Gemstone Tool Company (GTC) is a privately held firm that competes in the consumer and industrial...

 

   

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Gemstone Tool Company (GTC) is a privately held firm that competes in the consumer and industrial market for construction tools. In addition to its main manufacturing fa- cility in Seattle, Washington, GTC operates several other manufacturing plants located in the United States, Canada, and Mexico. These plants produce their entire range of products, including power drills and saws, nail guns, and hand-held tools such as hammers, screwdrivers, wrenches, and pliers. For the sake of simplicity, let us suppose that the Winnipeg, Canada plant only produces wrenches and pliers. Wrenches and pliers are made from steel, and the process involves molding the tools on a molding machine and then assembling the tools on an assembly machine. The amount of steel used in the production of wrenches and pliers and the daily availability of steel is shown in the first line of Table 7.3. On the next two lines are the machine utilization rates needed in the production of wrenches and pliers and the capacity of these ma- chines as well. Finally, the last two rows of the table indicate the daily market demand for these tools and their variable (per unit) contribution to earnings. For example, according to the second column of Table 7.3, each wrench produced by GTC uses 1.5 lbs. of steel, and spends 1.0 hour on the molding machine and 0.3 hour on the assembly machine. The demand for wrenches is 15,000 per day. Every 1,000 wrenches produced (and then sold) contributes $130 to the earnings of GTC. According to the fourth column of Table 7.3, the amount of steel available per day is 27,000 lbs. The amount of molding machine capacity is 21,000 hours per day and the amount of assembly machine capacity is 9,000 hours per day. GTC would like to plan for the daily production of wrenches and pliers at its Winnipeg plant so as to maximize the contribution to earnings. In particular, GTC would like to answer the following questions: Now suppose that GTC is planning for next quarter, and that they would like to de- termine how much steel to contract for with local suppliers for the next quarter. Sup- pose that steel contracts are typically arranged for daily deliveries over the entire quarter, and that the market price for such contracts is $58.00/1,000 lbs. of steel. Let us define the decision variable: S= the amount of steel to contract for, for next quarter, in 1,000 lbs./day. GTC would like to determine the optimal value of S. We suppose that the fol- lowing aspects of the problem are known: • The market price of steel is $58.00/1,000 lbs. • The utilization of steel, assemby machine hours, and molding machine hours in wrenches and pliers is the same as given in the original problem. • The molding machine capacity is the same as in the original problem, namely 21,000 hours/day. • The demand for wrenches and pliers is the same as in the original problem, namely 15,000 wrenches per day and 16,000 pliers per day. . The unit contribution to earnings of production of pliers is the same as in the original problem, namely $100/1,000 units. However, we also suppose that the following aspects of the problem are uncertain: • The assembly machine capacity for next quarter is uncertain. GTC has ordered new assembly machines to replace, as well as to augment, their existing assem- bly machines, but it is not known if these new machines will be delivered in time to be used next quarter. Let us suppose that the assembly machine capacity for next quarter will either be 8,000 hours/day (with probability 0.5) or 10,000 TABLE 7.3 Data for the Gemstone Tool Company Steel (lbs) Molding Machine (houn) Assembly Machine (hours) Demand Limit (tools/day) Contribution to Earnings (5/1,000 units) 3 Wrenches 1.5 1.0 0.3 15,000 $130 Pliers 1.0. 10 05 16,000 $100 Availability 27.300 lbs/day 21,000 hours/day 9,000 hours/day Please define two decision variables W as the number of wrenches to be produced per day (in 1000 units) and P as the number of pliers to be produced per day (in 1000 units). (1) Formulate a mathematical programming model that maximizes revenue per day. Clearly specify the objective function and constraints. -6pts (2) Use Excel or Lindo to solve for W and P. What are the optimal decisions & revenue? -5pts (3) Which resources are the least critical (FE) from an optimization standpoint? -4pts Gemstone Tool Company (GTC) is a privately held firm that competes in the consumer and industrial market for construction tools. In addition to its main manufacturing fa- cility in Seattle, Washington, GTC operates several other manufacturing plants located in the United States, Canada, and Mexico. These plants produce their entire range of products, including power drills and saws, nail guns, and hand-held tools such as hammers, screwdrivers, wrenches, and pliers. For the sake of simplicity, let us suppose that the Winnipeg, Canada plant only produces wrenches and pliers. Wrenches and pliers are made from steel, and the process involves molding the tools on a molding machine and then assembling the tools on an assembly machine. The amount of steel used in the production of wrenches and pliers and the daily availability of steel is shown in the first line of Table 7.3. On the next two lines are the machine utilization rates needed in the production of wrenches and pliers and the capacity of these ma- chines as well. Finally, the last two rows of the table indicate the daily market demand for these tools and their variable (per unit) contribution to earnings. For example, according to the second column of Table 7.3, each wrench produced by GTC uses 1.5 lbs. of steel, and spends 1.0 hour on the molding machine and 0.3 hour on the assembly machine. The demand for wrenches is 15,000 per day. Every 1,000 wrenches produced (and then sold) contributes $130 to the earnings of GTC. According to the fourth column of Table 7.3, the amount of steel available per day is 27,000 lbs. The amount of molding machine capacity is 21,000 hours per day and the amount of assembly machine capacity is 9,000 hours per day. GTC would like to plan for the daily production of wrenches and pliers at its Winnipeg plant so as to maximize the contribution to earnings. In particular, GTC would like to answer the following questions: Now suppose that GTC is planning for next quarter, and that they would like to de- termine how much steel to contract for with local suppliers for the next quarter. Sup- pose that steel contracts are typically arranged for daily deliveries over the entire quarter, and that the market price for such contracts is $58.00/1,000 lbs. of steel. Let us define the decision variable: S= the amount of steel to contract for, for next quarter, in 1,000 lbs./day. GTC would like to determine the optimal value of S. We suppose that the fol- lowing aspects of the problem are known: • The market price of steel is $58.00/1,000 lbs. • The utilization of steel, assemby machine hours, and molding machine hours in wrenches and pliers is the same as given in the original problem. • The molding machine capacity is the same as in the original problem, namely 21,000 hours/day. • The demand for wrenches and pliers is the same as in the original problem, namely 15,000 wrenches per day and 16,000 pliers per day. . The unit contribution to earnings of production of pliers is the same as in the original problem, namely $100/1,000 units. However, we also suppose that the following aspects of the problem are uncertain: • The assembly machine capacity for next quarter is uncertain. GTC has ordered new assembly machines to replace, as well as to augment, their existing assem- bly machines, but it is not known if these new machines will be delivered in time to be used next quarter. Let us suppose that the assembly machine capacity for next quarter will either be 8,000 hours/day (with probability 0.5) or 10,000 TABLE 7.3 Data for the Gemstone Tool Company Steel (lbs) Molding Machine (houn) Assembly Machine (hours) Demand Limit (tools/day) Contribution to Earnings (5/1,000 units) 3 Wrenches 1.5 1.0 0.3 15,000 $130 Pliers 1.0. 10 05 16,000 $100 Availability 27.300 lbs/day 21,000 hours/day 9,000 hours/day Please define two decision variables W as the number of wrenches to be produced per day (in 1000 units) and P as the number of pliers to be produced per day (in 1000 units). (1) Formulate a mathematical programming model that maximizes revenue per day. Clearly specify the objective function and constraints. -6pts (2) Use Excel or Lindo to solve for W and P. What are the optimal decisions & revenue? -5pts (3) Which resources are the least critical (FE) from an optimization standpoint? -4pts

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