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1. Solve the following LP problem using the corner point graphical method At the optimal solution. cal- culate the slack for each constraint: Maximize profit = 4X + 4Y subject to 3X 5Y s 150 X 2Y s 10 5X 3Y = 150 X. 0 2. Consider this LP formulation: Minimize cost = SX + 2Y subject to X 3Y 2 90 8X 2Y 160 3X 2Y 120 y 70 X. 0 Graphically illustrate the feasible region. and apply the isocost line procedure to indicate which corner point produces the optimal solution What is the cost of this solution? 3. Graphically analyze the following problem: Maximize profit = $4X $6Y subject to X 2Y s 8 hours 6X 4Y 5 24 hours (a) What is the optimal solution? (b) If the first constraint is altered to X + 3Y s 8. does the feasible region or optimal solution change? 4. Automobiles arrive at the drive-through window at a post office at the rate of four every 10 minutes. The average service time is 2 minutes. The Poisson dis- tribution is appropriate for the arrival rate and ser- vice times are exponentially distributed (a) What is the average time a car is in the system? (b) What is the average number of in the system? cars (c) What is the average time cars spend waiting to receive service? (d) What is the average number of cars in line be- hind the customer receiving service? (e) What is the probability that there are no cars at the window? (f) What percentage of the time is the postal clerk busy? (g) What is the probability that there are exactly two cars in the system? 5. Fromproblem #4 above, a second drive-through window is being considered A single line would be formed, and as a car reached the front of the line it would go to the next available clerk. The clerk at the new window works at the same rate as the current one. (a) What is the average time car is in the system? (b) What is the average number of cars in the system? (c) What is the average time cars spend waiting to receive service? (d) What is the average number of cars in line behind the customer receiving service? (e) What is the probability that there are no cars in the system? (f) What percentage of the time are the clerks busy? (g) What is the probability that there are exactly two cars in the system? 6. One mechanic services five drilling machines for a steel plate manufacturer Machines break down on an average of once every 6 working days, and break- downs tend to follow a Poisson distribution The mechanic can handle an average of one repair job per day. Repairs follow an exponential distribution (a) How many machines are waiting for service, on average? (b) How many are in the system. on average? (c) How many drills are in running order, on aver- age? (d) What is the average waiting time in the queue? (e) What is the average wait in the system?

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