10. accounting library and decision frame Ralph produces two products, with respective quantities denoted q1 and q2. Product costs are aggregated into direct material (DM), direct labor (DL), and overhead (OV ) categories. Relevant LLAs are summarized by TMC = 160, 000 + 420q1 + 480q2 and S&A =

10, 000+ 80q1 +20q2. (TMC denotes total manufacturing cost, and S&A is the total of all period costs.) No inventory is present, as all production is sold. Revenue is given by TR = 1, 000q1 + 700q2. In addition, Ralph’s output is limited by the following pair of departmental capacity constraints:

2q1 + q2 ≤ 1, 000 q1 + 2q2 ≤ 1, 000 For convenience, no other costs are present.

(a) Determine Ralph’s optimal production plan, and the shadow prices for each of the two constraints.

(b) After this original plan is formulated, a customer enquires about production of a single unit of a custom product. Ralph estimates incremental TMC will be 300, and incremental S&A will be 0.
This product will require two “hours” of capacity in each department.
Determine the cost of this custom product, assuming Ralph wants to compare the cost of the product to the offered price (
P) to decide whether to oblige the customer.

(c) Conversely, formulate a three variable optimization model to simultaneously determine the optimal quantities of the two original products as well as the custom product described in (b)
above.

(d) Carefully explain why you have two distinct cost figures for the same product, one in part

(b) and the other in part (c).

(e) Presuming production of this custom product, what unit cost for this product will be recorded in the accounting library?