Buyer's Direct Mills (BDM) Inc. is a century old textile company that takes raw cotton and spins it into cotton yarn. This yarn is in turn purchased by weaving companies which produce cotton cloth for downstream production into clothing. There are four grades of finished product - Extra Fine (e.g. used in expensive dress shirts), Fine, Medium, and Coarse (e.g. used to create denim cloth for blue jean production).

Through the recent recession Buyer's Direct was able to maintain its workforce and wasn't forced to lay-off staff. Now that the worst of the recession is over, the clothing market is heating up hence an increase in demand for BDM's products.

However, BDM's spinning machine capacity is unable to keep up with production orders. To remedy the situation, BDM has contracted with six other cotton yarn production companies (which were hit hard by the recession and are hungry for the work). Those companies are Able Fibres, Cotton Spin, Denim Source, Every Yarn, Fibres & Filaments and Grand River Milling Company. All of the contractors are able to produce all types of cotton yarn, except for Able Fibres and Denim Source which do not have the modern spinning machines required to produce Extra Fine cotton yarn.

BDM currently receives the raw cotton from its suppliers and does some initial pre-spinning preparation (opening/loosening, carding, drawing & roving) of the cotton. This intermediate product is shipped to BDM's contractors for spinning & winding. The finished product is then shipped back to BDM for final spooling & packaging before being shipped to BDM's customers.

Decision Variables Given the amount of each type of yarn that BDM needs to produce (or have produced by a contracting company) to meet demand, BDM's managers realized that the essence of their problem was how to allocate spinning production across the seven facilities in order to minimize costs. The decision variables for this optimization model are denoted Xij where i denotes the size of cotton yarn (i = 1 . . . 4; 1=Extra Fine, 2=Fine, 3=Medium, and 4=Coarse) and j represents the company that would be assigned

to produce it (j=A . . . G; [A]ble Fibres, [B]uyer's Direct Mills, [C]otton Spin, [D]enim Source, [E]very Yarn, [F]ibres & Filaments and [G]rand River Milling Company). For example, X3D would indicate the amount of Medium cotton yarn produced at Denim Source. DO NOT deviate from this scheme! The unit of measure for the decision variables is "kilograms of cotton per month (Kg/month)."

Variable Costs of Production The prices charged to BDM by their contractors are used for the costs of production calculations. BDM's managers used internal accounting reports to determine a reasonable cost for production on Buyer's Direct Mills own spinning machines. The unit of measure is "Dollars per kilogram of cotton ($/Kg)."

Transportation Costs As mentioned above, the yarn that is to be spun by the six contractors needs to be shipped from BDM to the contractor for spinning and back again to BDM for final preparation.

Again, BDM's managers consulted their accounting system and produced a report that gave them their transportation costs (in $/Kg/Km) and checked with their central dispatch system to determine the roundtrip distances between BDM and the contractor facilities (Km). This information was aggregated together. Transportation costs are in "dollars per kilometer ($/Km)."

Resource Consumption Additionally, it is important to understand the nature of the cotton spinning process. The production capacity of a facility is measured in "machine hours per month" and each size of cotton yarn requires a different amount of machine hours per kilogram of product. Because each mill has different spinning machines and different machine configurations, the number of machine hours to produce a kilogram of product differs from contractor to contractor and product to product. These conversion factors are detailed. For example, at the Grand River facility, it take 0.7 hours to produce one kilogram of Extra Fine cotton. Grand River has 2,500 machine hours available to contract out to BDM.

1.1 Questions

A spreadsheet, BDM.xls, is available to you; it has all of the data from this problem in it and using it as a starting point should reduce possible errors due to transcription. As succinctly as possible, answer the following questions:

1. Use the "equation editor" in Microsoft Word (or your favourite word processor) to:

(a) Use summation notation (Big) to write out the objective function of your model.

(b) Write down the capacity constraint for the Grand River Milling Company's spinning machines.

(c) Write down the demand constraint for the Coarse cotton yarn product.

(d) How are you going to model the inability for Able Fibres and Denim Source to produce Extra Fine cotton fibre (these cells are in gray on the spreadsheet)?

2. Enter the requisite parameters from the BDM.xls spreadsheet into the Solver dialog and run Solver.

(a) Which companies get assigned what work?

(b) What is the overall cost of this arrangement?

3. Sensitivity Analysis. You should be able to answer these questions from the Sensitivity Report produced by running Solver in part 2. above. Alternatively you could modify the BDM.xls model and re-run Solver. In any case. . .

(a) BDM can increase their in-house spinning capacity, for Medium size yarn only, by renting equipment at $1,500 per month. Should they rent the equipment? The machine has a production capacity of 300 hours per month and would run at the same rate of 0.425 hours/Kg. Suppose the estimated production cost of running this machine is less than for BDM's existing machines.

Suppose it is $5.70 instead of the current $11.40?

(b) The plant manager and the accounting department estimate that BDM's internal production costs could vary within a 5% range of the figures shown above. Would your recommendation change in the extreme cases?

(c) You estimate that the production capacity of one of your local mills, Denim Source, could vary within a 20% range of the figures above. Would your recommendation change in the extreme cases?

(d) A new client is interested in purchasing up to 6,000 Kg/month of Medium size yarn. What is the minimum price that BDM should quote to this new client? Assume that BDM has not decided to expand its spinning capacity, and does not want to change the prices that they currently charge their existing clients.

4. One of the local mills, Grand River Milling Company, has the possibility of running an ovetime shift (which would double their capacity) by paying its workers only 13% more than the normal wage). You know that the workers' salaries contribute t approximately 50% of the prices that the Grand River mill charges BDM. The transportation costs would not change. Modify the model to take account of this change. Does the optimal solution change?