Operations Research Assignments -
ASSIGNMENT 1 - Transportation Model
Q1. Find the initial basic feasible solution of the given cost matrix using NWC method.
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R1
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R2
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R3
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Supply
|
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W1
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2
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4
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3
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200
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W2
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1
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5
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6
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400
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W3
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3
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6
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5
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300
|
|
Demand
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250
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300
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350
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|
Q2. Find the initial basic feasible solution of the given cost matrics using least cost method.
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R1
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R2
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R3
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Supply
|
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W1
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2
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4
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3
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100
|
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W2
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1
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5
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6
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400
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W3
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3
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6
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5
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300
|
|
Demand
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250
|
300
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350
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|
Q3. Find the initial basic feasible solution of the given cost matrics using va method.
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R1
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R2
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R3
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Supply
|
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W1
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2
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4
|
3
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200
|
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W2
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1
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5
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6
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400
|
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W3
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3
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6
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5
|
300
|
|
Demand
|
200
|
300
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350
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|
Q4. Find the initial basic feasible solution of the given cost matrics using all five methods that have been discussed in class.
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R1
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R2
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R3
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Supply
|
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W1
|
2
|
3
|
4
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200
|
|
W2
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1
|
4
|
5
|
400
|
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W3
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3
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7
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5
|
300
|
|
Demand
|
250
|
300
|
350
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|
ASSIGNMENT 2 - Decision Theory and Games
Q1. A Big Bazar must decide on the level of supplies it must stock to meet the needs of its customers during Diwali days. The exact number of customers is not known, but it is expected to be in one of the four categories, 200, 250, 400 or 450 customers. For levels of supplies are thus suggested with level j being ideal (from the view point of incurred costs) if the number of customers falls in category j. Deviations from the ideal levels results in additional costs either because extra supplies are needlessly or because demand cannot be satisfied. The Table below provides these costs in thousands of rupees.
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Customer Category
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Supplies level
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A1
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A2
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A3
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A4
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E1
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2
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2
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3
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10
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E2
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4
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1
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2
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15
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E3
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1
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3
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1
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40
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E4
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20
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15
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30
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Find the best alternative under each discussed criterion.
Q2. The probability distribution of monthly sales of an item as follows:
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Monthly Sales
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0
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1
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2
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3
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4
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5
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6
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Probabilities
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0.01
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0.06
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0.25
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0.30
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0.22
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0.10
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0.06
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The cost of carrying inventory (unsold during the month) is Rs 30 per unit per month and cost of unit shortage is Rs 70. Determine optimum stock to minimize expected cost.
Q3. For the following pay off table, transform the zero-sum game into an equivalent linear programming problem and solve it by simplex method.
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Player B
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Player A
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B1
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B2
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B3
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A1
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10
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20
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5
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A2
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13
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9
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12
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A3
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4
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5
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7
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Q4. A and B play a game in which each has three coins, a 5p, a 10p and a 20p. Each selects a coin without the knowledge of the others choice. If the sum coins is odd amount, A wins B's coin and vice versa. Find the best strategy for each player and the value of the game.
Q5. Solve the following games by reducing them to 2x2 games by graphical method.
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Player A
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Player B
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I
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II
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III
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IV
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V
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I
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3
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0
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6
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-1
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7
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II
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-1
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5
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-2
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2
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1
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ASSIGNMENT 3 - Inventory Model
Q1. The probability distribution of monthly sales of an item as follows:
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Monthly Sales
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0
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1
|
2
|
3
|
4
|
5
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6
|
|
Probabilities
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0.01
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0.06
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0.25
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0.30
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0.22
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0.10
|
0.06
|
The cost of carrying inventory (unsold during the month) is Rs 30 per unit per month. The current policy is to maintain a stock of four items at the beginning of each month. Assume that the cost of shortage is proportional to both time and quantity short. Obtain the imputed cost of a shortage of one item for one unit of time.
Q2. A baking company sells cake by the kilogram. It makes a profit of 50 paise a kilo on every kilogram sold on the day it is backed. It disposes of all cakes not sold on the date it is baked at loss of 12 paise a kilogram. If demand is known to be rectangular, between 2000 and 3000 kg, determine the optimum daily amount baked.
Q3. An automobile factory manufactures a particular type of gear within the factory. This gear is used in the final assembly. The particulars of this gear are: demand rate r= 14000 units/year, production rate k = 35,000units/year. Set-up cost is 500 per set up and carrying cost is Rs 15/unit/year. Find the economic batch quantity and cycle time.
Q4. Anil's company buys 2000 bats annually. A fixed cost of Rs. 50 is incurred each time an order is placed. Inventory carrying cost is estimated at 20%. Supplier offers a 10% discount in price per bat of Rs 100 if orders are placed for more than or equal to 150 bats at a time. In what order size should the company purchase.
ASSIGNMENT 4 - Replacement Model and Queuing
Q1. For a certain type of light bulbs( 1000 Nos.), following mortality rates have been observed:
Week: 1, 2, 3, 4, 5
Percent failing by the end of week: 10, 25, 50, 80, 100
Each bulb costs Rs.10 to replace an individual bulb on failure. If all bulbs were replaced at the same time in group it would cost Rs. 4 per bulb. It is under proposal to replace all bulbs at fixed intervals of time, whether or not the bulbs have burnt out. And also it is to continue replacing immediately burnt out bulbs. Determine the time interval at which all the bulbs should be replaced? You will have to compare individual and group replacement.
Q2. The number of glasses of beer ordered per hour at Dick's Pub follows a Poisson distribution, with an average of 30 beers per hour being ordered.
1. Find the probability that exactly 60 beers are ordered between 10 P.M. and 12 midnight.
2. Find the mean and standard deviation of the number of beers ordered between 9 P.M. and 1 A.M.
3. Find the probability that the time between two consecutive orders is between 1 and 3 minutes.
Q3. Indiana Bell customer service representatives receive an average of 1,700 calls per hour. The time between calls follows an exponential distribution. A customer service representative can handle an average of 30 calls per hour. The time required to handle a call is also exponentially distributed. Indiana Bell can put up to 25 people on hold. If 25 people are on hold, a call is lost to the system. Indiana Bell has 75 service representatives.
1. What fraction of the time are all operators busy?
2. What fraction of all calls are lost to the system?
Q4. An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that the average service time for each customer is 4 minutes, and both inter arrival times and service times are exponential. Answer the following questions:
1. What is the probability that the teller is idle?
2. What is the average number of cars waiting in line for the teller? (A car that is beingserved is not considered to be waiting in line.)
3. What is the average amount of time a drive-in customer spends in the bank parking lot lot (including time in service)?
ASSIGNMENT 5 - Dynamic Programming and Simulation
Q1. What is Bellman's Principle of Optimality explain with an example.
Q2. Consider the following lpp.
Maximize Z = 3x1+5x2
Subject to
X1<=4,
X2<=12,
3X1+2X2<=18,
and X1,X2>=0
Solve this problem using Dynamic Programming.
Q3. Pierre's Bakery bakes and sells french bread. Each morning, the bakery satisfies the demand for the day using freshly baked bread. Pierre's can bake the bread only in batches of a dozen loaves each. Each loaf costs 25¢ to make. For simplicity, we assume that the total daily demand for bread also occurs in multiples of 12. Past data have shown that this demand ranges from 36 to 96 loaves per day. A loaf sells for 40¢, and any bread left over at the end of the day is sold to a charitable kitchen for a salvage price of 10¢/loaf. If demand exceeds supply, we assume that there is a lost-profit cost of 15¢/loaf (because of loss of goodwill, loss of customers to competitors, and so on). The bakery records show that the daily demand can be categorized into three types: high, average, and low. These demands occur with probabilities of .30, .45, and .25, respectively. The distribution of the demand by categories is given in Table below. Pierre's would like to determine the optimal number of loaves to bake each day to maximize profit (revenues - salvage revenues -cost of bread - cost of lost profits). Demand Distribution by Demand Categories
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Demand
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36
|
48
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60
|
72
|
84
|
96
|
|
High
|
0.05
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0.10
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0.25
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0.30
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0.20
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0.10
|
|
Average
|
0.10
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0.20
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0.30
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0.25
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0.10
|
0.05
|
|
Low
|
0.15
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0.25
|
0.35
|
0.15
|
0.05
|
0.05
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