Question 1 Which of the following is ALWAYS true when solving a feasible linear program by graph? The objective function line does not have the same slope as any of the constraints The feasible region is bounded by the vertical axis ($x_1 = 0$), the horizontal axis ($x_2 = 0$) or both. An optimal solution occurs at the intersection of two or more constraints The feasible region must be bounded. 1 pts
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The profit function for two products is Profit = -3x1^2 + 42x1 - 3x2^2 + 50x2 + 700, where x1 represents units of production of product 1 and x2 represents units of production of product 2. Producing one unit of product 1 requires 4 labor-hours and producing one unit of product 2 requires 6 labor-hours. Currently, 24 labor-hours are available. The cost of labor-hours is already factored into the profit function. However, it is possible to schedule overtime at a premium of $5 per hour. (a) Formulate an optimization problem that can be used to find the optimal production quantity of product 1 and the optimal number of overtime hours to schedule. (Let OT be the number of overtime hours scheduled.) Max s.t. <= 24 + OT x1, x2, OT >= 0 (b) Solve the optimization model you formulated in part (a). How much should be produced and how many overtime hours should be scheduled? What is the profit (in dollars)? (Round your answers to three decimal places.) Units of product 1 x1 = Units of product 2 x2 = Overtime hours OT = Profit P = $
Sri K.
Consider two firms, Firm A and B. Suppose a job only has two attributes: wages (W) and safety (S). Assume that Firm A earns a predetermined sales revenue of $500 when it employs one worker and it costs the firm $50 per unit of safety. Therefore, the firm’s total cost can be expressed as TC = W + 50S. Firm B’s predetermined sales revenue is $300 when employing one worker and the firm’s cost of providing one unit of safety is $20. Firm B’s total cost can be expressed as TC=W+20S. For simplicity, assume that each firm hires only one worker. a) (1.5 points) Draw the isoprofit schedule for both Firm A and B when both firms earn zero profits. Make sure to draw the two isoprofit schedules on the same graph (place W on the vertical axis and S on the horizontal axis). Graphically illustrate the employer’s offer curve. b) (3 points) Suppose there are three workers, David, Kate, and John with different preferences over wages and safety. Assume that their utility functions can be expressed as follows: • David: U(W,S)=WS • Kate: U(W,S) = min{W,20S} • John: U(W,S) = W+ 30S For each worker, do the following: 1) Solve for the optimal (W*, S*). 2) Explain which firm the worker would prefer to work at. 3) Graphically illustrate how the optimum is found. c) (1 point) Suppose the government implements a new policy such that all firms must provide a minimum of 6 units of safety to hire a worker. Graphically illustrate how the isoprofit schedules of the two firms and the employer’s offer curve change. d) (1.5 points) Assume that David, Kate, and John have the same utility function as in part b). Given the new policy described in part c), which firm will they work for? Also, solve for the optimal (W*,S*) for each individual.
Akash M.
The manufacturer makes two types of electric hammers, heavy-duty and regular. The heavy-duty model requires 3 hours to assemble and %2 hour to finish and package. Regular one requires 2 hours to assemble and 1 hour to finish and package. The maximum number of assembly hours is 24 per day, and the maximum number of hours available for finishing and packaging is 8 per day. If the profit on the heavy-duty model is $90 per unit and the profit on the regular model is $72 per unit, how many of each model will maximize profit? Steps (show work for credit). 1) Set up a system of inequalities, find constraints, and solve. 2) Graph the constraints and find the feasible region. 3) Find a point that produces the maximum profit. Show work and use full sentences to answer the question
Supreeta N.
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