ISyE 3104: Introduction to Supply Chain Modeling: Manufacturing and Warehousing

Instructor: Spyros Reveliotis

Spring 2001

Homework #2 Solutions

Problem 1: Experimentation with the Excel spreadsheet provided in Item 10 of the “course materials” list, indicates that, for the specified problem data, there is no feasible production schedule that can satisfy the entire product demand over the considered planning horizon. Hence, assuming that no additional production capacity (fermentors) can be installed during the considered time interval, we must consider how to accommodate the schedule infeasibility, by deciding which part of the demand should be left uncovered.

This decision should take into consideration the product phases w.r.t. (i) their entire life-cycles, as well as (ii) their seasonal cycles, and their implications for the company’s marketing and distribution policies. One way to reason about this problem is as follows:

  1. The pale ale seems to be the major company product (i.e., the most well-established and with the most extensive circulation), having reached its mature phase. Hence, the company should maintain a stable and responsive production for this product. Furthermore, the current product expansion to a new market (Northeast) implies that the company must be careful with its new undertaken obligations (deals and/or contracts with the new distributors and customers). Based on these considerations, it seems that the demand for pale ale should be met on its entirety.
  2. The stout is a brew that is in its growth phase, and it is developing to the second major product for the company. Therefore, it is to the company’s advantage to promote aggressively this product. Allowing for planned shortages does not seem to support the company interests.
  3. Similarly, the summer brew is a product in its growth phase. Furthermore, the product demand in the considered planning horizon corresponds to the “opening period” for the product. As a result, the company should try to meet the entire expected demand for this product, as well, since in this way it would promote and secure the product position in the market.
  4. Finally, the winter ale is a product that is rather well-established; the product has been around for a while, and its annual demand presents some stability. At the same time, this is a seasonal product for the company, and the estimated product demand for the considered planning horizon– especially the last few weeks - corresponds to the closing phase of the product seasonal cycle.The previous two remarks imply that the company might be able to “take a hit” w.r.t. this product, by terminating the product distribution a little earlier than planned for this year, without this fact affecting significantly the product marketability and the company image.

Based on the above considerations, the proposed production plan is as follows;

Problem 2: Taking into consideration the constraint on the maximal allowed shelf-life can be done according to the ideas and techniques discussed in class (c.f. the discussion on the 2ndformulation in the document of item 10 in the “Course Materials” list).

The variable fermentor availability implies that the value of parameter N, that models the number of fermentors available in the original formulation,now varies from period to period, over the considered planning horizon. Technically, this is modeled by indexing N with t: Nt denotes the number of fermentors available in period t.

Finally, to model production in lots of full or half fermentors, we employ another set of binary variables, Yit, such that Yit=1 impliesinitiation of a lot equal to half fermentor for product iin period t. Hence, the role and presence of variables Yit in the overall formulation is similar to that of variables Xitin the originalformulation, with some adjustments resulting from the fact that these variables now present production of half rather than full-fermentor lots. The corresponding formulation is provided in:

Two additional comments pertaining to the presented formulation are as follows:

  1. The constraint added before the variable sign restrictions enforces that we cannot start, both, a full and a half fermentor of the same product at the same period.
  2. Since, under the new regime, production is carried in lots of, both, full and half fermentors, the originally employed scheme for counting fermentors allocated to product iat period t due to scheduled receipts, is not working any more. Specifically, dividing a scheduled receipt corresponding to a half-fermentor lot by the fermentor capacity will return a fermentor occupancy of ½rather than (the desired value of) 1. Hence, to remedy this problem, now we provide the information regarding the scheduled receipts in two data sets (problem parameters), srf and srh, with the srf series representing scheduled receipts equal to full-fermentor lots, and the srhseriesrepresenting scheduled receipts equal to half-fermentor lots.

Extra Credit: The corresponding INPUT and OUTPUT LINGO files are provided below: