A project has six activities $$(A$$ to $$F)$$ with respective activity durations $$7,5,6,6,8,4$$ days. The network has three paths $$A-B,C-D$$ and $$E-F.$$ All the activities can be crashed with the same crash cost per day. The number of activities that need to be crashed to reduce the project duration by $$1$$ day is

The sales of a product during the last four years were $$860, 880, 870$$ and $$890$$ units. The forecast for the fourth year was $$876$$ units. If the forecast for the fifth year, using simple exponential smoothing, is equal to the forecast using a three period moving average, the value of the exponential smoothing constant $$\alpha $$ is

Consider a single server queuing model with Poisson arrivals $$\left( {\lambda = 4/hour} \right)$$ and exponential service $$\left( {\mu = 4/hour} \right)$$. The number in the system is restricted to a maximum of $$10.$$ The probability that a person who comes in leaves without joining the queue is

Consider a linear programming problem with two variables and two constraints. The objective function is: Maximize $${x_1} + {x_2}.$$ The corner points of the feasible region are $$(0,0), (0,2), (2,0)$$ and $$(4/3, 4/3).$$

If an additional constraint $${x_1} + {x_2} \le 5$$ is added, the optimal solution is