Constrained Optimization: The Method of Lagrange Multipliers:


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1 Constrained Optimization: The Method of Lagrange Multipliers: Suppose the equation p(x,) x 60x 7 00 models profit when x represents the number of handmade chairs and is the number of handmade rockers produced per week. The optimal (maximum) situation occurs when x = 5 and =. However due to an insufficient labor force the can onl make a total of 0 chairs and rockers per week ( x + = 0). So how man chairs and how man rockers will give the realistic maximum profit? We will come back to this questions shortl but first we will look at the following example. Using Level Curves and the constraint function to determine optimal points: A Compan has a determined that its production function is the CobbDouglas function f (x,) x where x is the number of labor hours and is the number of capitol units. The budget constraint for the compan is given b 00x a) If the compan decides to spend $00, 000 on x then how much can be spent on under this budget constraint? solution: If $00000 is spent on x then 00x = and x = 000. There is $00000 left to spend on therefore 00 = and = 000. In this case total production will be f(000, 000) = (000) (000) = (08)(0) = 080 units. Suppose the amount spent on x is changed to $50,000. How will that change production? Following the same procedure as above x = 500 and = 500. The production f(500, 500) then be 89 units. Is there a wa to determine values for x and that will give the optimal production possible given this budget constraint? we will first tr to find the optimal situation using level curves with the constraint function. It turns out that the global maximum or global minimum occurs where the graph of the constraint equation is tangent to one of the level curves of the original function. In order to plot the level curves we must solve for to get f (x,) > C x x where C now represents values of f(x,). Now we will plot for various values of C. (The choice of values for C is determined mostl b common sense.) In this case we
2 will let C = 000, C = 000 and C = 000. After looking at the resulting level curves and constraint, I decide to add another value C = 00. Now I can make a good guess After considering the graph below, we can guess the optimal value will occur for x = 600 and = 400. optim al solution 5000 C=000 C=000 C=00 C=00 constraint x There is also an algebraic approach to finding the optimal solution given a certain constraint. We will use the method of Lagrange Multipliers to find the maximum situation in the problem above. In the proceeding sections ou have learned how to use partial derivatives to find the optimal situation for a multivariable equation. Now we will expand on this process to find the optimal situation with a constraint. Below is an outline of the process we will use followed examples. Procedure for Appling the Method of Lagrange Multipliers: In order to maximize or minimize the function f(x,) which is subject to the constraint g(x, ) = k we will follow the following procedure. Step # First create the LaGrange Function. This function is composed of the function to be optimized combined with the constraint function in the following wa: L(x, ) = f(x, ) [g(x, )  k]
3 Step # Now find the partial derivative with respect to each variable x, and the Lagrange multiplier of the function shown: L(x, ) = f(x, ) [g(x, )  k] Step # Set each of the partial derivatives equal to zero to get L x = 0, L = 0 and L = 0 Using L x = 0, L = 0, proceed to solve for x and solve for in terms of. Now substitute the solutions for x and so that L = 0 is in terms of onl. Now solve for and use this value to find the optimal values x and. Note: If M is the max or min value of f(x, ) subject to the constraint g(x, ) = k, then the Lagrange multiplier is the rate of change in M with respect to k. (i.e. = dm/dk and therefore approximates the change in M resulting in a one unit increase in k.) Example We will revisit the CobbDouglas function f (x,) x and the budget constraint function 00x We will use the method of Lagrange multiplier described above to find the optimal solutions. First we will create the Lagrange equation L(x,) x (00x Now find L x = 0, L = 0 and L = 0. L x x 00 L x 00 L 00x Using L x = 0, L = 0 to solve for x and Set the first derivative L x = 0 to get and solve for = solve for again to get = x 00 x 00 and then set L = 0 and We now need to use the results to turn the equation L = 0 into an equation of one variable onl. To do this we will set the equations equal to each other and solve for as shown below.
4 x 00 = x 00 > x 00 = x 00 > multipl both sides b 00 and then b to get x x x x x x Now substitute the values of into L = 0 to get 00x 00( x) and solve for x = and then find to be =.. The value of x and that will maximize production is x = 666 and = and the maximum production is 6 units. Example : We will now revisit the profit equation introduced at the beginning of this section. p(x,) x 60x 7 00 where x represents the number of handmade chairs and is the number of handmade rockers produced per week. The constraint on this profit is g(x, ) = x + = 0. In order to find the optimal situation given this constraint we will follow the method given above. First we will create the LaGrange equation L(x, ) = p(x, ) [g(x, ) k] = ( x 60x 7 00) (x 0). Now find L x = 0, L = 0 and L = 0. L x 4 x 60 0 L L x 0 0 Using L x = 0, L = 0 to solve for x and to get x 5 4 and 6. Now substitute the values of x and into L = 0 to get L x 0 (5 4 ) ( 0 ) 0 ( ) and = 6.8 so x and Therefore the maximum profit will be achieved when 0 chairs and 9 rockers are produced. ( Remember that the maximum profit is x = 5 and = when there is not constraint.)
5 Notice that the value of = 6.8 and if the constraint was increased from 0 to then the maximum profit would increase b $6.80 Example : Amanda is getting a new dog. She wants to build a pen for her dog in the back ard. The pen will be rectangular using 00 feet of fence. Amanda plans to build the pen up against the wall of her house so that she will onl need three sides of fence. She wants to build a pen with the maximum amount of area. Therefore we need to maximize the area equation A = x. Since Amanda has a constraint in the amount of fencing she can use, we will use the LaGrange method and create the following equation. L(x,) x (x 00) Now find L x = 0, L = 0 and solve for x and to get that x = and =. Find L = 0 and substitute in the values for x and to put the equation in terms of. Now solve to get that = 50. Therefore the optimal values are x = 50 and = 00 feet of fencing. (Notice that if the constant 00 feet of fencing is increased b on foot to 0 feet then b marginal analsis the maximum value for the area will increase b 50 square feet. Determine what the maximum area would be for 0 feet of fencing) Answer is Example 4 The function U = f(x,) represent the utilit or customer satisfaction derived b a consumer from the consumption of a certain amount of product x and a certain amount of product. In the example below the maximum utilit is determined when the given budget constraint affects the amount of each product produced. The function f(x,) x is a utilit function with a budget constraint given to be g(x,) = x + 4 = 40. The first step is to create the LaGrange equation L(x,) x (x 4 40) L x L x x 0 and therefore = x L L x 4 0 and therefore = x The next step is to find L x and then use the partials above to help transform L into an equation with just one variable. It does not matter if it is terms of x onl, onl or onl ( as in the two examples above)
6 In this case it is easier to set the equations equal to each other and solve for either x or as seen below. set x x x x x Going back and substitute into L x 4 40 x 4( x ) 40 0 and solve to get x = 0 and then =5. ( You will also get = x = (0)(5) = 50. What is the meaning of this information? It means that if the constant is increased to 4 the maximum utilit will increase b 50 ) Practice Problems Problem ) Find the minimum value of the function f(x, ) = x +  x subject to the constraint x + =. Problem The highwa Department is planning to build a picnic area for motorist along a major highwa. It is rectangular with area of 5,000 square ards. It is to be fenced off on the three sides not adjacent to the highwa. What is the least amount of fencing needed to complete the job? Problem Using the information given in problem graph the level curves and the constraint function and indicate where the minimum point will occur. Problem 4 Two women have decided to start a cottage industr that will be run out of their homes. The are going to make two tpes of specialt shirts, rhinestone and embroider. The total gross profit is given b P(x, ) = 6x x  where x is the number of rhinestone shirts made and is the number of embroider shirts made per month. Together, the are able to make at most a total of six shirts per month. What is the number of each tpe of shirt that the should tr to make and sell in order to maximize their profit each month.? Problem 5 If the equation f(x,, z) = x + + z is subject to the constraint x  + z = 6, find the optimal point under these conditions.
7 Problem 5 A) An editor has been allotted $60000 to spend on the development and promotion of a new book. It is estimated that if x thousand dollars is spent on development and thousand on promotion, approximatel f(x, ) = 0x copies of the book will be sold. How much mone should the editor allocate to development and how much to promotion in order to maximize sales? C) Graph the level curves and the constraint function and determine the maximum point. B) Suppose the editor s allocation is increased to $6000 to spend on the development and promotion of the new book. Estimate how the additional $000 will affect the maximum sales level. Problem 6 A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is to be,00 square meters and no fencing is needed along the bank. Find the dimensions of the pasture that will require the least amount of fencing. Problem 7 A CobbDouglas production function for new compan is given b f (x,) 50x 5 5 where x represents the units of labor and represents the units of capital. Suppose units of Labor and capital cost $00 and $00 each respectivel. If the budget constraint is $0,000, find the maximum production level for this manufacture.
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