A company produces two products that are processed on two assembly lines. Assembly line$1$ has $100$ available hours, and assembly line $2$ has $42$ available hours. Each product requires $10$ hours of processing time on line $1,$ while on line $2$ product $1$ requires $7$ hours and product $2$ requires $3$ hours. The profit for product $1$ is $$6$ per unit, and the profit for product $2$ is $$4$ per unit. $(5$pts$)$A$.$ Formulate a linear programming model for this problem. $($Write only objective function and constraint functions$)2.$ What is the primary goal of linear programming? $(2$pts$)$a$.$ Maximize or minimize a linear objective functionb. Solve nonlinear equationsc. Optimize a quadratic objective functiond. Minimize the number of constraints $3.$ In linear programming, a constraint is: $(2$pts$)$a$.$ The objective function to be maximizedb. limitation or condition on decision variablesc. The lower and upper bounds on decision variablesd. The optimal solution to the problem $4.$ What does the feasible region in linear programming represent? $(2$pts$)$a$.$ The set of all possible objective function valuesb. The set of all integer solutionsc. The set of all infeasible solutionsd. The set of all feasible solutions that satisfy the constraints $($q$5-$q$10)$ Determine the following operating Characteristics for the Multiple$-$Server System in Queuing:Lambda$=4,$ mu$=5,$ and c$=2$ servers $($use multi$-$server equation to solve$)5.$ Probability that no customers are in the system $(2$ pts$)6.$ Number of customers, on average, in the system $(2$ pts$)7.$ Number of customers, on average, waiting to be served $(2$ pts$)8.$ Average time in the system $(2$ pts$)9.$ Average time spent waiting in the line $(2$ pts$)10.$ Probability that a customer must wait for service $(2$ pts$)11.$ What is the relationship between arrival rate $()$ and service rate $()$ for a stable queuing system? $(2$pts$)12.$ What does the "First$-$Come, First$-$Served $($FCFS$)"$ rule imply in queuing theory? $(2$ pts$)$ a$.$ Customers with higher priority are served firstb. Customers are served in the order they arrivec. Customers with lower priority are served firstd. Customers choose their preferred order of service $13.$ Given critical path $25811,$ find mean expected time and project variance and standard deviation. $(10$ pts$)14.$ The critical path helps project managers to: $(2$pts$)$a$.$ Determine the maximum number of team members required.b$.$ Allocate resources based on cost estimates.c$.$ Calculate the float time for non$-$critical activities.d$.$ Identify the minimum time required to complete the project. $15.$ What is float or slack in a project network? $(2$pts$)$a$.$ The amount of time an activity can be delayed without delaying the project.b$.$ The difference between the optimistic and pessimistic time estimatesc. The time required to complete an activity under ideal conditions.d$.$ The total time taken for an activity from start to finish. $16.$ Transshipment model: A concrete company transports concrete from three plants to three construction sites. The supply capacities of the three plants, the demand requirements at the three sites, and the transportation costs $($minimize$)$ per ton are as follows: Determine the linear programming model formulation for this problem. $(10$ pts$)$a$.$ Objective function:b$.$ Constraint functions $($must have $6,$ use X$1$A$,$ X$1$B$,$ X$3$A$,$ etc. notation when setting your constraint functions$)17.$ Suppose you are tasked with ranking three different cars based on three key criteria: $(1)$ Price$=$score of $10$ means very affordable, $1=$not affordable, $(2)$ Fuel Efficiency, and $(3)$ Comfort. Your goal is to employ the TOPSIS $($Technique for Order of Preference by Similarity to Ideal Solution$)$ method to determine the best$-$ranking car.A$.$ Fill out the best and worst numbers for each column below. Based on the best and worst numbers, solve S$+,$ S$-,$ and S$(+)+$ S$(-)$ in each cell below. $(15$ pts$)$CriteriaPriceFuel EfficiencyComfortS$(+)$S$(-)$S$(+)+$ S$(-)$Car $1471$ Car $2654$ Car $31098$ V$+($best$)$ V$-($worst$)$ B$.$ Rank each car based on p$-$ value.CriteriaPRankCar $1$ Car $2$ Car $318.$ The Rainwater Brewery produces beer. The annual fixed cost is $$150,$ and the variable cost per barrel is $$16.$ Price is related to demand, according to the following demand equation:v $=7500-53\mathrm{.}8$pDevelop the nonlinear profit function for the brewery and determine the price that will maximize profit and volume. $(10$ pts$)$A$.$ Price: B$.$ Volume: $19.$ What does Economic Order Quantity $($EOQ$)$ represent in inventory management? $(2$ pts$)$A$)$ The maximum quantity of a product a customer should order.B$)$ The minimum quantity of a product a company should keep in stock.C$)$ The optimal order quantity that minimizes total inventory costs.D$)$ The average quantity of products sold in a given time period.E$)$ The total quantity of products a company can produce in a year$\mathrm{.}20.$ What does the term "Reorder Point" $($ROP$)$ represent in inventory management? $(2$ pts$)$A$)$ The maximum number of units a supplier can deliver at one time.B$)$ The point at which the inventory level triggers a new order to be placed.C$)$ The minimum quantity of units a company should produce in a given time period.D$)$ The average number of units sold per day.E$)$ The total number of units a company aims to sell in a year$\mathrm{.}21.$ In a scenario where lead time is unpredictable and subject to sudden fluctuations, what adjustment to safety stock levels would be most appropriate? $(2$ pts$)$A$)$ Increase safety stock to buffer against potential delays.B$)$ Decrease safety stock to minimize carrying costs.C$)$ Keep safety stock constant and adjust reorder points.D$)$ Utilize a just$-$in$-$time $($JIT$)$ inventory system.E$)$ Implement a fixed$-$order quantity system$\mathrm{.}22.$ The I$-75$ Carpet Discount Store sells Super Shag carpet with an average daily customer demand of $30$ yards and a standard deviation of $5$ yards. The lead time for receiving a new order is $10$ days. The store aims for a service level of $95\%$ with a $5\%$ probability of stockout. What is the primary purpose of calculating the reorder point and safety stock in this scenario? $(2$ pts$)$A$)$ To estimate the average daily demand for Super Shag carpetB$)$ To minimize the lead time for receiving new carpet ordersC$)$ To determine the maximum order quantity for each shipmentD$)$ To ensure a sufficient inventory level to meet customer demand during the lead timeE$)$ To evaluate the effectiveness of the store's discount strategy$23.$ What is the primary purpose of a moving average in time series analysis? $(2$ pts$)$A$)$ To identify outliers in the datasetB$)$ To smooth out short$-$term fluctuations and highlight trends over timeC$)$ To quantify the correlation between variablesD$)$ To estimate the mean of the time seriesE$)$ To measure the volatility of the data points $24.$ Consider a weather model with two states: "Sunny" and "Rainy." The weather transitions between these states based on the following transition matrix: $(15$ pts$)$ A$.$ Create transition matrix below SunnyRainySunny Rainy B$.$ Use Markov chain to predict the possibility $(\%)$ of sunny and rainy for day $2.$