Linear Programming : Solution by Graphical MethodQ.1 XYZ firm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. Because of the need to ensure certain nutrient constituents, it is necessary to buy additional one or two products, which we shall call A and B. The nutrient constituents (vitamins and proteins) in each unit of the product are given below:

Nutrient constituentsNutrient constituent in the productMinimum requirement of nutrient constituents

AB

1366108

231236

32010100

Product A costs Rs.20 per unit and product B costs Rs.40 per unit. Determine how much of products A and B must be purchased so as to provide the pigs nutrients not less than the minimum required, at the lowest possible cost. Solve the LP problem graphically. Q.2 A firm makes two products- X and Y- and has a total production capacity of 9 tonnes per day, X and Y requiring the same production capacity. The firm has a permanent contract to supply at least 2 tonnes of X and at least 3 tonnes of Y per day to another company. Each tonne of X requires 20 machine hours production time and each tonne of Y requires 50 machine hours of production time. The daily maximum possible number of machine hours is 360. All the firms output can be sold, and the profit made is Rs.80 per tonne of X and Rs. 120 per tonne of Y. It is required to determine the production schedule for maximum profit and to calculate this profit. Solve graphically. Q.3 The manager of an oil refinery must decide on the optimal mix of two possible blending processes of which the inputs and outputs per production run are as follows:ProcessInput (units)Output

Grade AGrade BGasoline XGasoline Y

15358

24544

The maximum amount available of crudes A and B are 200 units and 150 units respectively. Market requirements show that at least 100 units of gasoline X and 80 units of gasoline Y must be produced. The profits per production run for process 1 and process 2 are Rs.300 and Rs.400 respectively. Solve the LP problem by the graphical method.Q.4 A manufacturer produces two different models, X and Y, of the same product. The raw materials r1 and r2 are required for production. At least 18 kg of r1 and 12 kg of r2 must be used daily. Also at most 34 hrs of labour are to be utilized. 2 kg of r1 are needed for each model X and 1 kg of r1 for each model Y. For each model of X and Y, 1 kg of r2 is required. It takes 3 hrs to manufacture a model X and 2 hrs to manufacture a model Y. The profit is Rs.50 for each model X and Rs.30 for each model Y. How many units of each model should be produced to maximize the profit? Solve the LP problem by the graphical method.Q.5 A person requires 10, 12 and 12 units of chemicals A, B and C respectively for his garden. A liquid product contains 5, 2 and 1 units of A, B and C respectively per jar. A dry product contains 1, 2 and 4 units of A, B and C per carton. If the liquid product sells at Rs.3 per jar and dry product sells for Rs.2 per carton, how many of each should be purchased in order to minimize cost and meet the requirement? Solve graphically. Q.6 A furniture manufacturer makes two products: chairs and tables. Processing of these products is done on two machines A and B. A chair requires 2 hrs on machine A 6 hrs on machine B. A table requires 5 hrs on machine A and no time on machine B. There are 16 hrs per day available on machine A and 30 hrs on machine B. Profit gained by the manufacturer from a chair and a table is Rs.2 and Rs.10 respectively. What should be the daily production of each of the two products? Solve graphically.

Linear Programming : Solution by Simplex MethodQ.1 A manufacturer produces four products A, B, C and D, each of which is processed on three machines X, Y and Z. The time required to manufacture one of each of the four products and the capacities of each of the three machines are indicated in the following table:Processing time

ProductMachine XMachine YMachine Z

A1.542

B213

C421

D312

Capacity (hrs)550700200

The profit contribution per unit of the four products A, B, C and D is Rs.4, 6, 3 and 1 respectively. The manufacturer wants to determine its optimal product mix:(a) Formulate the above as a linear programming problem.(b) Solve it with the simplex method.(c) Find out the optimal product mix and the total maximum profit contribution.(d) Which machine(s) has (have) excess capacity available? How much?(e) If the profit contribution from product B increases by Rs.2 per unit, will the optimal product mix change?(f) What are the shadow prices of the machine hours on the three machines?(g) If machine A is to be shut down for 50 hrs due to repairs, what will be the effect on profits?(h) If the company whishes to expand the production capacity, which of the three machines should be given priority?Q.2 A firm produces three products A, B and C each of which passes through three departments: Fabrication, Finishing and Packaging. Each unit of product A requires 3, 4 and 2; a unit of product B requires 5, 4 and 4, while each unit of product C requires 2, 4 and 5 hours respectively in the three departments. Everyday, 60 hrs are available in the fabrication department, 72 hrs in the finishing department and 100 hrs in the packaging department.The unit contribution of product A is Rs. 5, of product B is Rs.10 and the product C is Rs.8.Formulate the problem as an LPP and determine the number of units of each of the products that should be made each day to maximise the total contribution. Also determine if any capacity would remain unutilized.

Q.3 Maximise Z = 3x1 + 2x2 Subject to x1 + x2 15 2x1 + x2 28 x1 +2x2 20 x1, x2 0Solve using Simplex Method.Q.4 Solve the following LPPMaximize Z = 2x1 + 4x2Subject to 2x1 + x2 18 3x1 + 2x2 30 x1 +2x2 = 26 x1, x2 0Q.5 A company sells a fertilizer which is made from two chemical compounds, nitrate and phosphate. The mixture is sold in 50 kg bags. The fertilizer is widely used in agriculture. It has been established based on scientific studies that the nitrate content should be at least 20 kgs and the phosphate content should be not more than 40 kgs in a bag of 50 kgs. The nitrate compound costs Rs. 10 per kg and phosphate costs Rs. 25 per kg. The company desires to determine the mixture such that the costs of the ingredients in the mixture is minimum.Q.6 The WG company produces high quality glass products including windows and glass doors. It has three plants, A, B and C. Aluminium frames and hardware are made in plant A, wooden frames are made in plant B and plant C produces the glass and assembles the final products. Production capacities of A, B and C are 4, 12 and 18 hrs respectively.The top management has decided to revamp the product line in order to improve the earnings. Unproductive products are being discontinued and two new products having large sales potential are launched. These are product P1 and product P2.Profit per batch of P1 and P2 is Rs.3000 and Rs. 5000 respectively. One batch of P1 requires 1 and 3 hrs in plants A and C respectively, but none in plant B. one batch of product P2 needs 2 hrs each in plant B and C. The marketing division has concluded that both products could be produced by these plants. But, because both products would be competing for the same production capacity in plant C, it is necessary to determine the product mix which would be most profitable.Q.2 Use the simplex method to solve the following LP problem. Maximize Z= 10x1+15x2 Subject to constraints 2x1+ x2 26 2x1 + 4x2 56 -x1 + x2 5 And x1, x2, 0Q.2 Solve the linear programming problem by simplex method:

Maximize Z = 3x1 + 5x2 + 4x3 Subject to constraints 2x1 + 3x2 8 2x2 + 5x3 10 3x1 +2x2 + 4x3 15 x1, x2, x3 0

Transportation Problem

Q.1 A company has three factories, say F1, F2 and F3 that are feeding 5 zones- North, South, Eastern, Western and Central whose monthly demands are 20, 30, 25, 25 and 20 thousand units respectively. The cost of transporting one unit from each factory to each destination is given below:

FactoryZones

NorthSouthEasternWesternCentral

F15471012

F21310756

F31298710

If the factory capacities are 40, 30 and 50 thousand units respectively then find the optimal shipping schedule.

Q.2 The following table gives the cost of transporting material from supply point A, B, C, and D to demand points E,F,G,H and I.

FromTo

EFGHI

A810121715

B15131899

C142061313

D131971212

Find the optimal solution. If in the above problem, the transportation cost from A to G is reduced to 10, what will be the new optimal schedule.

Q.3 A cement company has three factories which manufacture cement which is then transported to four distribution centres. The quantity of monthly production of each factory, the demand of each distribution centre and the associated transportation cost per quintal are given as follows: Distribution CentresMonthly Production (in Quintal)

WXYZ

FactoriesA108547000

B791588000

C61014810000

Monthly Demand (in Quintals)6000600080005000

(i) Suggest the optimum transportation schedule.(ii) Is there any other transportation schedule which is equally attractive? If so, write that.Q.4 A product is produced by three factories A, B & C. Their production capacities are: Factory A- 50, B- 80 & C-110 (in 100units). These factories supply product to 4 stores, demand of which are 40, 60, 80 and 60 thousand units respectively. Per unit transportation cost in 100Rs. from each factory to each store is given in table. 1 2 3 4A2345

B5431

C1332

Determine the extent of deliveries from each of the factories to each of the stores so that the total transportation cost is minimum. Use VAM for initial solution.Q.5 A company has factories A, B and C with supply warehouses at D, E, F and G. Monthly factory capacities are 160, 150 and 190 units respectively. Monthly warehouse requirements are 80, 90, 110 and 160 units respectively. Unit shipping costs (in rupees) are as follows:To

FromDEFG

A42483837

B40495251

C39384043

Determine the optimum distribution for this company to minimize shipping costs.Q.6 The following table shows all the necessary information on the availability of supply to each warehouse, the requirement of each market and unit transportation cost (in Rs.) from each warehouse to each market.

WarehouseMarketSupply

PQRS

A635422

B592715

C578615

712179

The shipping clerk has worked out the following schedule from experience: 12 units from A to Q, 1 unit from A to R, 9 units from A to S, 15 units from B to R, 7 units from C to P and 1 unit from C to R.i) Check and see if the clerk has the optimal schedule.ii) Find the optimal schedule and minimum total transportation cost.

Q.7 A company has factories at F1, F2 and F3 which supply warehouses at W1, W2 and W3. Weekly factory capacities are 200, 160 and 90 units respectively. Weekly warehouse requirements are 180, 120 and 150 units respectively. Unit shipping costs (in rupees) are as follows:W1W2W3Supply

F1162012200

F214818160

F326241690

Demand180120150350

Determine the optimal distribution for this company to minimize shipping costs. Q.8 Determine an initial basic feasible solution to the following transportation problem using Vogels approximation method. Also test for the optimality using MODI method and find the optimal solution. (8 marks)

Destination

SourceD1D2D3D4Supply

S1121430

S2332150

S3425920

Demand20403010

Game TheoryQ.1 Two breakfast food manufacturers, ABC and XYZ are competing for an increased market share. The payoff matrix, shown in the following table, shows the increase in market share for ABC and decrease in market share of XYZ.ABCXYZ

Give CouponsDecrease PriceMaintain Present StrategyIncrease Advertisement

Give Coupons2-241

Decrease Price61123

Maintain Present Strategy-3206

Increase Advertisement2-371

Find the optimal strategies for both the manufacturers and value of the game.Q.2 Two companies A and B are competing for the same product. The different strategies are given in the following payoff matrix:Company ACompany B

B1B2B3

A13-42

A21-3-7

A3-247

Determine the best strategy for both the players.Q.3 Use dominance to reduce the size of the following game to 2x2 game and hence find the optimal strategies and value of the game.Player APlayer B

B1B2B3B4B5

A124384

A256378

A367987

A442843

Q.4 Given the following payoff matrix of a zero sum game, determine the optimal strategies for the players and the value of the game:Player APlayer B

B1B2B3B4

A15-458

A2620-5

A371287

A428-65

Inventory ManagementQ.1 A stockist has to supply 400 units of a product every month to his customers. He gets the product at Rs.100 per unit from the manufacturer. The cost of ordering and transportation from the manufacturer is Rs. 75 per order. The cost of carrying inventory is 8% per year of the cost of product. Find the economic lot size and the total optimum cost including capital cost. Also find the interval between orders and frequency of orders per unit time. Q.2 For one of the A class items, the following data are available:Annual demand = 1000; Ordering cost = Rs.400; Holding cost = 40% and cost per unit = Rs.20.The following three strategies are available for the procurement:(i) Place four orders of equal size every year.(ii) Place the order for 500 units at a time and avail a discount of 10% on the cost of items.(iii) Follow the EOQ policy.Which of the above strategy do you recommend? Justify your answer.Q.3 A hardware store procures and sells hardware items. Information on an item is given here:Expected annual sales = 8000 unitsOrdering cost = Rs 180 per orderHolding cost = 10 % of the average inventory valueThe item can be purchased according to the following schedule:Lot size Unit Price (Rs)1 999 22.001000 1499 20.001500 1999 19.002000 and above 18.50You are required to determine the best order size.

PERT and CPMQ.1 For a small project of 13 activities, the details are given below:

ActivityDurationPreceding activity

A4E

B2A

C1B

D12K

E14-

F2E

G3F

H2F

I4F

J3I,L

K4C,G,H

L2D

M2I,L

i) Construct the network diagram and find the earliest occurrence time and latest occurrence time. ii) Indicate the critical path and calculate the project completion time. iii) For each non-critical activity, find the total float, free float, interfering float and independent float

Q.2 For a small project of 12 activities, the details are given below:

ActivityDurationPreceding activity

A9-

B4-

C7-

D8B,C

E7A

F5C

G10E

H8E

I6D,F,H

J9E

K10I,J

L2G

a) Construct the network diagram and find the earliest occurrence time and latest occurrence time. b) Indicate the critical path and calculate the project completion time. c) For each non-critical activity, find the total float, free float, interfering float and independent float.

Q.3 The owner of a chain of fast food restaurants is considering a new computer system for accounting and inventory control. A computer company sent the following information about the system installation:

ActivityDescriptionImmediate PredecessorToTmTp

ASelect the computer model_468

BDesign input/output systemA5715

CDesign monitoring systemA4812

DAssemble computer h/wB152025

EDevelop the main programsB101826

FDevelop the input/output routinesC8916

GCreate databaseE4812

HInstall the systemD, F123

ITest and implementG, H678

a) Obtain in earliest and latest scheduling times of the various activities.b) Construct an arrow diagram for this problem, determine the critical path and state the expected project completion time.c) Determine the probability that the project will be completed in 55 days.d) If the company wants to be 90 % sure that the system will be installed by a certain due date , how many days prior to that should it start the work?e) Suppose the company agrees to install the computer system in 50 days, failing which it would pay a penalty of Rs.500 per day. What is the probability that a penalty but not exceeding Rs. 2000 will be paid?. Q.4 Following table gives the list of various activities involved in the launch of a new CREDIT CARD service by a company, their immediate predecessors and their expected durations (in days):ActivityImmediate PredecessorExpected Duration

ToTmTp

A-101214

BA141517

CB234

DC468

EC101214

FE202527

GC101720

HF567

ID71214

JH, I141720

KC123

LK101520

ML357

NM, J131517

ON202122

PO7914

QP234

RQ222

SP71013

TS579

UT, R, G4812

a) Draw an arrow diagram for the project.b) Find the expected project completion timec) Determine the probability of completing the project in 165 days.

Q.5 A project consists of nine activities whose time estimates (in weeks) and other characteristics are given below?ActivityPreceding ActivityTime estimates (weeks)

ToTmTp

A-246

B-666

C-61224

DA258

EA111423

FB, D81012

GB, D369

HC, F91527

IE41016

a) Draw the network diagramb) Identify the critical activitiesc) What is the expected project completion time and its variance?d) What is the probability of completing the project one week before the expected time.e) A penalty of Rs.15000 per week is to be imposed on the contractor if the project is not completed in 36 weeks. What is the probability that he has to pay a penalty? A penalty of Rs.45000?