sundays
IOE 202: Operations Modeling, Fall 2009 Page 4 Space LP model for Postal Employee schedule min x1 s.t. x1 x1 x1 x1 x1 +x2 +x3 +x4 +x4 +x2 +x2 +x3 +x2 +x3 +x4 +x2 +x3 +x4 x2 +x3 +x4 x3 +x4 x 1 , x2 , x3 , x 4 , +x5 +x6 +x5 +x6 +x5 +x6 +x6 +x7 +x7 +x7 +x7 +x7 +x5 +x5 +x6 +x5 +x6 +x7 x 5 , x6 , x7 (Number hired) ≥ 17 (Monday) ≥ 13 (Tuesday) ≥ 15 (Wednesday) ≥ 19 (Thursday) ≥ 14 (Friday) ≥ 16 (Saturday) ≥ 11 (Sunday) ≥0 Let’s solve with Excel Solver! IOE 202: Operations Modeling, Fall 2009 Page 5 Space Integer values needed! The optimal solution of the Post Oﬃce staﬃng problem given by Excel: 1 1 1 1 x1 = 1 , x2 = 5 , x3 = 0, x4 = 7 , x5 = 0, x6 = 3 , x7 = 5, 3 3 3 3 with the objective value of 22 1 . 3 Since hiring parttime employees (or fractional people!) is not allowed, we need to ensure that only integer values of the decision variables are considered. To do so in Excel: In the Solver dialog box, add a group of constraints on the decision variables to be integer. New optimal solution: x1 = 6, x2 = 6, x3 = 0, x4 = 7, x5 = 0, x6 = 4, x7 = 0, with the objective value of 23.
IOE 202: Operations Modeling, Fall 2009 Page 6 Space Postal employee scheduling: modiﬁed min x1 s.t. x1 x1 x1 x1 x1 +x2 +x3 +x4 +x4 +x2 +x2 +x3 +x2 +x3 +x4 +x2 +x3 +x4 x2 +x3 +x4 x3 +x4 x 1 , x2 , x3 , x 4 , +x5 +x6 +x5 +x6 +x5 +x6 +x6 +x7 +x7 +x7 +x7 +x7 +x5 +x5 +x6 +x5 +x6 +x7 x 5 , x6 , x7 (Total number hired) ≥ 17 (Monday) ≥ 13 (Tuesday) ≥ 15 (Wednesday) ≥ 19 (Thursday) ≥ 14 (Friday) ≥ 16 (Saturday) ≥ 11 (Sunday) ≥ 0, integer Note that the above is still a linear model, but with integer variables; problems of this type are referred to as “(Linear) Integer Programs,” or IPs. Solver tip: Under “Options,” set Tolerance to 0 to get the optimal solution.
IOE 202: Operations Modeling, Fall 2009 Page 7 Space Use of integer and binary variables
Integer variables are used in models in which values of some or all variables represent nondivisible quantities (number of employees hired, number of airplanes manufactured, etc.) Binary variables (i.e., variables that are allowed to take on only values of 0 and 1) are useful when we need to decide whether or not to undertake an activity. Contrast this with divisible, or general integer variables, which model the quantity, or level of the activity. Through various “tricks of the trade,” many complex situations involving both “yesno” and quantitative decisions can be represented and solved as Integer Programming models. IOE 202: Operations Modeling, Fall 2009 Page 8 Space Selecting courses to satisfy requirements To get an IOE minor, a student must take at least two Math courses, at least three IOE courses, and at least two EECS courses.
Course Math IOE EECS Math214 IOE310 EECS280 IOE265 IOE373 IOE366 EECS283 Math214 is a prerequisite for IOE310 IOE265 is a prerequisite for IOE366 Credit is not given for both EECS280 and EECS283 What is the least number of courses the student can take to satisfy the major requirements? Note: Here, we need to make a “yesno” decision about taking each course. IOE 202: Operations Modeling, Fall 2009 Page 9 Space Formulation of a mathematical model of course selection Decision variables: represent decisions by variables. Objective function: express the performance criterion in terms of the decision variables; should it be minimized of maximized? IOE 202: Operations Modeling, Fall 2009 Page 10 Space Formulation of a model of course selection – cont. Constraints: express all (explicit and implicit) constraints and restrictions on the values of the decision variables. Optimal solution:
IOE 202: Operations Modeling, Fall 2009 Page 11 Space Combining binary and divisible variables To clean up a polluted river, the state is going to build some pollution control stations. Three sites are under consideration. Two diﬀerent pollutants need to be controlled, and the state legislature requires that at least 80,000 tons of pollutant 1 and at least 50,000 tons of pollutant 2 be removed. The relevant data is indicated in the table below (the last two columns indicate how much of each pollutant each station removes by processing 1 ton of water). Cost of building Cost of treating Pollutant 1 Pollutant 2 1 ton of water removed removed Site 1 $120,000 $20 0.30 0.30 Site 2 $60,000 $30 0.50 0.20 Site 3 $40,000 $40 0.10 0.50 What is the cheapest way of satisfying the legislation requirements? IOE 202: Operations Modeling, Fall 2009 Page 12 Space Operational decisions in the pollution problem
What decisions do you need to make? What performance measure are you using to compare diﬀerent decisions? What constraints (restrictions) must your decisions satisfy? What assumptions are being made? IOE 202: Operations Modeling, Fall 2009 Page 13 Space Observations about the pollution problem We are not required to build all three stations! We get to decide which ones to build. These decisions are represented by binary variables. We get to decide how much water to process at each station. These decisions are represented by divisible variables. We cannot process water at nonexistent stations! Need to link the values of the divisible and binary variables via constraints. IOE 202: Operations Modeling, Fall 2009 Page 14 Space Formulation of a mathematical model for the pollution problem Decision variables: represent decisions by variables. Objective function: express the performance criterion in terms of the decision variables; should it be minimized of maximized? IOE 202: Operations Modeling, Fall 2009 Page 15 Space Formulation of a model for the pollution problem – cont. Constraints: express all (explicit and implicit) constraints and restrictions on the values of the decision variables. Optimal solution:
IOE 202: Operations Modeling, Fall 2009 Page 16 Space An investment model Presently, you have $1,000 to invest. Cash ﬂows associated with 5 available investments are shown in the table; you can put no more than $500 in any investment. In addition to these investments, you can invest as much money as you want into 12month CDs, which pay 6% interest. How should you invest to maximize your cash at hand at the end of year 3? Investment A B C D E Now $1.00 $1.00 $1.00 Year 1 +$1.15 +$1.28 $1.00 $1.00 +$1.15 +$1.32 Year 2 Year 3 +$1.40 Added twist: For each investment you put money in, you need to pay a brokerage fee of $50 at the time of investment. Note: each year, you can only invest cash available on hand!
IOE 202: Operations Modeling, Fall 2009 Page 17 Space Formulation of a mathematical model for the investment problem Decision variables: Objective function: Maximize cash on hand at the end of Year 3: IOE 202: Operations Modeling, Fall 2009 Page 18 Space Mathematical model for the investment problem – continued Constraints: Cash invested ≤ cash on hand:
Year 0: Year 1: Year 2: At most $500 in any investment and can’t invest unless brokerage fee is paid! Restrictions on variables: IOE 202: Operations Modeling, Fall 2009 Page 19 ...
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 Fall '09
 MarinaEpelman

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