1.Solve the following linear model graphically to findoptimal values ofthe decision variables𝑥1and 𝑥2that maximize the objective function, z. (FYI, we designate the optimal objective function value as z* and the values of the decision variable that produce this 𝑥1∗and𝑥2∗.)Max𝑧=2𝑥1+𝑥2s.t.𝑥2≤103𝑥1+4𝑥2≤602𝑥1+𝑥2≤30𝑥1≤14𝑥1≥0,𝑥2≥02. Consider the following linear model where 𝑐1has not yet been defined.Max𝑧=𝑐1𝑥1+𝑥2s.t.𝑥1+𝑥2≤6𝑥1+2.5𝑥2≤10𝑥1≥0,𝑥2≥0Use the graphical approach that we covered to find the optimal solution, x*=(𝑥1∗,𝑥2∗)for all values of −∞≤𝑐1≤∞. Hint: First draw the feasible region and notice that there are only a few corner points that can be the optimal solution.Also remember that if the objective function line reaches a maximum and it includes two adjacent corner points, both of these points and all of the points that create the line that connects them are optimal. There is literally an infinite number of optimal solutions … but only one optimal objective function value because if you plug in the values of x1and x2you get the same (max) value.Now,𝑐1controls that slope of the objective function lineand this is the critical concept. Go to any of the corner points and some values of c1that make that point the optimal solution. Then ask yourself how much can the slope increase (or decrease) that will move the optimal solution to an adjacent corner point. That range is the answer to this question.

3. A fabrication shop makes two types of valve actuators for explosive environments–call them A and B. This actuator requires special manufacturing processes so there is one production line dedicatedexclusive to producing them.The line consists of three workstations(w/s 1, w/s 2,andw/s 3).Demand for actuators is robust so you can assume that all aresold as soon as they are produced. Your task is to find the right mix of actuators to produce that maximizesprofit. Production of both types of actuators uses the same linear routing: raw materialsgoes to w/s 1; output of w/s 1 goes to w/s 2; output of w/s 2 goes to w/s 3. At this point the actuators are completed and go to pack and ships. Details of the processing timesareprovided in the technology table below along with the total number of hours available at each w/s and the profitrealized for selling anactuatorof each type.

w/s1(hours)w/s2(hours)w/s3(hours)Profit/actuator($)Type A16.510.01.6875Type B8.513.01.7800Time available(hours)15013722.5Onemember of the management team doesn’t trust math modelsand loudly proclaim the obvious answer is to produce the maximum number of Type A actuators and discontinue Type Bbecause A’s are the most profitable. He argues that even if the model says something different, it is because the analystis “playing with the numbers to make it say what they want itto say.”You need to address thisbecause this person is somewhat influential in addition to being loud.

a) Compute the maximum number of Type A actuators that can be made and stay within the time constraintsof all workstations. What is the profit? For completeness, do the same calculations if you only produce Type B actuators?

b)Now, use linear programming to find the solution that maximizes profit.For now, assume that we can produce fractions of actuatorsso the number to produce can be a continuous variable and we can use the methodology covered in the lessons.To find the number of Type A and Type B actuators that will maximize profit, perform the following step:

•Write the model in equation form.Let xAand xBbe the number of actuators of type A and B to produce, respectively.Hint: You model will look something like problems 1 and 2. If z is the total profit the it will be equal to the profit gained from producing xAunits of Type A plus the profit gained from producing xBunits of Type B. There will be constraints for the each of the w/s’s that ensure the time used to produce xAand xBis less than the total time available.•Solve the model graphically for x*= (*Ax, *Bx)•Compute the optimal profit (z*)that corresponds to x*.•Compare this answer with your answersin part a) and note what you find. Which solution has the higher profit? What differences do you see in how the solutions utilize the w/s’s?c)Sometimes rounding up and down from the optimal solution of the “relaxed”problem(i.e., assuming variables that are really integers are continuous)doesn’t give you the optimal solution to the real/integer problem…and sometimes it does. Conceptually, the reason is because you might be able utilize the constrained resources better a bit further from the optimal solution to the relaxed problem thanjust rounding up and down. In practical problems, it frequently doesn’t matter much but it is an importantpoint for you to understand because when profit margins are close, a little improvement can mean a lot. Anyway,you are now being asked to find the optimal integer solution using brute force enumerationusing a spreadsheet that has six columns:Type AType Bw/s 1w/s 2w/s/3ProfitUse the first two columns to identify all possible solutions. Vary the number of Type A actuators from 3to 10 and, for each number of Type Aactuators, varythe number of Type B

actuatorsfrom 3to 11. That is, there will be 72 rows with the first two columns (Type A, Type B) = (3,3), (3,4), (3,5), … (10,11). Use columns 3 through 5to determine if the solution is feasible. To do this, compute the time required for the workstation to produce the actuations in the first two columnsrequires. Then check to see each is less than the maximum available. (You can use an IF statement if you know how they work or do the checking by hand.) You should be able to do this for the first row/solution, then copy and paste for the rest. Identify any rowsthat areinfeasible by highlighting the cells in those rows.For each of the feasible solutions, compute the profit and identify the one that is the maximum. Is this solution the same as rounding the one is part b up and/or down?

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