Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function: S = 15 L0.2 C 0.8 In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100. (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 50,000 tons of steel at minimum cost. Min L + C L0.2 C 0.8 L, C (b) Solve the optimization problem you formulated in part a. Hint: When using Excel Solver, start with an initial L > 0 and C > 0. If required, round your answers to two decimal places. L = $ C = $ Cost = $

Solution

S = 15 x x

Total cost, T = wL + rC = 50L + 100C

Total revenue, R = Output price (P) x Quantity = P x 15 x x

(a)

Optimization problem will be:

Max R = P x 15 x x

Subject to T = 50L + 100C

(b) When S = 50,000

Cost is minimized when (MPL / MPC) = w / r

MPL = R / L = P x 15 x 0.2 x = P x 3 x

MPC = R / C = P x 15 x 0.8 x = P x 12 x

MPL / MPC = (3/12) x (C / L) = 50/100

C / 4L = 1/2

4L = 2C

2L = C

Substituting in production function,

15 x x = S

15 x x = 50,000

15 x x x = 50,000

L = 50,000 / (15 x 20.8)

L = 1,914.50

C = 2L = 3,829.00

Total cost ($) = 50 x 1,914.50 + 100 x 3,829.00 = 95,725.00 + 382,900 = 478,625.00

Note: This optimization problem can be solved without using Solver too, as shown here.

in 2001, dell computer became the world’s largest personal computer vendor, continuing to

gain market share and post profits in an industry struggling with slumping sales and billions of

dollars in losses. dell sells 90% of its pcs directly to the final customer, largely bypassing the

reseller channel that accounts for most of the world’s pc sales. this direct customer relationship

is the key to dell’s business model, and provides distinct advantages over the indirect sales

model. dell’s direct relationship with the customer allows it to tailor its offerings to customer

needs, offer add-on products and services, and use the internet to offer a variety of customer

services. in addition, dell’s pcs are built to customers’ specifications upon receipt of an order,

giving dell additional advantages over indirect pc vendors who must try to forecast demand and

ship products based on those forecasts. dell’s direct sales and build-to-order model has achieved

superior performance in the pc industry in terms of inventory turnover, reduced overhead, cash

conversion, and return on investment (kraemer, et al., 2000).

explanation: