or practiceproblems 2015

8/10/2019 Or PracticeProblems 2015

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OR Problems ( All Topics )

First Topic Linear Programming Problems for practice

Q.1 A manufacturing company is engaged in producing three types of products: A,B and C. The production department produces, each day, components sufficient tomake 50 units of A, 25 units of B and 30 units of C. The management is confrontedwith the problem of optimizing the daily production of products in assemblydepartment where only 100 man hours are available daily to assemble the

products.The following additional information is available.

Type of product Profit contribution perunit of product (Rs)

Assembly time perproduct (hrs)

A 12 0.8B 20 1.7

C 45 2.5

The company has a daily order commitment for 20 units of product A and atotal of 15 units of products B and C. Formulate this problem as an LP model so asto maximize the total profit.

Q.2 A company has two plants, each of which produces and supplies two products

A and B. The plants can each work up to 16 hours a day. In plant 1, it takes threehours to prepare and pack 1,000 gallons of A and one hour to prepare and pack onequintal of B. In plant 2, it takes two hours to prepare and pack 1,000 gallons of Aand 1.5 hours to prepare and pack one quintal B. In Plant 1 it costs Rs. 15,000 to

prepare and pack a thousand gallons of A, and RS. 28,000 to prepare and pack aquintal of B, whereas these costs are Rs 18,000 and Rs 26,000, respectively in

plant 2. The company is obliged to produce daily at least 10 thousand gallons of Aand 8 quintals of B.

Formulate this problem as an LP model to find out as to how the company

should organize its production so that the required amounts of the two products beobtained at minimum cost.

Q.3 An electronic company is engaged in the production of two components C1and C2used in radio sets. Each unit of C1costs the company Rs 5 in wages and RS5 in material, while each of C2costs the company Rs 25 in wages and RS 15 inmaterial. The company sells both products on one period credit terms, but the


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companys labour and material expenses must be paid in cash. The selling price of

C1is Rs 30 per unit and of C2it is Rs 70 per unit. Because of the strong monopolyof the company for these components, it is assumed that the company can sell atthe prevailing prices as many units as it produces. The companys productioncapacity is, however, limited by two considerations. First, at the beginning of

period , the company has an initial balance of Rs 4,000 (cash plus bank credit pluscollections from past credit sales). Second, the company requires production ofeach C1 , 2 hours of machine time and 2 hours of assembly time, whereas the

production of each C2requires 2 hours of machine time and 3 hours of assemblytime. Total machine time available is 2000 hrs whereas total assembly timeavailable 1400 hrs in the given period. Formulate this problem as an LP model soas to maximize the total profit to the company.

Q.4 A company has two grades of inspectors 1 and 2, who are to be assigned for a

quality control inspection. It is required that at least 2,000 pieces be inspected per 8hour day. Grade 1 inspector can check pieces at the rate of 40 per hour, with anaccuracy of 97 per cent. Grade 2 inspector checks at the rate of 30 pieces per hourwith an accuracy of 95 per cent.

The wage rate of a Grade 1 inspector is Rs 5 per hour while that of a Grade 2inspector is Rs 4 per hour. An error made by an inspector costs Rs 3 to thecompany. There are only nine Grade 1 inspectors and eleven Grade 2 inspectorsavailable in the company. The company wishes to assign work to the availableinspectors so as to minimize the total cost of the inspection. Formulate this

problem as an LP model so as to minimize daily inspection cost.


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Q.5 An electronic company produces three types of parts for automatic washingmachines. It purchases casting of the parts from a local foundry and then finishesthe part on drilling, shaping and polishing machines.

The selling prices of parts A, B and C, respectively are Rs 8, Rs 10 and Rs14. All parts made can be sold. Castings for parts A, B and C, respectively cost Rs5, Rs 6 and Rs 10.

The shop possesses only one of each type of machine. Costs per hour to runeach of the three machines are Rs 20 for drilling, Rs 30 for shaping and RS 30 for

polishing. The capacities (parts per hour) for each part on each machine are shownin the following table :

Machine Capacity per hour

Part A Part B Part C

Drilling 25 40 25

Shaping 25 20 20

Polishing 40 30 40The management of the shop wants to know how many parts of each type it

should produce per hour in order to maximize profit for an hours run. Formulate

this problem as an LP model so as to maximize total profit to the company.

Q.6 A pharmaceutical company produces two pharmaceutical products. A and B.Production of both products requires the same process, I and II. The production ofB results also in a byproduct C at no extra cost. The product A can be sold at a

profit of Rs 3 per unit and B at a profit of Rs 8 per unit. Some of this byproduct

can be sold at a unit profit of Rs 2, the remainder has to be destroyed and thedestruction cost is Re 1 per unit. Forecasts show that up to 5 units of C can be sold.The company gets 3 units of C for each unit of B produced. The manufacturingtimes are 3 hours per unit for A on process I and II, respectively, and 4 hours and 5hours per unit for B on process I and II, respectively. Because the product C resultsfrom producing B, no time is used in producing C. The available times are 18 and21 hours of process I and II, respectively. Formulate this problem as an LP modelto determine the quantity of A and B which should be produced, keeping C inmind, to make the highest profit to the company.

Q.7 A tape recorder company manufacturers models A, B and C which have profit

contributions per unit of Rs 15, Rs 40 and RS 60, respectively. The weekly

minimum production requirements are 25 units for model. A, 130 units for model

B and 55 units for model C. Each type of recorder requires a certain amount of


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time for the manufacturing of component parts, for assembling and for packing.

Specifically a dozen units of model A require 4 hours for manufacturing, 3 hours

for assembling and 1 hour for packaging. The corresponding figures for a dozen

units of model B are 2.5, 4 and 2 and for a dozen units of model C are 6, 9 and 4.

During the forthcoming week, the company has available 130 hours of

manufacturing, 170 hours of assembling and 52 hours of packaging time.

Formulate this problem as an LP model so as to maximise total profit to the

company.

Q.8 ABC company manufactures three grades of paint Venus, Diana and Aurora.

The plant operates on a three shift basis and the following data is available fromthe production records :

Requirmentof resource

Grade Venus Diana Aurora Availability(capacity/month)

Specialadditive(kg/litre)

0.30 0.15 0.75 600 tonnes

Milling(kilolitres per

machineshift)

2.00 3.00 5.00 100 machineshifts

Packing(kiloliters pershift)

12.00 12.00 12.00 80 shifts

There are no limitations on other resources. The particulars of sales forecasts andestimated contribution to overheads and profits are given below.

Venus Diana Aurora

Maximumpossible sales permonth (kilolitres)

100 400 600

Contribution(Rs/kilolitre)

4,000 3,500 2,000


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Due to commitments already made, a minimum of 200 kiloliters per monthof Aurora has to be necessarily supplied the next year.

Just as the company was able to finalise the monthly production programmefor the next 12 months, an offer was received from a nearby competitor for hiring40 machine shifts per month of milling capacity for grinding Dianna paint, thatcould be spared for at least a year. However, due to additional handling at thecompetitors facility, the contribution from Dianna will get reduced by Re 1 per

litre.

Formulate this problem as an LP model for determining the monthlyproduction programme to maximise contribution.

Q.9 A garment manufacturer has production line making two styles of shirts. Style 1 require 200

grams of cotton thread, 300 grams of dacron thread, and 300 grams of liner thread. Style IIrequire 200 grams of cotton thread, 200 grams of dacron thread and 100 gram of liner thread.The manufacturer make the net profit of Rs. 19.50 on Style I ,Rs. 15.90 on Style II. He has inhand inventory of 24kg. Of cotton thread ,26kg of dacron thread, and 22 kg of liner thread. Hisimmediate problem is to determine the production schedule given the current inventory to makea maximum profit. Formulate the LPP Model

Q.10 A firm makes two types of furniture: Chairs and tables. The contribution of each productas calculated by accounting department is Rs. 20 Per chair and Rs 30 per table. Both products areprocessed on three machines M1,M2 and M3.The time required by each product and total timeavailable per week on each machine are as follow:

Machine Chair Table Available Hours

M1 3 3 36

M2 5 2 50

M3 2 6 60

How should the manufacturer his production in order to maximize contribution?

Q.11 The ABC manufacturing company can make two products p1 and p2.Each of the productrequires time on a cutting machine and a finishing machine. Relevant data are:

P1 P2

Cutting Hours(Per Unit) 2 1

Finishing Hours(Per Unit) 3 3

Profit (Rs. Per unit) 6 4

Maximum Sales(Unit PerWeek)

310 200


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The number of cutting hours available per week is 390 and number of finishing hoursavailable per week is 810.How much should be produced of each product in order to achieve theprofit for the company?

First Topic Linear Programming Additional Problems for Practice

PROBLEM 1:XYZ factory manufactures two articles A and B.To manufacture the article A,a certain machinehas to be worked for 1.5 hours and in addition a craftsman has to work for 2 hours.Tomanufacture the article B, the machine has to be worked for 2.5 hours and in addition thecraftsman has to work for 1.5 hours in a week the factory can avail of 80 hours of machine timeand 70 hours of craftsman time.The profit on each article A is Rs.50 and that on each article B isRs.40. If all the articles produced can be sold away,find how many of each kind should beproduced to earn the maximum profit per week.Formulate the problem as LP model.

PROBLEM 2:An electric company is engaged in the production of two components C1 and C2 used in T.V.sets. Each unit of C1 costs the company Rs. 25 in wages and Rs. 25 in material,while each unitof C2 costs the company Rs. 125 in wages and Rs 75 in material.The company sells bothproducts on one-period credit terms, but the companys labour and material expenses must bepaid in cash.The selling price of C1 is Rs.150 per unit and of C2 it is Rs 350 per unit.Because ofthe strong monopoly of the company for these components ,it is assumed that the company can

sell at the prevailing prices as many units as it produces. The companys production capacity is,however, limited by two considerations. First, at the beginning of period 1, the company has aninitial balance of Rs. 20,000(cash plus bank credit plus collections from past cedit sales). Second, the company has available in each period 4,000 hours of machine time and 2,800 hours ofassembly time.The production of each C1 requires 6 hours of machine time and 4 hours ofassembly time,whereas the production of each C2 requires 4 hours of machine time and 6 hoursof assembly time.Formulate this problem as an LP model so as to maximize the total profit of thecompany.

PROBLEM 3:IMC manufactures a variety of computing and computer- related equipment.One such productis a monitor for use with business computer systems and IMC currently has plans to producetwo models of the same monitor: Model A which is the basic ,low-price monochrome monitorand Model B which is a more sophisticated and expensive colour graphics monitor.The companyis not actually involved in manufacture directly but rather buys the various component partswhich are required for the two models from outside suppliers.The components are then


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assembled by IMC to produce Model A and Model B and each unit produced is then thoroughlyinspected for quality and performance.IMC then sells the two models under its own brand name.There are, therefore, two basic stages to the production process within the firm- the assembly ofthe components and the inspection of the final product. Information about the resources requiredto produce the two models has been obtained from the production department and the accounts

department.Model A requires 28 hours of labour to assemble from component parts,while ModelB requires 42 hours.After assembly each computer is then tested in the inspection department toensure it is working satisfactorily.Because of the technical complexity of the product- and thefirms desire to maintain good quality the control- the inspection test is time-consuming,withModel A requiring 12 hours of inspection although Model B requires only 6 hours as more careand time is taken in the assembly stage . At present the company employs 400 people in theassembly department,each working a 7-hour day; 100 people are presently employed in theinspection department but they work an 8 hour day.The company presently operates a 6-dayworking week.Current wage rates are Rs. 20 per hour in assembly and Rs. 15 per hour ininspection.The accounts department has calculated that in terms of the components and parts,Model A costs Rs.355 and Model B Rs.565 to produce.Currently the two models sel for Rs.

1,295 and Rs. 1,745 respectively.An additional aspect of the problem ,the firm faces , is that eachmodel requires a particular component- a microchip that forms part of the monitors memory.The supplier of these chips can provide no more than 600 in any one working week.FORMULATE a linear programming problem which allows the production manager todetermine how many units of Models A and B should be produced weekly in order to maximizeprofits.

PROBLEM 4:XYZ Electronics company produces three types of parts for automatic washing machine .Itpurchases casting of the parts from a local foundry and then furnishes the part of drilling,shaping and polishing machines.

The selling prices of part A,B and C respectively are Rs.40, Rs.50 and Rs.70.All parts made can be sold. Castings for part A,B and C respectively cost Rs. 25,Rs. 30 andRs.50.

The shop possesses only one of each type of machine. Costs per hour to runeach of the three machines are Rs.100 for drilling,Rs.150 for shaping and Rs. 150 for polishing.The capacities(parts per hour) for each part on each machine are shown in the adjoining table:

Machine Capacity per hour

Part A Part B Part C

Drilling 25 40 25

Shaping 25 20 20Polishing 40 30 40

The management of the shop wants to know how many parts of each type itshould produce per hour in order to maximize profit for an hours run. Formulate the problem asan LP model.


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PROBLEM 5:

Ex-servicemen Airport Services Company is considering the purchase of new vehicle for thetransportation between the Delhi airport and hotels in the city.There are three vehicles underconsideration: station wagons, minibuses and large buses.The purchase price would be Rs.1,45,000 for each station wagon, Rs. 2,50,000 for the minibus and Rs. 4,00,000 for the large bus.The Board of Directors has authorized a maximum amount of Rs. 50,00,000 for these purchases.Because the heavy air travel, the new vehicles would be utilized at maximum capacity regardlessof the type of vehicles purchased. The expected net annual profit would be Rs. 15,000 for thestation wagon ,Rs. 35,000 for the minibus and Rs. 45,000 for the large bus. The company hashired 30 new drivers for the new vehicles. They are qualified drivers for all three types ofvehicles. The maintenance department has the capacity to handle an additional 80 wagonstations. A minibus is equivalent of 5/3 wagon stations and each large bus is equivalent to 2

station wagons in terms of their use of the maintenance department. Determine the optimalnumber of vehicles to be purchased in order to maximize profit. FORMULATE the problem asLP model.

PROBLEM 6:Vitamins V and W are found in two different foods F1 and F2.One unit of food F1 contains 2units of vitamin V and 5 units of vitamin W. One unit of food F2 contains 4 units of vitamin Vand 2 units of vitamin W. One unit of food F1 and F2 cost Rs. 30 and 25 respectively. Theminimum daily requirements(for a person) of vitamin V and W is 40 and 50 units respectively.Assuming that anything in excess of daily minimum daily requirement of vitamin V and W is notharmful, find out the optimal mixture of food F1 and F2 at the minimum cost which meets thedaily minimum requirement of vitamins V and W. FORMULATE this as a Linear Programmingmodel.

PROBLEM :7A company is making two products A and B .T he cost of producing one unit of A and B is Rs60 and Rs 80 respectively .As per the agreement, the company has to supply at least 200 units ofproduct B to its regular customers. One unit of product A requires one machine hour whereasproduct B has machine hours available abundantly within the company.Total machine hoursavailable for product A are 400 hours. One unit each of product A and B requires one labourhour each and total of 500 labour hours are available. The company wants to minimize the costof production b satisfying the given requirements. FORMULATE the problem as a linearprogramming problem.


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PROBLEM :8Frontier bakery has received order from a company M/s Bodhraj Ltd., for the supply of highprotein biscuits. The order will require 1000 Kg. of biscuits mix which is made from 4ingredients R,S T and U which cost Rs.16,Rs. 4,Rs. 6 and Rs.2per Kg. respectively. The batch

must contain a minimum of 400 Kilos of protein, 250 kilos of fat, 300 kilos of carbohydrates and50 Kilos of sugar. The ingredients contain the following percentage by weight:

Ingredients Protein Fat Carbohydrates Sugar Filler

RSTU

50% 30% 15% 5% 0%10% 15% 50% 15% 10%30% 5% 30% 30% 5%0% 5% 5% 30% 60%

Only 150 kilos of S and 200 kilos of T are immediately available.Draft a suitable LP model.

PROBLEM :9A 24-hour supermarket has the following minimal requirements for security officers:

Table 1: STAFFING REQUIREMENTSTIME OF DAY MINIMUM NUMBER

OF CASHIERS REQUIRED

MIDNIGHT4AM 74 AM8 AM 208 AM - NOON 14NOON4PM 204 PM8 PM 108 PM - MIDNGHT 5

TABLE 2: SHIFT SCHEDULESHIFT STARTING

TIMEENDINGTIME

1

23456

MIDNIGHT

4AM8AMNOON4PM8PM

8AM

NOON4PM8PMMIDNIGHT4AM


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Shift 1 follows immediately after shift 6. An officer works 8 consecutive hours, starting at thebeginning of one of the six periods. The personnel manager wants to determine how manyofficers should work each shift in order to minimize the total number of officers employed whilestill satisfying the staffing requirement.Formulate the problem as a linear programming problem.

PROBLEM :10A media specialist plans to allocate advertising expenditure in three media whose unit costs of amessage are Rs. 1500; Rs.1250 and Rs.1000 respectively. The total advertising budget availablefor the year is Rs. 50000.The first medium is a monthly magazine and it is desired to advertisenot more than once in one issue. At least five advertisements should appear in the secondmedium and the no of advertisement in the third medium should strictly lie between 6 and10.The effective audience for unit advertisement in the three media is given below:Medium: 1 2 3Expected effective audience: 50,000 40,000 25,000

Formulate a linear programming problem to find the optimal allocation of advertisement in threemedia that would maximize the total effective audience.

PROBLEM :11A person is interested in investing Rs.5,00,000 in a mix of investments. Theinvestment choices and expected rates of return on each one of them are :-

Investment Projected Rate of ReturnsMutual Fund A 0.12Mutual Fund B 0.09Money Market Fund 0.08Government Bonds 0.085Share Y 0.16Share X 0.18

The investor wants at least 35 per cent of his investment in government bonds.Because of the higher perceived risk of the two shares, he has specified that the combinedinvestment in these not to exceed Rs.80,000. The investor has also specified that atleast 20 percent of investment should be in the money market fund and that the amount of money invested inshare should not exceed the amount invested in mutual fund. His final investment condition isthat the amount invested in mutual fund A should be no more than the amount invested in mutualfund B. the problem is to decide the amount of money to invest in each alternative so as to obtainthe highest annual total return. FORMULATE the above as linear programming problem.

PROBLEM :12


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In a chemical industry two products A and B are made involving two operations. The productionof B also results in a by-product C. the product A can be sold at a profit of Rs. 3 per unit and B ata profit of Rs. 8 per unit. The by-product C has a profit of Rs. 2 per unit. Forecasts show thatupto 5 units of C can be sold. The company gets 3 units of C for each unit of B produced. Themanufacturing times are 3 hrs per unit for A on each of the operation one and two and 4 hrs and

5 hrs per unit for B on operations one and two respectively. Because the product C results fromproducing B, no time is used in producing C. the available times are 18 hrs and 21 hrs ofoperation one and two respectively. The company desires to know that how much A and Bshould be produced keeping C in mind to make the highest profit. FORMULATE LP model forthis problem.

PROBLEM :13

A public limited company is planning its capital structure that will consist of equity capital, 15 %debentures and term-loans. Debentures are to be repaid on face value, interest rate is payable halfyearly and annualized cost of issue of debenture is %. Interest on term-loan is 18% p.a. to bepaid annually while the cost of equity is estimated as 20%. It is decided not to have outsiders

funds not more than 2 times of equity fund; also the amount of term-loan must be at least 50% ofthe debenture amount. FORMULATE a suitable LP model so as to minimize average cost ofcapital of the company.

PROBLEM.14

A mutual fund has cash resources of Rs. 200 million for investment in diversified portfolio.Table below shows the opportunities available, their estimated annual yields, risk factors andterm period details.FORMULATE a suitable LP model to find the optimal portfolio that will maximize returns,considering the following policy guidelines:

Investment type Annual Yield (%) Risk factors Time period (years)

Bank Deposits 9.5 0.02 6

Treasury notes 8.5 0.01 4

Corporate Deposits 12.0 0.08 3

Blue-chip Stocks 15.0 0.25 5

Speculative Stocks 32.5 0.45 3

Real Estate 35.0 0.40 10

All the funds available may be investedWeighted average period of at least 5 years as planning horizon.Weighted average risk factor not to exceed 0.20.Investment in real estate and speculative stocks to be not more than 25 % of the money investedin total.


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PROBLEM.15

A company has the following independent projects available:

Project cash flows (Rs.000)

Year A B C D E F

0 (100) - - (40) - (30)1 (50) (60) - (60) (120) (10)

2 (10) (70) (40) 50 100 20

3 70 10 (80) 10 (10) 10

Cash flows extend beyond year 3 but all are cash inflows for each project

NPV(Rs.000)

20 15 10 30 10 5

New capital for these projects is limited to :Year 0 Rs. 1,20,000Year 1 Rs. 2,00,000Year 2 NilYear 3 Nil

Cash generated from these investments can be re-invested in other projects in the same year.EXPRESS the above problem in LP format, assuming the objective of the company is tomaximize NPV and the projects are divisible.

PROBLEM :16

A ship has three cargo loads--- Forward, after and center. The capacity limits are :

Weight (tons) Volume (in cubic Ft. )

Forward 2,000 1,00,000

Center 3,000 1,35,000

After 1,500 30,000

The following cargos are offered. The shipowner may accept all or any part of each commodity.

Commodity Weight (tons) Volume (in cubic Ft. ) Profit per tonne ( Rs.)

A 6,000 60 150

B 4,000 50 200C 2,000 25 125

In order to preserve the trim of the ship, the weight in each load must be proportional to thecapacity in tones. The cargo is to be distributed so as to maximize the profit. FORMULATE theproblem as LP model.


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PROBLEM :17The PQR stone company sells stones secure from any of the three adjacent quarries. The stone

sold by the company must conform to the following specifications:Material X equal to 30%Material Y equal to or less than 40%Material Z between 30% and 40%

Stone from quarry A costs Rs. 100 per tonne and has the following properties:Material X20%Material Y60%Material Z20%

Stone from quarry B costs Rs. 120 per tonne and has the following properties:

Material X40%Material Y30%Material Z30%

Stone from quarry C costs Rs. 150 per tonne and has the following properties:Material X10%Material Y40%Material Z50%From what quarries should PQR stone company secure rocks in order to minimize cost per tonneof rocks?

PROBLEM 18

Use Simplex Method to solve the following L.P. Problem

Max.Z = 6x1+ 8x2

Subject to : 30x1+ 20x2 0

PROBLEM 19

Use simplex method to solve the following L P problem :

Max.Z. = 6 x1 + 8 x2

Subject to Constraints :

2 x1+ 3 x2 < 16

4 x1+ 2 x2 < 16


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PROBLEM 20

Use simplex method to solve the following LP ProblemMax Z = 4 x1+ 5x2+8x3Subject to : X1+ X2+ X3+ < 1003 x1+ 2x2+ 4x3< 500

x1, x2, x3, >0

Second Topic Transportation Model Problems for practice

Q.1 A dairy firm has their plants located in a state. The daily milk production at each plant isas follows:

PLANT 1 2 3

MILK SUPPLY 6 1 10

Each day, the firm must fulfill the needs of its four distribution centers. Minimumrequirement at each center is as follows:

CENTER 1 2 3 4

MILKSUPPLY

7 5 3 2

Cost in the hundreds of rupees of shipping one million litre from each plant to eachdistribution center is given in the following table:

Distribution center

D1 D2 D3 D4

P1 2 3 11 7

P2 1 0 6 1

P3 5 8 15 9

Find initial basic feasible solution for given problem by using:

(a)NorthWest corner rule(b) Least cost method and(c) Vogels approximation method

If the objective is to minimize the total transportation cost.


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Q.2 A Company has factories at F1, F2, and F3 which supply to warehouses at W1, W2, and W3.Weekly factory capacities are 200,160,and 90 units, respectively. Weekly warehouserequirement are 180,120and 150 units respectively. Unit shipping costs (in rupees) are asfollows:

Ware house

W1 W2 W3 SUPPLYF1 16 20 12 200

F2 14 8 18 160

F3 26 24 16 90

DEMAND 180 120 150 450

Determined the optimal distribution for this company to minimize total shipping cost.

Second Topic Transportation Model Additional Problems for Practice

Q.1 A company has four manufacturing plants and five warehouses. Each plant manufacturesthe same product which is sold at different prices in each warehouse area. The cost ofmanufacturing and cost of raw material are different in each plant to various factors. Thecapacities of the plants are also different. The data are given in the following table:

PLANT ITEM 1 2 3 4Manufacturing cost (RS) per unit 12 10 08 08Raw material cost (RS)per unit 08 07 07 05

Capacity per unit time 100 200 120 80

The company has five warehouses. The sales prices, transportation costs and demands aregiven in the following table:

ware house transportation cost per unit sale price demandper unit(Rs.)

1 2 3 4

A 4 7 4 3 30 80B 8 9 7 8 32 120C 2 7 6 10 28 150D 10 7 5 8 34 70E 2 5 8 9 30 90

(a) Formulate this problem as a transportation problem to maximize profit.(b) Find the solution using VAM method.


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Q.2 Obtain an optimal solution to the transportation problem given below by UV method. UseVAM method for obtaining initial feasible solution

D1 D2 D3 D4 CAPACITY

S1 19 30 50 10 7S2 70 30 40 60 9

S3 40 8 70 20 18

DEMAND 5 8 7 14 34

Q.3 Consider a firm having 2 factories. The firm is to ship its products from the factories tothree- retail stores. The number of units available at factories X and Y are 200 and 300,respectively while those demanded at retail stores A ,B and C are 100,150 and 250,respectively. Rather than shipping directly from factories to retail stores, it is asked toinvestigate the possibility of Trans- shipment. The transportation cost (in Rupees) per unit isgiven in the table .

FACTORY RETAIL STORE

X y A B C

Factory X 0 8 7 8 9Y 6 0 11 9 10

Retail store A 7 2 0 5 1B 1 5 1 0 4C 8 9 7 8 0

Find out the optimal shipping schedule.

Q.4 ABC limited has three production shops supplying a product to five warehouses. The coat ofproduction varies from shop to shop and cost of transportation from one shop to a warehousealso varies. Each shop has a specific production capacity and each warehouse has certain amountof requirement. The cost of transportation are given below:

Ware house

I II III IV V SUPPLY

A 6 4 4 7 5 100

B 5 6 7 4 8 125C 3 4 6 3 4 175

DEMAND

60 80 85 105 70 400


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The Cost Of Manufacturing The Product At Different Production Shop Is

shop Variable cost Fixed cost

A 14 7000

B 16 4000

C 15 5000

Find the optimum quantity to be supplied from each shop to different warehouses at minimumtotal cost.

Third Topic Assignment Model Problems for practice

Q.1 A job production unit has four jobs A,B,C,D which can be manufactured on eachof the four machines P,Q,R and S. The processing cost of each job on each machine is given inthe table below:

Jobs MachineP Q R S

Processing cost (Rs.)

A 31 25 33 29

B 25 24 23 21

C 19 21 23 24

D 38 36 34 40

To achieve minimum processing cost, which job will you process on which machine?

Q.2 A workshop has four machines and four tasks for completion. Each of the machines can

perform each of four tasks. Time taken at each of the machines to complete each task is given inthe matrix below:

How should the tasks be assigned to machines requirement of machine hours?

Tasks Machine

A B C D

Processing times (Hrs.)

I 51 77 49 55

II 32 34 59 68

III 37 44 70 54

IV 55 55 58 55


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Q.3 A pharmaceutical company has four branches, one each at city A, B, C and D. A branchmanager is to be appointed one at each city, out of four candidates P, Q, R and S. The monthly

business depends upon the city and the effectiveness of the branch manager in that city.

Branch

manager

City

A B C D

Monthly business (Rs. lakhs)

P 11 11 9 9

Q 13 16 11 10

R 12 17 13 8

S 16 14 16 12

Which manager should be appointed at which city so as to get maximum total monthly business?

Q.4 Darda oil mills have four plants each of which can manufacture anyone of the fourproducts. The manufacturing costs differ from plant to plant and so do the sales revenues. Therevenue and cost details are as given below:

Sales revenue (Rs. Lakhs)

Plants Products

I II III IV

ABCD

70807578

88908785

69717374

82948089

Manufacturing cost (Rs. Lakhs)

Plants Products

I II III IV

ABCD

59656265

70737274

55555958

71796876

Suggest which plant should produce which product to maximize profit?


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Q.5 A company has four territories open, and four salesman available for an assignment. Theterritories are not equally rich in its sales potential. It is estimated that, a typical salesman,operating in each territory would bring in the following annual sales.

Territory I II III IV

AnnualSales(Rs.) 126000 105000 84000 63000

The four salesmen also differ in their ability. It is estimated that, working under the sameconditions, their yearly sales would be proportionately as follows:

Salesman A B C D

Proportion 7 5 5 4

Assign the salesmen to each territory if the criterion is maximum expected total sales.

Third Topic Assignment Model Additional Problems for Practice

Q.1 A departmental head has four subordinates and four tasks for completion. The subordinatesdiffer in their capabilities and tasks differ in their work contents and intrinsic difficulties. Hisestimate of time for each subordinate and each task is given in matrix below:

Tasks Subordinates

I II III IV

Processing cost (Rs.)

A 17 25 26 20

B 28 27 23 25C 20 18 17 14

D 28 25 23 19

How should the tasks be assigned to minimize requirements of man-hours?

Q.2 A departmental head has three subordinates and four tasks for completion. The employeesdiffer in their capabilities and the tasks differ in their work contents. With the performancematrix given below, which three of four tasks should be assigned to subordinates?

Tasks Subordinates

I II III

ABCD

982021

12131215

11171317


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Q.3 A gear manufacturer requires 2000 numbers per month of each of the six types of gears. Sixhobbing machines are available to process these gears. The gears differ in their work contents-gear with, number of teeth, module etc-and machine differ in their capabilities-speeds, feeds andability to take depth of cut. The production control department has prepared the machine wise

cost matrix as shown in the matrix below:

Gear Hobbing machines

M1 M2 M3 M4 M5 M6

IIIIIIIVVVI

15201930613

181616-812

131215421016

1014-381214

-181935915

141520361018

Gear I can be assigned to machine M5 because of steep helix angle. Gear III can not be assignedto machine M4 as it is not within the capacity of this machine. And gear IV can not be loaded onmachine M2 because of limitations of process capability of the machine. Find the optimumassignment schedule.

Q.4 A salesman has to visit five cities. He wishes to start from a particular city, visit each cityonce and return to starting city. The cost of going from a city to another in Rs. is given below:

FromCity

To city

A B C D E

ABCDE

016182111

120171413

151301812

171814018

111217160

Determine the least cost route.


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Fourth Topic Decision Analysis Problems for practice

Q.1 A departmental store purchases Christmas trees, which can be orderedonly in lots of 100. Each tree selling price Rs. 40 each. Unsold trees, however,have no salvage value. The purchase price of the trees is Rs. 25 each The

probability distribution obtained from analysis of past data is given below:

Trees sold probabilities100 0.20200 0.35300 0.25400 0.15500 0.05

(a)Setup a payoff table(b)How much quantity should the departmental store buy to maximize its

profit?

Q.2 A manufacturer of sewing machines is faced with the problem of selecting one of

the two models for manufacturing. The profit depends on the market acceptability

of the model which are present is uncertain but is had been broadly classified into

four categories: excellent, good, fair and poor. ( Take alpha = 0.6)

The profits or losses (losses are indicated by negative sign) expected by the management

from the different levels of market acceptability of the models are as follows:

__________________________________________________________________Market Profit (Rs.) for the model for the

Indicated market acceptability__________________________

Probability Deluxe Janata__________________________________________________________________Excellent 0.2 60,000 78,000Good 0.3 28,000 38,000Fair 0.4 18,000 8,000Poor 0.1 8,000 -12,000__________________________________________________________________Which product should the company select from the standpoint of maximin (gain)criterion?


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Q.3 A company is making a large boiler installation. A certain automatic monitoring unit is

critical for the operation of the whole system. At the time of original order, the spares for this

unit can be purchased for Rs. 2,000 per unit. The probability distribution for the failure of the

unit during the life time of installation is given as :

_________________________________________________________________Failure_________________________Probability____________

0 0.351 0.252 0.203 0.154 0.05

___________________________________________________________If a spare is needed and is not available, the total cost of idle time and replacement cost

will be Rs. 15,000. Unused spares have no salvage value.

Determine the optimal no. of spares to be ordered.

Q.4 A newspaper boy is thinking of selling a special one time edition of a sports magazine to his

regular newspaper customers. Based on his knowledge of his customers, he believes that he can

sell between 9 to 12 copies.

The magazines can be purchased at Rs. 8 each and can be sold for Rs. 12 each.

Magazine that are not sold can be returned to the publisher for a refund of 50%.

(a) Construct the decision matrix for the above inventory problem

indicating possible monetary consequences.

(b) Determine the best decision from the stand point of

(i) Maximin criteria (ii) Maximax criteria

(iii)Hurwiez a-criterion assuming a=0.40

(iv) Minimax regret criteria (v) Laplace criteria

Q.5 Agent Corner, an authorized dealer in domestic appliances find that the cost ofholding refrigerator in stock for a week is Rs. 50.

Customers who cannot get a new refrigerator immediately wanted to go to another dealerfor which expected profit is Rs. 350 per customer.

Probability distribution of demand is as follows:

No. of refrigerator: 0 1 2 3 4 5 6Probability : 0.05 0.10 0.20 0.30 0.20 0.10 0.05


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Assuming that there is no time lag between ordering and delivery, how manyrefrigerators should we order per week?

Q.6 A departmental store buys Christmas tree at a landing cost of Rs 25 each and sells themat an average of Rs 40. Any tree left over after the selling season has no resale value.

The productivity distribution of sale of trees derived from analysis of pas t sales data isunder:

Tree (sale) Probability

100 0.10200 0.15300 0.35400 0.20

500 0.10600 0.05700 0.05

a) How many trees should be department store buy to maximize its profit?

b) If trees left after the selling season cost Rs 5 each to remove ,does it affect theinventory decision?

Q.7 A Ship building company has launched a program for the construction of new class ofships. Certain spare units like prime over, each costing 200000 have to be purchased. If theseunits are not available when needed, a serious loss is incurred which is in order of Rs 10000000each instance requirements of spares with the corresponding probabilities are given below.

Nos of spares: 0 1 2 3 4 5

Probability of 0.876 0.062 0.041 0.015 0.005 0.001requirementHow many spares should the company buy in order to optimize inventory decision?


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Fourth Topic Decision Analysis Additional Problems for Practice

Q.1 Aperishable item is ordered only once each demand period. Acquisition cost is $3,selling price is $5, and salvage value is $1.50. What is optimal order quantity? Given:

Demand Probability100 0.1110 0.2120 0.2130 0.3140 0.1150 0.1

Q.2 A newspaper boy buys papers for Rs 1.30 each and sells them for Rs 1.40 each. He

cannot return unsold newspapers. Daily demand has the following distribution.

No. of customers: 23 24 25 26 27 28 30 31 32

Probability: 0.01 0.03 0.06 0.10 0.20 0.25 0.10 0.05 0.05

If each day's demand is independent of the previous days, how many papers he should ordereach day?

or practiceproblems 2015

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