NAME_______________________________________ OIM410 Business Process Optimization Fall 2015, Exam 2, 100 points You are to work on this exam independently. This exam has 4 questions on 4 pages. Answer all questions using single sides of 8.511 paper. Begin each question on a new page. Due 8pm, December 14, 2015 SOM 315 (slip under the door). Please staple all pages. [1] Ali Baba Distribution has three manufacturing plants = 1 ... 3 and five distribution centers (DC), = 1 ... 5. The capacity of each plant is and demand (total requirement) at distribution center is . Assume that total supply exceeds total requirement at DCs. Variable transportation costs are per unit shipped from plant to DC . [5 points] Write the linear programming model that determines the minimum cost shipment plan of product from plants to DCs. b) [5 points] Alter the model in part a) to incorporate the following: Product is transported in trucks that have a capacity of K units of product. The fixed cost of operating a truck from plant to DC is . c) [5 points] Alter the model in part a) to incorporate the following: Assume that total demand exceeds total capacity and, consequently, the first priority is to minimize the total shortfall in meeting demand and the second priority is to minimize the cost of shipment. The model is now a goal program. d) Reverting to the model in part a): Ali Baba wants to address system wide distribution from plants to distribution centers to market regions. Thus, DCs are points of transshipment of product to meet demand , at market regions = 1, ... ,20. The capacity of distribution centers is assumed to be unlimited. The variable cost of shipping one unit from DC = 1, ... ,5 to market region = 1, ... ,20 is , . i) [1 points] What are the additional variables? ii) [4 points] Write the new constraints for each DC (the demand constraints in the above mode are altered to ensure conservation of flow at DCs (what flows into each DC must equal what flows out of each DC). iii) [5 points] Write the constraints that ensure that demand is satisfied at market regions. iv) [5 points] Write the new objective function. a) [2] A paper mill produces so-called linear board in jumbo reels having a standard width of 68\" and each reel has a fixed length. The company's customers, however, order reels having smaller widths (and the same fixed length as the larger reel). Today's orders are for: (i) 90 reels of 22-inch width; (ii) 110 reels of 20-inch width; and (iii) 70 reels of 12-inch width. These smaller widths are to be cut on a slitting machine from the larger standard size reel. For example, if a jumbo reel is slit into two reels of 22\" width and one reel of 20\" width, then the residual reel of 4\" width is trim waste. Also, any excess production of the required size reels is also deemed to be waste. The production scheduler would like to manufacture today's orders so as to determine the number of times a 68\" jumbo reel is cut using pattern , = 1 ... 10 in a manner that will minimize total waste in inch-reels. A linear program that minimizes total inch-reels of waste has been formulated as follows: 21 + 42 + 03 + 64 + 25 + 106 + 87 + 48 + 09 + 810 +221 + 202 + 123 31 + 22 + 23 + 14 + 15 + 16 + 07 + 08 + 09 + 010 = 90 + 1 01 + 12 + 03 + 24 + 15 + 06 + 37 + 28 + 19 + 010 = 110 + 2 01 + 02 + 23 + 04 + 25 + 36 + 07 + 28 + 49 + 510 = 70 + 3 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 1 , 2 , 3 0 min OIM410/Fall 2015/EXAM 2/100 Points 2/4 [10 points] Identify a feasible solution to the above model. Specify values for all variables and the value of the objective function. b) [10 points] Suppose that setting up a pattern for slitting (on a slitting machine) is disruptive and consequently, a management directive is that no more than 2 patterns be used. Alter the above model so that this directive is incorporated. (Hint: You will need to introduce binary variables.) c) [5 points] Management is not satisfied with the cutting plan obtained from the model in part b). The paper mill operates the slitting machine 8 hours per day. The time required for slitting a jumbo reel depends on the number of blades the pattern required. Further, when a particular pattern is used, there is a set-up time. The set-up time and the slitting time per jumbo reel for a pattern are given in the following table. For example, pattern 5 requires a 40 minute set-up and each jumbo reel slit using the pattern requires 4 minutes. Write the constraint that, when appended to the model in part b) would ensure that the total time that the slitting machine is used is no more than 480 minutes. a) PATTERN Set-up time in minutes Time per jumbo reel in minutes 1 30 3 2 30 3 3 30 3 4 30 3 5 40 4 6 40 4 7 30 3 8 40 4 9 40 4 10 50 5 [3] The Maynard Wire Company produces color-coded wire in two coatings: standard inexpensive Plastic, and high-quality Teflon. Planning at Maynard Wire is done on a quarterly basis, and for the next quarter, the demand, in tons, denoted , for product = , , in month = 1, ... ,3 is provided in the following tables. Demand for Plastic Wire (tons) Demand for Teflon Wire (tons) Month 1 1200 2 1400 Month 2 1300 1 800 2 905 3 1055 Maynard wire has two production lines (wire trains) called, Kolbert and Loomis. The number of days the lines are available varies from month to month due to scheduled maintenance and is denoted , for line = , in month, = 1, 2, 3 and provided in the following table. Days of Production Availability in Month Month Line 1 2 Kolbert 26 26 Loomis 28 27 3 29 30 Both types of wire can be produced on either line. The production rates, in tons per day, are denoted , on line = , for product = , . The per day variable cost of producing one ton of product = , on line = , is denoted , . The set-up time, in days, for = , on line = , is denoted , . The values for these parameters are given below. Production Rates in Tons/Day Line Plastic Teflon Kolbert 40 35 Loomis 50 42 Daily Variable Operating Costs Line Plastic Teflon Kolbert $100 $102 Loomis $105 $108 Line Kolbert Loomis Setup times in days Plastic Teflon 0.50 0.75 1.00 0.50 Both types of wire may be stored for future delivery in Maynard Wire's warehouse which has, for all intents and purposes, unlimited capacity. The warehousing cost, which includes the cost of transportation to the warehouse, in dollars per ton of product is $8 per ton. The following variables have been defined and a model that produces an optimal production and warehousing plan follows: OIM410/Fall 2015/EXAM 2/100 Points 3/4 denotes number of days product is produced on line in month indicates the setup of product on line in month denotes the number of tons of product in inventory at the end of month ,, ,, , Subject to: = = =1 ,, , ,, 0 ,1 + , ,, , = , ,, 0 , 0 ,, = 0,1 min , ,, + 8 , ,, + , ,, , =1 =1 =1 = = , ; = 1, ... ,3 = , ; = , ; = 1, ... ,3 ,0 = 0 = , ; = , ; = 1, ... ,3 = , ; = 1, ... ,3 = , ; = , ; = 1, ... ,3 = , ; = 1, ... ,3 = , [5 points] Write the specific objective function for the data. That is, without parameters and without summation notation. (No partial credit.) b) [5 points] Write the specific capacity constraint for available days in month 2 on the Kolbert line. The constraint should have only variables and numbers in the specification. That is, no parameters. (No partial credit.) c) [5 points] Write the specific constraint that ensures that Teflon can be produced on Loomis in month 1 only if the set-up for production has occurred. The constraint should have only variables and numbers in the specification. That is, no parameters. (No partial credit.) d) [5 points] Write the specific constraint that ensures that demand for Plastic in month 2 is satisfied. The constraint should have only variables and numbers in the specification. That is, no parameters. (No partial credit.) e) [10 points] Assume now that Maynard's warehouse has a capacity limitation of 10 tons at the end of any month. Maynard can, when required, lease additional warehouse space. The \"leased warehouse\" requires a fixed cost of $37,500 in any month that space is leased, and a per ton variable cost (which includes transportation to the warehouse) of $12. Alter the specified model to reflect this enhanced problem scenario. a) [4] SourceEast (SE) is a manufacturer of signature handbags for major labels of the world. Two of SE's customers are Fende and Cooch. The following table specifies the required labor hours to manufacture one lot-unit of the Fende brand and the Cooch brand for three major operations required in the manufacture of product. Product Fende Cooch Labor Hours for 1 Lot-Unit Cutting Sewing Finishing 3 6 2 7 4 5 OIM410/Fall 2015/EXAM 2/100 Points 4/4 The available capacity, in hours, for the three operations is as follows: 850 for cutting; 970 for sewing; and, 1420 for finishing. Due to process inefficiencies, manufactured handbags may or may not be within a customer's quality specifications. Product that is within quality specifications is termed 1st-Grade and that which is not is termed 2nd-Grade. Historically, 80% of produced lots for Fende are 1st-Grade and 70% of produced lots for Cooch are 1st-Grade. Both grades of a product can be sold. 1st-Grade product can be sold as 2nd-Grade but not vice versa. Fende 1st-Grade is sold (to Fende) for $100 per lot-unit and Cooch 1st-Grade is sold (to Cooch) for $90 per lot-unit. Fende lot-units are sold in the secondary market for $55 and Cooch lot-units are sold in the secondary market for $30. Production costs are $50 for a Fende lot-unit, and $20 for a Cooch lot-unit. SE's customers do not place orders for a specific number of lot-units, but expect rapid delivery. There is an upper limit on how much of SE's products can be sold in both markets as well as a minimum that SE knows will be sold (which we assume to be zero). SE has, from historical data, concluded that there is a range of values for the number of units of both grades of product it can sell and this is provided in the following table. Product Fende Cooch 1st-Grade Minimum Maximum 50 60 25 40 2nd-Grade Minimum Maximum 0 100 0 100 Write a linear program that will determine the number of units of both product and should be made and sold in the two markets in order to maximize revenue. a) [1 points] Define the variables (Hint: Number of units of each product to be made; Number of units of each product that are 1st-grade; Number of units of each product that are 2nd-Grade; Number of 1st-grade units of each product to be sold as 1st-grade; Number of 1st-grade units of each product to be sold as 2nd grade.) b) [4 points] Write the objective. c) [10 points] Write the constraints