A fertilizer manufacturer uses four ingredients to make two fertilizers. It has limited supplies of the ingredients and wants to determine the optimal mix to maximize profits based on the contribution amounts of each fertilizer. A toothpick manufacturer makes two types of toothpicks with different production capacities in cutting and packing. It wants to determine the optimal production level to maximize profits. A feed company markets two cattle feed mixes with different ingredient requirements. It wants to determine the optimal mix of ingredients to use to maximize profits based on available supplies and selling prices of the mixes.

1. Linear Programming Problems A. S. Narag

2. Product Mix : Profit Maximization 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.

3. Product Mix: Profit Maximization 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?

4. Product Mix: Profit Maximization 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 .

5. Product Mix: Profit Maximization 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

6. Product Mix: Profit Maximization 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.

7. Product Mix: Profit Maximization 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.

8. Product Mix : Cost Minimization A state agricultural federation manufactures and markets cattle feed. It uses two ingredients which have the following characteristics :- Ing.A Ing.B Cost/kg. Rs.10 Rs.40 Fiber content 25% 75% Fat content 20% 10% 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?

9. Cost Minimization 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.

10. Transportation Problem: Minimal Operating Costs 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.

11. Product Mix: Minimize Cost of Production 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

12. Product Mix: Cost Minimization 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

13. Media Planning Problem: Cost Minimization EXPOSURE RATES TA TB MAGAZINE 2% 1% (Rs.80,000) TV COMMERCIAL 1% 3% (Rs.3,20,000) MIN. EXPOSURE 50% 30%