11/13/2023 0 Comments Optimization problems minimize costLike this article? Check out more posts about Calc 1.=1. University of Maryland student? Get a tutor in College Park for in-person, one-on-one help! This works out to be about $25 in "overhead" per tub - not including, of course, the cost of materials, labor, water, and energy. If the box must have a volume of 50ft 3 determine the dimensions that will minimize the cost to build the box. So, at this point we can write our first, very general equation: The material used to build the top and bottom cost 10/ft 2 and the material used to build the sides cost 6/ft 2. Next, students use calculus to find this minimum cost by finding critical values and applying the first or second derivative test. The dimensions of the enclosure that minimize the cost are 20 feet by 30 feet. Cool (or should I say.hot?) This cost, apparently, is going to depend on how many hot tub shells we make at a time - too many at once and we'll have to pay for the extra space to keep them around while we assemble them, too few at once and we'll have to pay the "start-up cost" for the machine more often. In many cases, there are two (or more) variables in the problem. Ok, so we see that they are asking us to minimize cost. If Hot Bod wants to minimize their total production cost while meeting their quota of 10,000 units, how many tub shells should they mold at a time? Once you've read it through once, we can focus on the sentences at the end of the problem and see what they're actually asking for: As usual, you should first read the problem to get a general picture of the scenario (I'll wait). If Hot Bod wants to minimize their total production cost while meeting their quota of 10,000 units, how many tub shells should they mold at a time? How many batches in total will they run? What will be their total production cost? This start-up process is expensive - about an extra $1000 per batch! So, there is a tradeoff: if they run many small batches, they will incur a lot of start-up costs, but if they run fewer, larger batches, they will have to pay for more space to store the shells while they're waiting to be assembled. Since the variables are non-negative, include the constraints: x 0 x 0 y 0 y 0. For standard minimization linear programming problems, constraints are of the form: ax + by c a x + b y c. The injection-molding machine on the other hand is very fast, but there is a catch - each time they start up the machine to make a fresh batch of hot tub shells, they have to carefully heat up and calibrate the machine, and get approved by the plant safety manager. Minimization Linear Programming Problems. The right combination is the one that minimize the cost of producing the given target level of output q0. So the most efficient way in this context refers to what is the 'right' combination of (L,K) so achieve q0. Differentiate both sides of the equation with respect to the variable in the. The only decision the firm controls at this point is how much of inputs it uses. Write an equation in terms of a single variable for the quantity to be maximized/minimized. Identify the quantity to be maximized/minimized and all given values. Berry Plastics also requires that you specify how much storage space you will need in your contract - you have to pay for the space up front even if you're not using it the entire time.Īssembly is a slow and delicate process, so Hot Bod workers can only finish about 10 tubs per day. The IWDDSP method to solving optimization problems: 1. Berry Plastics charges a base cost of $50 per day for use of their facility, plus an extra $1 per day per shell for the space needed to store the shells while they're waiting to be finished. Once the shell has cooled, assembly workers finish the product by attaching the hoses and motors and installing insulation.īeing a small company, Hot Bod doesn't have their own factory - they will have to rent space from the Berry Plastics Corporation. The hot tub shells are made using injection-molding, in which molten plastic is squirted into metal molds at high pressure, and then allowed to cool. They have an exclusive deal with Gallmart to supply the retail giant with 10,000 units over the next several years. Hot Bod Jacuzzi & Spa Company is launching a new hot tub - the Neverleak Massage-o-matic DeLux. The inventory cost problem, however, is something that comes up in real-life manufacturing scenarios all the time - how can I minimize my operating costs? In fact, the problem we see here today is a simplified version of a problem I covered in a DETC conference paper that I published a few years back. A lot of the "word problems" that come up in calculus seem silly and contrived, because they are.
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