Qm linear programming as narag 1

Role Of Operations Research In Managerial Decision Making

Management Science Operations Research Operational Research Operation Analysis It is sometimes believed that operations research refers to constant monitoring of an organization's ongoing activities such as production scheduling and inventory control, facility maintenance and repair, staffing of service facilities etc. Whereas many management science studies treat other kind of decisions that bear on daily operations only indirectly. These studies usually have planning orientation. For example determining the breadth of a firm’s product line, developing a long term plan for plant expansion, designing a network of warehouses for a wholesale distribution system, entering a new business by merger or acquisition .

Some Definitions Of Operations Research O.R.Deals with systematic and scientific research on the operations of a system .(Mccloskey andTrefethen) 2. O.R. Is the application of scientific methods, techniques, and tools to problems involving the operations of a system so as to provide those in control of the system with optimum solution to the problem. (Churchmann,Ackoff and Arnoff). 3. O.R. Is a scientific approach to problem solving for executive management. (H.M.Wagner). 4. O.R. Is the art of giving bad answers to problems to which ‘worst’ answer are sought otherwise. (T.L.Saaty).

Major O.R.Techniques Linear Programming Transportation Models Assignment Models Other Mathematical Programming Techniques Like Goal, Integer, Dynamic Etc Network Techniques Like PERT/CPM Etc. Queuing Theory Inventory Management Simulation Markov Chains Game Theory

Characteristics Of O.R. 1. Inter-disciplinary Team Approach 2. Holistic Approach To The System 3. Focus On Decision Making

Phases Of An O.R. Study Identifying The Problem Building A Model Finding The Solution Testing The Model And The Solution 5. Establishing Control Limits 6. Implementation

Areas Of Applications Of Operations Research (Schumacher-Smith Survey) Forecasting 73 Production Scheduling 90 Inventory Control 90 Quality Control 51 Transportation 54 Advertising And Sales Research 27 Maintenance And Repairs 32 Accounting Procedures 17 Plant Location 32 Equipment Replacement 27 Packaging 7 Capital Budgeting 39 Percent Of Companies Area Of Application Reporting Activities

Frequency Of Use Tool/Tech Never Moderate Frequent Statistical Analysis 20 30 75 Computer Simulation 16 57 52 PERT/CPM 22 45 58 Inventory Mgt. Techniques 22 39 64 Linear Programming 30 47 48 Queuing Theory 46 46 33 Non-linear Programming 68 39 18 Dynamic Programming 62 37 26 Game Theory 64 37 24 (No. Of Companies) Source:- An Unpublished MBA (Part Time )Project, 1999. Usage Of OR Techniques Among 125 Companies

Operations Research Software Available Bernard W.Taylor, Management Science, Prentice-Hall,Inc., Englewood Cliffs, New Jersey (Contains AB: QM 4.0 By Sang Lee), 1996. Hamdy A.Taha, Operations Research : An Introduction, Prentice-Hall Of India, New Delhi, (Contains Tora And SimmnetI1),1995. Sang M.Lee and Jung P.Shim, Micro Management Science, Allyn and Bacon, (Contains Micro Manager Version2.0),1990. Yih-long Chang, Quantitative Systems (QS) Version 3.0, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1995. Yih-long Chang, Quantitative Systems For Business Plus (QSB+) Version 3.0, Prentice-Hall, Inc., Englewood Cliffs,NewJersey,1994.

The Five Generations Of OR Generation/ Time Frame Description First/ 1930s-1940s Emphasized The Applications Of The Scientific Methods To Problems Involving Operations Of Systems. OR Teams Were Interdisciplinary And Addressed Complex Problems. Second / 1950s Emphasized Mathematical/Optimization Techniques. Scientific Methods Still Employed And Computers Began To Be Integrated Into Tool Box. Third/ 1960s Those Who Emphasized Heuristics Programming And Ai-based Methodology Separated From OR, Thus Creating Two Approaches To “Satisficing.” Multiple Objective Optimization And Goal Programming Reflected The Change In OR. OR Associated With Quantitative; AI With Qualitative. Fourth/ 1970s- Present DSS And ES Methodologies Symbolize This Generation. ES Were Developed By AI In Response To Difficulties It Had Encountered; DSS By MS/OR. Decision Maker In Both Cases Becoming More Involved. Fifth/? There Is Now A Realization That To Solve A Larger Set Of “Real-World "problems, And So Make A Significant Impact On Decision Making Technology, One Needs To Employ A Much Larger Combination Of Techniques And Tools, Dependent Upon The Situation. Artificial Barriers And Constraints Of The Discipline; Should Not Be Inhibiting.

OR’S FORMULATION APPLIED SCIENTIFIC METHOD TO PROBLEMS INVOLVING OPERATIONS OF COMPLEX SYESTEMS(CS) ATTEMPTED TO USE MATHEMATICAL/ OPTIMIZATION TECHNIQUES TO SOLVE PROBLEMS INVOLVING OPERATIONS OF CS QUANTITATIVE METHODOLOGY APPLIED TO SOME SUBSET OF PROBLEMS INVOLVING OPERATIONS OF CS FIRST GENERATION DSS DEVELOPED IN RESPONSE TO THE LIMITED SUCCESS ACHIEVED BY THE PREVIOUS GENERATION QUALITATIVE METHODOLOGY APPLIED TO LARGE CLASS OF PROBLEMS(GENERAL PROBLEM SOLVERS-GPS) ES DEVELOPED IN RESPONSE TO THE LACK OF SUCCESS IN CREATING GPS IN PREVIOUS GENERATION CLASSICAL OR THIRD GENERATION EARLY OR SECOND GENERATION THE FIVE GENERATIONS OF OR MS/OR FOURTH GENERATION SOR (SYNERGISTICOR) FIFTH GENERATION ( 1930s-1940s) ( 1950s) ( 1970s)-PRESENT AI/ES ( 1960s) AI (?) SYNERGY OF ALL SCHOOLS OF THOUGHT QUANTITATIVE QUALITATIVE CLASSICAL OR AI MS/OR/DSS AI/ES OTHER RELEVANT PARADIGMS

Some Definitions Of Linear Programming 1. A Method To Allocate Limited Resources In An Optimal Manner So As To Satisfy The Laws Of Supply And Demand For The Firm’s Products. 2. A Method To Optimize A Linear Function Subject To Set Of Linear Constraints. 3. Yet Another Class Of Optimization Techniques.

Janata fertilizers uses nitrates, phosphates, potash and an inert filler material for fertilizer manufacture. The firm mixes these four ingredients to make two basic fertilizers, 5-10-5 and 6-5-10 (nos. represent % wt. Of nitrates, phosphates and potash in each tonne of fertilizers). The contribution towards profits and overheads is Rs.300 and Rs.400 per tonne for (5-10-5) and (6-5-10) respectively. The firm has 120 tonnes of nitrates ,200 tonnes of phosphates, 150 tonnes of potash and unlimited filler material. They will not receive additional chemicals until next month. They believe that they can sell or store at negligible cost all fertilizer produced during the month. Determine optimal product mix.

Ing.A I ng.B Cost/kg. Rs.10 Rs.40 Fiber content 25% 75% Fat content 20% 10% A state agricultural federation manufactures and markets cattle feed. It uses two ingredients which have the following characteristics Specifications are that fiber content should be at least 50% and fat content should not exceed 18% in every one kg. Of the feed. What amounts of A and B should be used so that cost/kg. is minimized?

PROBLEM-I 5-10-5 6-5-10 Available NITRATES 5% 6% 120 TONNES PHOSPHATES 10% 5% 200 TONNES POTASH 5% 10% 150 TONNES CONTRIBUTION Rs.300 Rs.400 Per Tonne

Problem-2 Ing.A Ing.B Cost/Kg. Rs.10 Rs.40 Fiber 25% 75% Fat 20% 10% Required Fiber Content – At least 50% Fat Content - Not More Then 18%

A MEDIA PLANNING PROBLEM EXPOSURE RATES TA TB MAGAZINE 2% 1% (Rs.80,000) TV COMMERCIAL 1% 3% (Rs.3,20,000) MINIMAL EXPOSURE 50% 30% A PRODUCT MIX PROBLEM A COSTS RS.150/- PER PAIR B COSTS RS.100/- PER PAIR BUYER WANTS:- EXACTLY 5000 PAIRS NOT MORE THAN 2000 PAIRS OF A FOR EVERY ONE PAIR OF A NOT MORE THAN TWO PAIRS OF B OBJECTIVE: MINIMIZE THE COSTS

A Product Mix Problem Cont/Pair Rs 150 Rs.100 HIKING BOOTS SKING BOOTS AVAILABLE SEWING 2 3 13 HOURS STRETCHING 5 2 16 HOURS SOLUTION :- HB-2, SB-3 TC = Rs 600. Another Product Mix Problem Cont/Pair Rs.100 Rs.150 A( moccasin) B (derby) Cutting & 100 Or 150 Upper Closing Lasting 50 Or 200 Sole Adhesion 250 Or 125 Solution :- A - 21.42857 B - 114.2857 TC – Rs 19285.71

PRODUCTS MATERIALS CONT/UNIT A B C P 1 1 2 3 Rs.3 P 2 2 1 1 Rs.4 P 3 3 2 1 Rs.5 AVAILABLE 10 12 15 PRIMAL PROBLEM MAX Z = 3X 1 +4X 2 +5X 3 X 1 +2X 2 +3X 3 < 10 2X 1 +X 2 +2X 3 < 12 3X 1 +X 2 +X 3 < 15 X 1 , X 2 , X 3 > 0 SV CONT Q 3 4 5 0 0 0 COEFF. X 1 X 2 X 3 S 1 S 2 S 3 S 2 0 1 0 0 4/5 -1/5 1 -3/5 X 1 3 4 1 0 -1/5 -1/5 0 2/5 X 2 4 3 0 1 8/5 3/5 0 -1/5 OC -4/5 -9/5 -2/5 DUAL PROBLEM Mm Z* = 10Y 1 +12Y 2 +15Y 3 Y 1 +2Y 2 +3Y 3 > 3 2Y 1 +Y 2 +Y 3 > 4 3Y 1 +2Y 2 +Y 3 > 5 Y 1 , Y 2 ,Y 3 > 0

SV COST Q 10 12 15 0 0 0 COEFF. Y 1 Y 2 Y 3 S 1 S 2 S 3 Y 3 15 2/5 0 3/5 1 -2/5 1/5 0 S 3 0 4/5 0 -4/5 0 1/5 -8/5 1 Y 1 10 9/5 1 1/5 0 1/5 -3/5 0 OC 1 4 3

Applications Of Linear Programming In Production Management Product Mix 2 Production Smoothing 3. Assembly Line Balancing 4. Sub Contracting 5. Some Purchasing Decisions 6 . Location Of Production Facilities Applications Of Linear Programming In Marketing Management 1. Media Planning 2. Routing Of Salesmen 3. Physical Distribution 4. Warehousing Decisions

APPLICATIONS OF LINEAR PRORAMMING IN FINANCIAL MANAGEMENT 1. Capital Budgeting 2. Financing Decisions 3. Portfolio Selection 4. Profit Planning 5 . Financial Audit Applications Of Linear Programming In Personnel Management 1. Job Assignment 2. Manpower Scheduling 3. Manpower Planning 4. Equitable Salaries 5. Manpower Deployment

Limitations Of Linear Programming 1. Linearity 2. Additivity 3. Continuity 4. Certainty 5. Single Objective

A TOOTHPICK MANUFACTURER MAKES TWO KINDS OF TOOTHPICKS: ROUNDS AND FLATS. MAJOR PRODUCTION FACILITIES INVOLVED ARE CUTTING AND PACKING. THE CUTTING DEPARTMENT CAN PROCESS 300 BOXES OF ROUNDS OR 600 BOXES FLATS PER HOUR. THE PACKING DEPARTMENT CAN PACKAGE 600 BOXES OF ROUNDS OR 300 BOXES OF FLATS PER HOUR. – IF THE CONTRIBUTION OF PROFIT FOR A BOX OF ROUNDS IS EXACTLY SAME AS THAT FOR A BOX OF FLATS, WHAT IS THE OPTIMUM PRODUCTION LEVEL? – UNDER WHAT CIRCUMSTANCES WOULD THE MANUFACTURER BE BETTER OFF TO PRODUCE ONLY ROUNDS? 2 A TRUCKING FIRM HAS RECEIVED AN ORDER TO MOVE 3,000 TONNES OF INDUSTRIAL MATERIAL TO A DESTINATION 1,000 KM. AWAY. THE FIRM HAS AVAILABLE AT THE MOMENT A FLEET OF 150 CLASS A 15 TONNE TRAILER TRUCKS AND ANOTHER FLEET OF 100 CLASS B 10 TONNE TRAILER TRUCKS. THE OPERATING COSTS OF THESE TRUCKS ARE RS. 3 AND RS. 4 PER TONNE KM RESPECTIVELY. BASED ON PAST EXPERIENCE, THE FIRM HAS A POLICY OF RETAINING AT LEAST ONE CLASS-A TRUCK WITH EVERY TWO CLASS-B TRUCKS IN RESERVE. IT DESIRES TO KNOW HOW MANY OF THE TWO CLASSES OF VEHICLES SHOULD BE DISPATCHED TO MOVE THE MATERIALS AT MINIMAL OPERATING COSTS.

3. A FARMER HAS A 100- HECTARE FARM. HE CAN SELL ALL THE TOMATOES, LETTUCE, OR RADISHES HE CAN RAISE. THE PRICE HE CAN OBTAIN IS RE.1 PER KILOGRAM FOR TOMATOES, RE. 0.75 A HEAD FOR LETTUCE AND RS. 2 PER KILOGRAM FOR RADISHES. THE AVERAGE YIELD PER HECTARE IS 2,000 KILOGRAM OF TOMATOES, 3000 HEADS OF LETTUCE AND 1,000 KILOGRAMS OF RADISHES. FERTILIZER IS AVAILABLE AT RE. 0.50 PER KILOGRAM AND THE AMOUNT REQUIRED PER HECTARE IS 100 KILOGRAMS EACH FOR TOMATOES AND LETTUCE AND 50 KILOGRAMS FOR RADISHES. LABOUR REQUIRED FOR SOWING, CULTIVATING AND HARVESTING PER HECTARE IS 5 MAN-DAYS EACH FOR TOMATOES AND RADISHES AND 6 MAN-DAYS FOR LETTUCE. A TOTAL OF 400 MAN-DAYS OF LABOUR ARE AVAILABLE AT RS.20 PER MAN-DAY. FORMULATE THIS PROBLEM AS A LINEAR PROGRAMMING MODEL TO MAXIMIZE THE FARMER’S TOTAL PROFIT.

4. FOUR PRDUCTS HAVE TO BE PROCESSED THROUGH THE PLANT, THE QUANTITIES REQUIRED FOR THE NEXT PRODUCTION PERIOD BEING: PRODUCT 1 2,000 UNITS PRODUCT 2 3,000 UNITS PRODUCT 3 3,000 UNITS PRODUCT 4 6,000 UNITS THERE ARE THREE PRODUCTION LINES ON WHICH THE PRODUCTS COULD BE PROCESSED. THE RATES FOR PRODUCTION IN UNITS PER DAY AND THE TOTAL AVAILABLE CAPACITY IN DAYS ARE GIVEN IN THE FOLLOWING TABLE. THE COST OF USING THE LINES IS RS. 600, RS. 500, RS. 400 PER DAY,RESPECTIVELY, ASSIGNMENT OF FOUR PRODUCTS: RATES OF PRODUCTION IN UNITS PER DAY. PRODUCTION PRODUCT MAX.LINE LINE 1 2 3 4 CAPACITY(DAYS) A 150 100 500 400 20 B 200 100 760 400 20 C 160 80 890 600 18 FORMULATE THE ABOVE AS A LINEAR PROGRAMMING PROBLEM TO MINIMIZE THE COST OF PRODUCTION .

5. PRQ FEED COMPANY MARKETS TWO FEED MIXES FOR CATTLE. THE FIRST MIX, FERTILEX, REQUIRES AT LEAST TWICE AS MUCH WHEAT AS BARLEY. THE SECOND MIX, MULTIPLEX REQUIRES AT LEAST TWICE AS MUCH BARELY AS WHEAT. WHEAT COSTS RS.1.50 PER KG. AND ONLY 1,000 KG. ARE AVAILABLE THIS MONTH. BARLEY COSTS RS. 1.25 PER KG. AND 1200 KG. ARE AVAILABLE. FERTILEX SELLS FOR RS.1.80 PER KG. UPTO 99 KG, AND EACH ADDITIONAL KG. OVER 99 KG. SELLS FOR RS.1.65. MULTIPLEX SELLS AT RS. 1.70 PER KG. UPTO 99 KG. AND EACH ADDITIONAL KG.OVER 99 KG. SELLS FOR RS.1.55. BHARAT FARMS WILL BUY ANY AND ALL AMOUNTS OF BOTH MIXES PQR FEED COMPANY WILL MIX. SET UP THE LINEAR PROGRAMMING PROBLEM TO DETERMINE THE PRODUCE MIX THAT RESULTS IN MAXIMUM PROFITS .

A MANUFACTURER HAS CONTRACTED TO PRODUCE 2,000 UNITS OF A PARTICULER PRODUCT OVER THE NEXT EIGHT MONTHS. DELIVERIES ARE SCHEDULED AS FOLLOW: JANUARY 100 FEBRUARY 200 MARCH 300 APRIL 400 MAY 100 JUNE 100 JULY 500 AUGUST 300 TOTAL 2,000 THE MANUFACTURER HAS ESTIMATED THAT IT COSTS HIM RE. 1 TO STORE ONE UNIT OF PRODUCT FOR ONE MONTH. HE HAS A WAREHOUSE CAPACITY OF 300 UNITS. THE MANUFACTURER CAN PRODUCE ANY NUMBER OF UNITS IN A GIVEN MONTH, SINCE THE UNIT CAN BE PRODUCED MOSTLY WITH PART-TIME LABOUR, WHICH CAN BE EASILY OBTAINED. HOWEVER,THERE ARE THE COST OF TRANING NEW PERSONNEL AND COSTS ASSOCIATED WITH LAYING OFF PERSONNEL WHO HAVE BEEN HIRED. THE MANUFACTURER HAS ESTIMATED THAT IT COSTS APPOXIMATELY 75 PAISE PER UNIT TO INCRESE THE PRODUCTION LEVEL FROM ONE MONTH TO THE NEXT(E.G. IF PRODUCTION IN JANUARY IS 200 AND IS INCREASED TO 300 IN FEBRUARY, THE COST IS RS.75 FOR TRANING THE ADDITIONAL PEOPLE REQUIRED TO PRODUCE AT THE 300 UNIT LEVEL.)SIMILARLY IT COSTS 50 PAISE PER UNIT TO REDUCE PRODUCTION FROM ONE MONTH TO THE NEXT. (AT THE END OF EIGHT MONTHS, ALL EMPLOYEE WILL BE LAID OFF WITH THE CORRESPONDING PRODUCTION-REDUCTION COSTS). ASSUME THE PRODUCTION LEVEL BEFORE JANUARY IS ZERO . FORMULATE THE ABOVE AS A LINEAR PROGRAMMING PROBLEM.

Transportation Models There are a number of availability centres such as factories, warehouses etc. With known availabilities. There are a number of consumption centres such as warehouses, markets etc. With known requirements. The distribution cost (freight, handling costs etc.) Per unit from an availability centre to a consumption centre is given The problem is to find an optimal distribution plan i.e. how many units should be allocated from which availability centre to which consumption centre so that the total distribution cost is minimized . Single commodity Direct shipment

The Assignment Model The assignment model is a special type of linear programming problem in which ‘n’ items are assigned among ‘n’ receivers, one item to a receiver, such that the total return resulting from the assignment is optimized. The returns associated with each assignment are assumed to be known and independent of each other. Ex-1 A marketing manager may have four salesmen and four sales territories. Which assignment will help in maximizing sales? Ex-2 A foreman may have five mechanics and five jobs, how the assignment should be made so as to minimize the total time taken for completing the jobs.

AN ASSIGNMENT PROBLEM WORKERS I II III IV A 16 15 18 JOBS B 13 16 14 C 14 13 11 D 16 18 17 A ROUTING PROBLEM TO CITY A B C D E A - 4 7 4 FROM B 4 - 3 4 CITY C 7 6 - 7 D 3 7 - 7 E 4 5 7 - 160 10 60 80 80 110 A TRANSPORTATION PROBLEM WAREHOUSES 14 12 3 5 12 15 6 3 4 FACTORIES TO/ FROM D E F G CAPACITY A 42 48 38 37 160 B 40 49 52 51 150 C 39 38 40 43 190 REQD. 80 90 110 220 500/500

For Maximization Case The numerical value of a given objective function with respect to the integer solution can never exceed the numerical value of that objective function with respect to the non-integer solution. For Minimization Case The numerical value of a given objective function with respect to the integer solution can never be less than the numerical value of that objective function with respect to the non-integer solution.

The cutting division of the photo film corporation requisitions from stock control department plastic films of 85 feet (fixed unit length) which can be cut according to two patterns. First pattern will cut each film length into 35 feet pieces with the remaining 15 feet to scrap. Second pattern will cut each film length into a 35 feet piece and two 25 feet pieces with nothing to scrap. The present order from a customer is for 8 pieces of 35 feet length and 6 pieces of 25 feet length. What minimum number of plastic films of 85 feet should be cut to meet customer requirement?

Problems On Integer Linear Programming 1. The ABC company requires an output of at least 200 unit of a product per day and to accomplish this target it can buy machine A or B or both. Machine A costs Rs.20,000 while machine B costs Rs.15,000 and company has a budget of Rs.2,00,000 for the same. Machines A and B will produce 24 and 20 units respectively of this product per day. However, machine A will require a floor space of 12 square feet while machine B will require 18 square feet and company has total floor space of 180 square feet only. Determine the minimum number of machines that should be purchased.

2. ABC COMPANY HAS 4 INDEPENDENT INVESTMENT PROJECTS AND MUST ALLOCATE A FIXED CAPITAL TO ONE OR MORE OF THEM SO THAT THE COMPANY’S NET PRESENT VALUE IS MAXIMIZED. THE ESTIMATED NET PRESENT VALUE AND THE ANTICIPATED CASH OUTFLOWS ASSOCIATED WITH THESE PROJECTS IS GIVEN IN THE FOLLOWING TABLE: NPV CASH OUTFLOWS(RS.1000) PR. NO (RS. 1000) YEAR(I) YEAR(II) 1 100 50 150 2 50 105 30 3 140 318 143 4 90 100 68 IN SELECTING THESE PROJECTS, THE COMPANY IS CONSTRAINED TO LIMIT ITS EXPENDITURE IN THE FIRST YEAR TO RS.5, 15,000 AND IN THE SECOND YEAR TO RS. 6,38,000. IF PROJECTS 1 AND 3 ARE MCTUALLY EXCLUSIVE, HOW SHOULD THE INVESTMENT BE MADE SO THAT THE TOTAL NET PRESENT VALUE IS MAXIMIZED?

SET-UP COST PER MAX. MACHINE COST(Rs ) UNIT (Rs) PRODUCTION 1 8000 5 4000 2 5000 4 3000 3 4000 8 1000 QUANTITY REQUIRED:5000 UNITS AT MINIMUM COST. PRODUCT PLANT P Q R CAPACITY A 35 24 20 600 B 30 28 25 1,000 C 20 25 37 800 D 24 32 28 800 DEMAND 500 800 600

GOALS HAVE PREEMPTIVE PRIORITIES. MINIMIZE DEVIATIONS FROM GOALS. 3. ATTACH WEIGHTS TO DIFFERENT ACTIVITIES AT THE SAME GOAL LEVEL. BASIC CONCEPTS OF GOAL PROGRAMMING

NTC PRODUCES TWO TYPES OF MATERIALS, A STRONG UPHOLSTERY MATERIAL AND A REGULAR DRESS MATERIAL. THE UPHOLSTERY IS PRODUCED ACCORDING TO DIRECT ORDERS FROM FURNITURE MANUFACTURERS. THE DRESS MATERIAL ON THE OTHER HAND, IS DISTRIBUTED TO RETAIL FABRIC STORES. AVERAGE PRODUCTION RATES FOR THE TWO MATERIALS ARE IDENTICAL; 1000 METRES/HR. BY RUNNING TWO SHIFTS, NET OPERATIONAL CAPACITY OF THE PLANT IS 80 HOURS/WK. THE MARKETING DEPARTMENT REPORTS THAT THE MAXIMUM ESTIMATED SALES FOR THE FOLLOWING WEEK IS 70,000 M. OF UPHOLSTERY AND 45,000 M. OF DRESS MATERIAL. ACCORDING TO THE ACCOUNTING DEPARTMENT, THE APPROXIMATE PROFIT FROM A METRE OF UPHOLSTERY MATERIAL IS RS.2.50 AND FROM A METRE OF DRESS MATERIAL IS RS.1.50. THE M.D. OF THE COMPANY BELIVES THAT A GOOD EMPLOYER- EMPLOYEE RELATIONSHIP IS IMPORTANT IN BUSINESS. HENCE HE DECIDES THAT A STABLE EMPLOYMENT LEVEL IS A PRIMARY GOAL FOR THE FIRM. THEREFORE, WHENEVER THERE IS EXCESS DEMAND OVER NORMAL PRODUCTION, HE SIMPLY EXPANDS PRODUCTION CAPACITY BY PROVIDING OVERTIME. HOWEVER HE FEELS THAT OVERTIME OF MORE THAN 10 HOURS/WK. SHOULD BE AVOIDED – BECAUSE OF ACCELERATING COSTS. CONTD….P/2.

FIRST GOAL : AVOID UNDER UTILISATION OF PRODUCTION CAPACITY, I.E. MAINTAIN STABLE EMPLOYMENT AT NORMAL CAPACITY SECOND GOAL : LIMIT OT OPERATION TO 10 HOURS. THIRD GOAL :ACHIEVE SALES GOALS OF 70,000 M. OF UPHOLSTERY AND 45,000 M. OF DRESS MATERIAL. FOURTH GOAL : MINIMISE OT OPERATION AS MUCH AS POSSIBLE. FORMULATE AND SOLVE THIS PROBLEM AS A GOAL PROGRAMMING PROBLEM.

BHARAT TELEVISION COMPANY PRODUCES CTV SETS. IT HAS TWO PRODUCTION LINES. PRODUCTION RATE OF LINE-1 IS 2 SETS/HR. AND IT IS 1.1/2 SETS/ HR. IN LINE-2. THE REGULAR PRODUCTION CAPACITY IS 40 HR./WK. FOR BOTH LINES. EXPECTED PROFIT FROM AN AVERAGE CTV SET IS RS.1000/- . THE TOP MANAGEMENT OF THE FIRM HAS THE FOLLOWING GOALS FOR THE WEEK (IN ORDINAL RANKING): GOAL -1: PRODUCTION GOAL –180 SETS. GOAL –2 : LIMIT OT OF LINE-1 TO 10 HOURS. GOAL –3 : AVOID UNDERUTILISATION OF REGULAR WORKING HOURS FOR BOTH LINES. GOAL –4 : LIMIT THE SUM OF OT FOR BOTH LINES. (ASSIGN WTS. ACCORDING TO RELATIVE COST OF OT HOUR. ASSUME COST OF OPREATIONS IS IDENTICAL FOR BOTH PRODUCTION LINES) FORMULATE THE PROBLEM AS A GLP MODEL. IF THE TOP MANAGEMENT DESIRES TO PUT A PROFIT GOAL OF RS.1,90,000 FOR THE WEEK AS THE TOP PRIORITY GOAL ABOVE THE STATED FOUR GOALS, HOW WOULD THE MODEL CHANGE.

ACADEMIC PLANNING AND ADMINISTRATION ACCOUNTING ANALYSIS ADVERTISING MEDIA SCHEDULING BLOOD COLLECTION AND DISTRIBUTION CAPITAL BUDGETING COMPUTER RESOURCE PLANNING AND ALLOCATION DECISION-SUPPORT SYSTEM DESIGN ECONOMIC POLICY ANALYSIS EDUCATIONAL SYSTEM PLANNING ENVIRONMENTAL PROTECTION PORTFOLIO DETERMINATION PRODUCTION SCHEDULING PROJECT SCHEDULING QUALITY CONTROL FACILITIES LOCATION AND LAYOUT PLANNING FINANCIAL PLANING HEALTH-CARE DELIVERY SYESTEM DESIGN INVENTORY MANAGEMENT LOCATION AND ALLOCATION DECISIONS MANPOWER PLANNING MARKETING LOGISTICS MILITARY STRATEGIES AND PLANNING NETWORK SCHEDULING ORGANIZATIONAL ANALYSIS PERSONNEL ADMINISTRATION POLICY ANALYSIS RESEARCH AND DEVELOPMENT TRANSPORTATION LOGISTICS URBAN PLANNING WATER RESOURCES PLANNING APPLICATION AREAS OF GOAL PROGRAMMING GOAL PROGRAMMING HAS BEEN WIDELY APPLIED TO VARIOUS DECISION PROBLEMS IN BUSINESS FIRMS, GOVERNMENT AGENCIES, AND NON PROFIT INSTITUTIONS. SOME OF THE BEST KNOWN APPLICATIONS OF GOAL PROGRAMMING INCLUDE THE FOLLOWING PROBLEM AREAS:

LOCATION 1 2 3 4 5 6 7 1 - 12 27 14 45 36 15 2 - 10 25 32 M 22 3 - 28 50 28 10 4 - 16 20 32 5 - 26 35 6 - 20 7 - . TOTAL OF 30, 50 & 20 TONNES OF THIS COMMODITY ARE TO BE SENT FROM LOCATIONS 1, 2 & 3 RESPECTIVELY. A TOTAL OF 15, 30 25 & 30 TONNES ARE TO BE SENT TO LOCATIONS 4, 5, 6 & 7 RESPECTIVELY. SHIPMENTS CAN BE SENT THROUGH INTERMEDIATE LOCATIONS AT A COST EQUAL TO THE SUM OF THE COSTS FOR EACH OF THE LEGS OF THE JOURNEY. THE PROBLEM IS TO DETERMINE THE OPTIMAL SHIPPING PLAN. A CERTAIN CORPORATION MUST SHIP A CERTAIN PERISHABLE COMMODITY FROM LOCATIONS 1, 2, 3, TO LOCATIONS 4, 5, 6 &7. A THE AIR FREIGHT PER TONNE (IN 100 RS.) BETWEEN SEVEN LOCATIONS IS GIVEN IN THE FOLLOWING TABLE. WHERE NO DIRECT AIR FREIGHT SERVICE IS AVAILABLE, A VERY HIGH COST M HAS BEEN USED.

A PROBLEM ON DYNAMIC PROGRAMMING A GOVERNMENT SPACE PROJECT IS CONDUCTING RESEARCH ON A CERTAIN ENGINEERING PROBLEM THAT MUST BE SOLVED BEFORE MAN CAN FLY TO THE MOON SAFELY. THREE RESEARCH TEAMS ARE CURRENTLY TRYING THREE DIFFERENT APPROACHES FOR SOLVING THIS PROBLEM THE ESTIMATE HAS BEEN MADE THAT UNDER PRESENT CIRCUMSTANCES, THE PROBABILITY THAT THE RESPECTIVE TEAMS-CALL THEM A,B AND C –WILL NOT SUCCEED IS 0.40,0.60 AND 0.80 RESPECTIVELY. THUS THE CURRENT PROFITABLITY THAT ALL THE THREE TEAMS WILL FAIL IS= 0.192. SINCE THE OBJECTIVE IS TO MINIMIZE THIS PROBABILITY THE DECISION HAS BEEN MADE TO ASSIGN TWO MORE TOP SCIENTISTS AMONG THE THREE TEAMS IN ORDER TO LOWER IT AS MUCH AS POSSIBLE. THE FOLLOWING TABLES GIVES THE ESTIMATED PROBABILITIES OF FAILURE WHEN 0, 1 AND 2 .SCIENTISTS ARE ADDED TO THE TEAMS. TEAM A B C NO.OF 0 0.40 0.60 0.80 NEW SCIENTISTS 1 0.20 0.40 0.50 2 0.15 0.20 0.30 HOW SHOULD THE TWO SCIENTISTS BE ALLOCATED TO THE TEAMS?

Qm linear programming as narag 1

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Qm linear programming as narag 1