CO 370 Deterministic OR Models Winter 2016 Project 1: 50 marks Due: Wednesday, March 16, in class, before lecture BestChip: Production Management BestChip (BC) is a large company that produces low-fat snack products for an expanding market. Basically, BC takes raw material (corn, wheat, and potatoes), and turns them into three types of snacks: regular chips, green-onion chips, and party mix. BC currently owns three plants in Yuma, Fresno, and Tucson. Each plant can produce any of the three types of snacks, although each plant has its own individual constraints and unit costs. These constraints cover restrictions on raw material, labor and blending time on a food-processing machine, and the specic values are given in Tables 1-3. Note that even though all plants produce the same products, dierent production processes are used and thus the products use dierent amounts of resources in dierent plants. However, the makeup of a snack does not depend on the production plant (as indicated by the last 3 columns of Tables 1-3). BestChip has 3 major customers located at Albuquerque, Phoenix, and San Digeo. The demand for each customerproduct pair is given below. All demand must be met. Product Demand (tons) Product Regular chips Green-onion chips Party mix Customers Albuquerque Phoenix 1,400 1,200 1,100 800 1,700 1,800 San Diego 1,700 1,800 2,200 Raw material is available only at the Yuma plant, and is shipped from there to the other plants for use in production. The snack products are shipped from the plants to the customers, with the exception of regular chips produced in Fresno. These regular chips are rst shipped to the Yuma plant (and then to the customers from the Yuma plant), where both these regular chips and the regular chips produced by the Yuma plant go through a special inspection facility. The inspection facility can process at most 6,000 tons. The shipping cost is $0.15 per ton per mile (for any product). The distances between the various locations are listed in Table 5. Table 6 contains the production costs per ton snacks produced, and production capacity in tons of snacks, for each plant. Product sales prices are given in Table 4. Assume that any excess production can be stored at no extra costs. 1 Table 1: Product-Resource Constraints: Yuma plant Product Regular chips Green-onion chips Party mix Total available Labor (hours/ton) 10 10 15 80,000 Resources Machine Corn (hours/ton) (tons/ton) 25 0.7 30 0.3 25 0.2 150,000 9,000 Wheat (tons/ton) 0.2 0.15 0.5 6,000 Potato (tons/ton) 0.1 0.55 0.3 4,500 Table 2: Product-Resource Constraints: Fresno plant Product Regular chips Green-onion chips Party mix Total available Labor (hours/ton) 5 8 10 50,000 Machine (hours/ton) 12 35 32 125,000 Resources Corn (tons/ton) 0.7 0.3 0.2 Wheat (tons/ton) 0.2 0.15 0.5 Potato (tons/ton) 0.1 0.55 0.3 Your Task Your job is to determine a recommendation for the company that maximizes its net prot. A recommendation must include a plan for production and shipping as well as the cost and revenue generated from each plant. Ignore the constraint that the number of units produced, shipped, and sold must be integers. Problem 1. Formulate BestChip's problem as a linear program. Be sure to describe what each of the decision variables and constraints represent. Implement your LP model in AMPL, and solve the LP. Specify the production, shipping, and sales plan that you recommend to BestChip based on the LP solution. You do not need to \"copy\" your AMPL model into a separate LP; it is sucient to submit a suitably annotated AMPL mod le with comments indicating what the variables and constraints represent. In coding up your LP model using AMPL, you should separate the model from the data. Submit a printout of your AMPL mod and dat les. (35 marks) Remarks. 1. For the AMPL implementation, you are likely to need 3D variables. See Q7 of the AMPL FAQ for more information about declaring such arrays. 2. In your AMPL implementation, you may nd it convenient to omit specifying the values of certain parameters that are never used. For example, suppose you include set plants; and set resource; in your model le to declare respectively the set of plants, and the set of resources. You also include param avail {plant,resource} >= 0; to declare the 2D-array specifying the availability of each resource at each plant. Since the Fresno and Tucson don't have any raw material, in your data le, you can use . to omit specifying the values of these entries. For example, your dat le could read as follows: 2 Table 3: Product-Resource Constraints: Tucson plant Product Regular chips Green-corn chips Party mix Total available Labor (hours/ton) 10 6 9 70,000 Resources Machine Corn (hours/ton) (tons/ton) 30 0.7 25 0.3 20 0.2 180,000 Wheat (tons/ton) 0.2 0.15 0.5 Potato (tons/ton) 0.1 0.55 0.3 set resource := labor, machine, corn, wheat, potato; set plant := yuma, fresno, tucson; param avail: labor machine corn wheat potato := yuma 80000 150000 9000 6000 4500 fresno 50000 125000 . . . tucson 70000 180000 . . . ; # # Other stuff # Problem 2. In addition, address the following questions in your recommendation. Where possible, you should use sensitivity analysis information gathered from AMPL to answer these questions (as opposed to solving a new LP). Submit a transcript of your AMPL session showing how you extract the relevant information from AMPL. (a) If you could get more raw materials at Yuma, how much would you like, and what would you be willing to pay? (3 marks) (b) At what plant(s) would you like to add extra machine hours and/or extra labor hours? How much would you be willing to pay per hour? How many extra hours would you like? (4 marks) (c) If the shipping cost for regular chips sent from Fresno to Yuma (i.e., we are not changing the shipping cost for other items) decreases by $0.1 per ton per mile, would this change the amount of regular chips produced at the Fresno plant? (3 marks) (d) If the shipping cost increased to $0.17 per ton per mile, what would be the decrease in BestChip's prot? (5 marks) 3 Table 4: Product Sales Price ($/ton) Product Regular chips Green-onion chips Party mix Customers Albuquerque Phoenix 130 150 120 110 150 190 San Diego 180 130 230 Table 5: Shipping Distances (miles) Yuma Fresno Tucson Yuma 0 490 240 Fresno 490 0 700 Tucson 240 700 0 Albuquerque 600 915 450 Phoenix 185 590 120 San Diego 170 340 412 Table 6: Production Costs and Capacity Plant Yuma Fresno Tucson Production cost ($/ton of snacks) 3.5 2 3 4 Capacity (tons of snacks) 7,000 8,000 7,000