Question
Brand X Cannery produces canned whole tomatoes andtomato sauce. This season, the company has available3,000,000 kilograms of tomatoes for these two products. Tomeet the demands of regular customers, it must produce at least80,000 kilograms of sauce and 800,000 kilograms of wholetomatoes. The cost per kilogram is $4 to produce canned wholetomatoes and $3.25 to produce tomato sauce. Labor agreemetnsrequire that at least 110,000 person-hours be used. Each1-kilogram can of sauce requires 3 minutes for one worker toproduce, and each 1 kilogram can of whole tomatoes requires 6minutes for one worker. How many kilograms of tomatoes shouldBrand X use for each product to minimize cost? (Forsimplicity, assume that the production of y1kilograms of canned whole tomatoes and y2 kilograms of tomato sauce requiresy1 + y2 kilograms oftomatoes.)
Explanation / Answer
Question Details: Brand X Cannery produces canned whole tomatoes[T] and tomatosauce[S]. This season, the company has available 3,000,000kilograms of tomatoes for these two products. ALL UNITS ARE IN THOUSAND KG. T+S=80..........................................2 T>=800........................................3 The cost per kilogram is $4 to produce canned whole tomatoes and$3.25 to produce tomato sauce. TOTAL COST = C =4T+3.25S...............................................4 Labor agreemetns require that at least 110,000 person-hours [H]beused. H>=110................................................5 [ INTHOUSAND HOURS....] Each 1-kilogram can of sauce requires 3 minutes for one worker toproduce, and each 1 kilogram can of whole tomatoes requires 6minutes for one worker. H=S*3*1000/(60*1000)+T*6*1000/(60*1000).......THOUSAND HOURS H=0.05S +0.1T>=110.................................................6 How many kilograms of tomatoes should Brand X use for eachproduct to minimize cost? (For simplicity, assume that theproduction of y1 kilogramsof canned whole tomatoes and y2 kilogramsof tomato sauce requires y1 +y2 kilogramsof tomatoes.) SO WE HAVE CONSTRAINTS AS T+S=80..........................................2 T>=800........................................3 H=0.05S +0.1T>=110.................................................6 AND OPTIMIZATION FUNCTION AS C =4T+3.25S...............................................4 POSSIBLE OPTIMUM POINTS ARE 1 & 2 INTERSECTION.............T=2920....S=80..........C=4*2920+3.25*80=11940 1 & 3........................................T=800....S=2200........C= 1 &6..........................................T=-800.....S=3800....NOTFEASIBLE 2 & 3...........................................NO 2 & 6........................................S=80..........T=1060.....C=4*1060+3.25*80=4500 3 &6.........................................T=800.........S=600......C=4*800+3.25*600=5150 How many kilograms of tomatoes should Brand X use for each productto minimize cost? INTERSECTION OF 2 & 6 IS THE MINIMUM COST ALTERNATIVE...THATIS T=1060 THOUSAND KG=1060000 KG AND S=80 THOUSAND KG =80000 KG BUT THE QUESTION IS SOME WHAT FLAWED! IN THAT IT ASKS FOR MINIMUMCOST OF PRODUCT SUBJECT TO CONSTRAINT ON T AND S GIVEN BY EQNS 2AND 3 .. SO EMPLOYING MINIMUM HRS TO CATER FOR REQUIREMENT IS OK, THOUGH ITIS NOT GOOD TO LEAVE UN UTILIZED 3000-1060-80=1860 THOUSAND KG OFTOMATOS CHECK BACK ON TYPOS IF ANY ?