Operations Research

A queueing system has one single server workstation that admits an infinitely long queue. The rate of arrival of jobs to the queueing system follows the Poisson distribution with a mean of 5 jobs/hour. The service time o...

A set of jobs $U, V, W, X, Y, Z$ arrive at time $t = 0$ to a production line consisting of two workstations in series. Each job must be processed by both workstations in sequence (i.e., the first followed by the second)....

In a supplier-retailer supply chain, the demand of each retailer, the capacity of each supplier, and the unit cost in rupees of material supply from each supplier to each retailer are tabulated below. The supply chain ma...

At the current basic feasible solution (bfs) $v_0 (v_0 \in \mathbb{R}^5)$, the simplex method yields the following form of a linear programming problem in standard form: minimize $z = -x_1 - 2x_2$ s.t. $x_3 = 2 + 2x_1 -...

In a linear programming problem, if a resource is not fully utilized, the shadow price of that resource is

A manufacturing unit produces two products Pl and P2. For each piece of P1 and P2, the table below provides quantities of materials M1, M2, and M3 required, and also the profit earned. The maximum quantity available per...

Parts P1 - P7 are machined first on a milling machine and then polished at a separate machine. Using the information in the following table, the minimum total completion time required for carrying out both the operations...

An assignment problem is solved to minimize the total processing time of four jobs (1, 2, 3 and 4) on four different machines such that each job is processed exactly by one machine and each machine processes exactly one...

For a single server with Poisson arrival and exponential service time, the arrival rate is $$12$$ per hour. Which one of the following service rates will provide a steady state finite queue length?

Two models, $$P$$ and $$Q,$$ of a product earn profits of Rs. $$100$$ and Rs. $$80$$ per piece, respectively. Production times for $$P$$ and $$Q$$ are $$5$$ hours and $$3$$ hours, respectively, while the total production...

Maximize $$\,\,\,\,\,\,\,\,\,Z = 5{x_1} + 3{x_2}$$ Subject to $$\,\,\,\,\,\,\,\,\,\,{x_1} + 2{x_2} \le 10,$$ $$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_1} - {x_2} \le 8, \cr & \,\,...

Maximize $$\,\,\,\,Z = 15{x_1} + 20{x_2}$$ Subject to $$\eqalign{ & 12{x_1} + 4{x_2} \ge 36 \cr & 12{x_1} - 6{x_2} \le 24 \cr & \,\,\,\,\,\,\,\,\,{x_1},\,\,{x_2} \ge 0 \cr} $$ The above linear programming problem has

In a single-channel queuing model, the customer arrival rate is $$12$$ per hour and the serving rate is $$24$$ per hour. The expected time that a customer is in queue is _______ minutes.

A manufacturer has the following data regarding a product: Fixed cost per month $$=$$ Rs. $$50,000$$ Variable cost per unit $$=$$ Rs. $$200$$ Selling price per unit $$=$$ Rs. $$300$$ Production capacity $$=$$ $$1500$$ un...

For the linear programming problem: $$\eqalign{ & Maximize\,\,\,\,\,Z = 3{x_1} + 2{x_2} \cr & Subject\,\,to\,\,\,\, - 2{x_1} + 3{x_2} \le 9 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...

In the notation $$(a/b/c) : (d/e/f)$$ for summarizing the characteristics of queuing situation, the letters $$‘b’$$ and $$‘d’$$ stand respectively for

A minimal spanning tree in network flow models involves

Jobs arrive at a facility at an average rate of $$5$$ in an $$8$$ hour shift. The arrival of the jobs follows Poisson distribution. The average service time of a job on the facility is $$40$$ minutes. The service time fo...

The jobs arrive at a facility, for service, in a random manner. The probability distribution of number of arrivals of jobs in a fixed time interval is

Consider an objective function $$Z\left( {{x_1},{x_2}} \right) = 3{x_1} + 9{x_2}$$ and the constraints $$\eqalign{ & {x_1} + {x_2} \le 8, \cr & {x_1} + 2{x_2} \le 4, \cr & {x_1} \ge 0,{x_2} \ge 0, \cr} $$ The maximum val...

If there are $$m$$ sources and $$n$$ destinations in a transportation matrix, the total number of basic variables in a basic feasible solution is

The total number of decision variables in the objective function of an assignment problem of size $$n\,\, \times \,\,n$$ ($$n$$ jobs and $$n$$ machines) is

At a work station, $$5$$ jobs arrive every minute. The mean time spent on each job in the work station is $$1/8$$ minute. The mean steady state number of jobs in the system is __________

Customers arrive at a ticket counter at a rate of $$50$$ per hr and tickets are issued in the order of their arrival. The average time taken for issuing a ticket is $$1$$ $$min.$$ Assuming that customer arrivals form a P...

A linear programming problem is shown below. $$\eqalign{ & Maximize\,\,\,\,3x + 7y \cr & Subject\,\,to\,\,\,3x + 7y \le 10 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4x + 6y \le 8 \cr & \,\,\,\,\,\,\,\,\,\,\,\...

One unit of product $${P_1}$$ requires $$3$$ $$kg$$ of resource $${R_1}$$ and $$1$$ $$kg$$ of resource $${R_2}$$. One unit of product $${P_2}$$ requires $$2$$ $$kg$$ of resource $${R_1}$$ and $$2$$ $$kg$$ of resource $${...

One unit of product $${P_1}$$ requires $$3$$ $$kg$$ of resource $${R_1}$$ and $$1$$ $$kg$$ of resource $${R_2}$$. One unit of product $${P_2}$$ requires $$2$$ $$kg$$ of resource $${R_1}$$ and $$2$$ $$kg$$ of resource $${...

Cars arrive at a service station according to Poisson's distribution with a mean rate of $$5$$ per hour. The service time per car is exponential with a mean of $$10$$ minutes. At state, the average waiting time in the qu...

Little’s law is relationship between

Simplex method of solving linear programming problem uses

Consider the following Linear Programming problem $$(LLP)$$ Maximize: $$Z = 3{x_1} + 2{x_2}$$ $$\,\,$$ Subject $$\,\,$$ to $$\eqalign{ & \,\,\,\,\,\,\,{x_1} \le 4 \cr & \,\,\,\,\,\,\,{x_2} \le 6 \cr & 3{x_1} + 2{x_2} \le...

In an $$M/M/1$$ queuing system, the number of arrivals in an interval of length $$T$$ is a Poisson random variable (i.e., the probability of there being $$n$$ arrivals in an interval of length $$T$$ is $${{{e^{ - \lambda...

Consider the Linear programme $$(LP)$$ Max $$4x$$ + $$6y$$ Subject to $$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,3x + 2y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,2x + 3y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x,y \ge 0 \c...

Capacities of production of an item over $$3$$ consecutive months in regular time are $$100,$$ $$100$$ and $$80$$ and in overtime are $$20,$$ $$20$$ and $$40.$$ The demands over those $$3$$ months are $$90,$$ $$130$$ and...

The number of customers arriving at a railway reservation counter is Poisson distributed with an arrival rate of eight customers per hour. The reservation clerk at this counter takes six minutes per customer on an averag...

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).$...

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).$...

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 $$10.$$...

A company has two factories $${S_1},$$ $${S_2}$$ and two warehouses $${D_1},$$ $${D_2}$$ . the supplies from $${S_1}$$ and $${S_2}$$ are $$50$$ and $$40$$ units respectively. Warehouse $${D_1},$$ requires a minimum of $$...

A company produces two types of toys: $$P$$ and $$Q.$$ Production time of $$Q$$ is twice that of $$P$$ and the company has a maximum of $$2000$$ time units per day. The supply of raw material is just sufficient to produc...

A maintenance service facility has Poisson arrival rates, negative exponential service time and operates on a ‘first come first served’ queue discipline. Breakdowns occur on an average of $$3$$ per day with a range of ze...

A manufacturer produces two types of products, $$1$$ and $$2,$$ at production levels of $${x_1}$$ and $${x_2}$$ respectively. The profit is given is$$2{x_1} + 5{x_2}.$$ The production constraints are $$$\eqalign{ & {x_1}...

The supply at three sources is $$50, 40$$ and $$60$$ units respectively whilst the demand at the four destinations is $$20, 30, 10$$ and $$50$$ units. In solving this transportation problem

A furniture manufacturer produces chairs and tables. The wood-working department is capable of producing $$200$$ chairs or $$100$$ tables or any proportionate combinations of these per week. The weekly demand for chairs...

Arrivals at a telephone booth are considered to be Poisson, with an average time of $$10$$ minutes between successive arrivals. The length of a phone call is distributed exponentially with mean $$3$$ minutes. The probabi...

Production flow analysis $$(PFA)$$ is a method of identifying part families that uses data from

Solve the following linear programming problem by simplex method $$\eqalign{ & Maximize\,\,\,\,\,\,4{x_1} + 6{x_2} + {x_3} \cr & Subject\,\,to\,\,\,\,\,\,2{x_1} - {x_2} + 3{x_3}\, \le 5 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,...

Cellular manufacturing is suitable for

In a single server infinite population queuing model, arrivals follow a Poisson distribution with mean $$\lambda = 4$$ per hour. The service times are exponential with mean service time equal to $$12$$ minutes. The expec...

At a production machine, parts arrive according to a Poisson process at the rate of $$0.35$$ parts per minute. Processing time for parts have exponential distribution with mean of $$2$$ minutes. What is the probability t...

The cost of providing service in a queuing system increases with

People arrive at a hotel in a Poisson distributed arrival rate of $$8$$ per hour. Service time distribution is closely approximated by the negative exponential. The average service time is $$5$$ minutes. Calculate (a) th...

In an assembly line for assembling toys, five workers are assigned tasks which take times of $$10, 8, 6, 9$$ and $$10$$ minutes respectively. The balance delay for the line is

If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that

The manufacturing area of a plant is divided into four quadrants. Four machines have to located one in each quadrant. The total number of possible layouts is

On the average $$100$$ customers arrive at a place each hour, and on the average the server can process $$120$$ customers per hour. What is the proportion of time the server is idle?