First, some explanation and partial example of the Inventory Problem. Start by assuming a situation where...

 

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First, some explanation and partial example of the Inventory Problem. Start by assuming a situation where a company needs 1000/year of a product. They will use this product up in consistent amounts, modeled nicely by a decreasing linear function over time, as in the graph below. Suppose now that there is a cost of $80 to place an order for this product, no matter how many are ordered, 1000 or 200 or just 5. There is also, though, a cost of storing this product, so the more they order at once, like 1000, the more it costs to store the product, say $4 per item per year. However, as we use up our 1000, we are storing less and less over the year, an average of (1/2)*1000 = 500, shown on the graph as a horizontal blue line. Case 1: Order 1000 once per year, costing $80. Storage of an average of 500 costs $4*500 = $2000. Our total costs are $2080. Case 2: Order 200 each of 5 times per year, costing $80*5 = $400. Storage of an average of 100(half of the 200 we order) costs us $4*100 = $400. Our total costs are now only $800. The smaller line segments in the graph show us ordering 200, using them up and ordering a new 200 just as we finish the previous quantity. Case 3: Order 100 items each of 10 times per year, costing $80*10 = $800. Storage of an average of 50(half of the 100 we order) costs us $4*50 = $200. Our total costs are now back up, to $1000. 500 1000 So, how do we find the right amount to order each time that provides us a minimized total cost, the perfect balance between ordering and storing? Start by letting X= Lot Size, which is how many we order each time. Discuss and decide how to use X to build representations for "number of orders per year" and "average amount stored". Construct a Total Cost Function modeled on the general idea: C(x) = Ordering Costs + Storage Costs, and then use Calculus and Algebra to locate the Critical Point and Minimum Total Costs. %3D 1) Company A needs 640 of a product per year, with an ordering cost of $30, and a per item per year storage cost of $1.50. 2) Company B needs 600 of a product per year with a $15 ordering cost and a $5 per item per year storage cost. For each company: a) Construct a Cost function b) Determine the Critical Point and use it to find minimum total costs. First, some explanation and partial example of the Inventory Problem. Start by assuming a situation where a company needs 1000/year of a product. They will use this product up in consistent amounts, modeled nicely by a decreasing linear function over time, as in the graph below. Suppose now that there is a cost of $80 to place an order for this product, no matter how many are ordered, 1000 or 200 or just 5. There is also, though, a cost of storing this product, so the more they order at once, like 1000, the more it costs to store the product, say $4 per item per year. However, as we use up our 1000, we are storing less and less over the year, an average of (1/2)*1000 = 500, shown on the graph as a horizontal blue line. Case 1: Order 1000 once per year, costing $80. Storage of an average of 500 costs $4*500 = $2000. Our total costs are $2080. Case 2: Order 200 each of 5 times per year, costing $80*5 = $400. Storage of an average of 100(half of the 200 we order) costs us $4*100 = $400. Our total costs are now only $800. The smaller line segments in the graph show us ordering 200, using them up and ordering a new 200 just as we finish the previous quantity. Case 3: Order 100 items each of 10 times per year, costing $80*10 = $800. Storage of an average of 50(half of the 100 we order) costs us $4*50 = $200. Our total costs are now back up, to $1000. 500 1000 So, how do we find the right amount to order each time that provides us a minimized total cost, the perfect balance between ordering and storing? Start by letting X= Lot Size, which is how many we order each time. Discuss and decide how to use X to build representations for "number of orders per year" and "average amount stored". Construct a Total Cost Function modeled on the general idea: C(x) = Ordering Costs + Storage Costs, and then use Calculus and Algebra to locate the Critical Point and Minimum Total Costs. %3D 1) Company A needs 640 of a product per year, with an ordering cost of $30, and a per item per year storage cost of $1.50. 2) Company B needs 600 of a product per year with a $15 ordering cost and a $5 per item per year storage cost. For each company: a) Construct a Cost function b) Determine the Critical Point and use it to find minimum total costs.