Question

Company XYZ produces and sells widgets. Production is done at two separate factories, one in Mississauga and one in Scarborough. An analysis reveals that in the Mississauga location, it costs Cₘ​(q)=1/50q³+10000 (dollars) to produce q widgets in a month, while in the Scarborough location it costs Cₛ​(q)=q²+20000 (dollars) to produce q widgets in a month. The company needs to fulfill orders for a total of 400 widgets in the month of March 2023. (a) How many widgets should each location be assigned to produce so that the total cost of production is minimized? (b) Due to differences in quality control and shipping costs, company XYZ decides to sell widgets produced at each factory at different price levels: widgets produced in Mississauga are to be sold for $2000 each, and widgets produced in Scarborough for $2800 each. Show that it is most profitable for the company to allocate all production to Scarborough.

Answer 1

To minimize total production cost, we need to find the optimal allocation of widget production between the Mississauga and Scarborough facilities by optimizing the combined cost functions. For profitability, multiplying the quantity by the respective selling prices and subtracting production costs demonstrates selling all widgets produced in Scarborough maximizes profit due to the higher selling price per widget.

Step-by-step explanation:

To determine how many widgets each location should produce to minimize total cost of production, we need to solve the problem using the method of optimization. The total cost function C(q) for producing q widgets can be found by combining the cost functions Cm(q) for Mississauga and Cs(q) for Scarborough. We set up the problem with the constraint that the total quantity of widgets, qm + qs, must be the total order of 400 widgets. We use calculus to find the minimum of the total cost function subject to this constraint.

For part (b), analyzing the profitability of producing widgets at each factory, we can compare the profits by multiplying the production quantity by the selling price at each location, and subtracting the respective production costs. Assuming fixed selling prices of $2000 and $2800 each for Mississauga and Scarborough, respectively, we can create a profit function and determine which production allocation yields the highest profit. Through calculation, we can show that allocating all production to Scarborough will result in higher profit as each widget there sells for $800 more compared to widgets from Mississauga, outweighing any differences in production cost.

Note that the provided sample figures related to economies of scale and toy models of production costs (such as cost per toaster oven or per hour wages for widget workers) are not directly relevant to the specific cost functions given for Company XYZ, so these details are not used in the solution.