Mary Kelly is a scholarship soccer player at State University. During the summer she works at a youth all-sports camp that several of the university’s coaches operate. The sports camp runs for eight weeks during July and August. Campers come for a one-week period, during which time they live in the State dormitories and use the State athletic field and facilities. At the end of the week a new group of kids comes in. Mary primarily serves as one of the camp soccer instructors. However, she has also been placed in charge of arranging for sheets for the beds the campers will sleep on in the dormitories. Mary has been instructed to develop a plan for purchasing and cleaning sheets each week of camp at the lowest possible cost.
Clean sheets are needed at the beginning of each week, and the campers use the sheets all week. At the end of the week, the campers strip their beds and place the sheets in large bins. Mary must arrange either to purchase new sheets or to clean old sheets. A set of new sheets costs $10. A local laundry has indicated that it will clean a set of sheets for $4. Also, a couple of Mary’s friends have asked her to let them clean some of the sheets. They have told her they will charge only $2 for each set of sheets they clean. However, while the laundry will provide cleaned sheets in a week, Mary’s friends can only deliver cleaned sheets in two weeks. They are going to summer school and plan to launder the sheets at night at a neighborhood Laundromat.
The accompanying table lists the number of campers that have registered during each of the eight weeks the camp will operate. Based on discussions with camp administrators from previous summers and on some old camp records and receipts, Mary estimates that each week about 20% of the cleaned sheets that are returned will have to be discarded and replaced. The campers spill food and drinks on the sheets, and sometimes the stain will not come out during cleaning. Also, the campers occasionally tear the sheets or the sheets get torn at the cleaners. In either case, when the sheets come back from the cleaners and are put on the beds, 20% are taken off and thrown away.
The Essay on Mary Chestnuts Civil War
Mary Boykin Chesnut was born on her grandparents' estate at Mount Pleasant, South Carolina on March 31, 1823. She learned early about the workings of a plantation by observing her grandmother. Grandmother Miller rose early to assign the cleaning and cooking duties for her servants. Besides keeping the mansion clean and prepared for the frequent guests, Mary's grandmother also took charge of making ...
WeekRegistered Campers
1. 115
2. 210
3. 250
4. 230
5. 260
6. 300
7. 250
8. 190
At the beginning of the summer, the camp has no sheets available, so initially sheets must be purchased. Sheets are thrown away at the end of summer.
Mary’s major at State is management science, and she wants to develop a plan for purchasing and cleaning sheets using linear programming. Help Mary formulate a linear programming model for this problem and solve it using the computer.
You will find the solution in the attached Excel file. The basic idea is the following. At the beginning of each week, Mary must choose how many sheets she’ll purchase, send to the laundry, or send to her friends. Let’s call ‘x’ to the number of sheets that are purchased, ‘y’ to the number of sheets that are sent to the laundry and ‘z’ to the number of sheets that are sent to her friends. We’ll use a sub index to show which week we’re talking about: for example, [pic]is the number sheets purchased at the beginning of week 1, [pic]is the number of sheets sent to the laundry at the beginning of week 4, and so on.
We must minimize total cost choosing the variables we mentioned above. The total cost function will be: [pic]
Notice that I deliberately excluded [pic] and[pic]: these must be equal to zero, as we have no dirty sheets at the beginning of the camp (beginning of week 1) to send to clean.
Now let’s see the constraints. At the beginning of each week, the number of clean sheets must be equal to or greater than the number of sheets that will be used that week. The number of clean sheets in the beginning of a week equals the number of sheets just purchased, plus the number of sheets received from the laundry and friends, plus the number of clean sheets the previous week, minus the number of sheets that were used the previous week. This number must be equal to or greater than the number of sheets that will be used during this week.
The Term Paper on Week Five Exercise Assignment
Liquidity ratios. Edison, Stagg, and Thornton have the following financial information at the close of business on July 10: Edison Stagg Thornton Cash $6,000 $5,000 $4,000 Short-term investments 3,000 2,500 2,000 Accounts receivable 2,000 2,500 3,000 Inventory 1,000 2,500 4,000 Prepaid expenses 800 800 800 Accounts payable 200 200 200 Notes payable: short-term 3,100 3,100 3,100 Accrued payables ...
Let’s define a variable that represents the number of clean sheets at the beginning of any week (call it “Clean”).
Clearly, the number of clean sheets at the beginning of week 1 is the number of sheets purchased on week 1:[pic]. The number of clean sheets at the beginning of week 2 will simply be the number sheets purchased on week 1 minus the amount used on week 1, plus the number of sheets just purchased (on week 2): [pic]
The number of clean sheets at the beginning of week 3 will be equal to the number of clean sheets on week 2, minus the ones that were used during week 2, plus the number that were just purchased (on week 3), plus 80% of the sheets sent to the laundry at the beginning of week 2 (since they took a week to clean them): [pic]
Similarly, for week 4:
(Notice that we now also receive 80% of the sheets we sent to Mary’s friends on week 2).
The following variables are defined analogously to the one in week 4. These are shown in the “Clean sheets available” column in the Excel file.
The constraint on these variables is that the number of clean sheets must be equal to or greater than the number of sheets that will be needed. For example, since we need 115 clean sheets on week 1, 210 clean sheets in week 2, and so on, then the constraints must be: [pic]
That’s one set of constraints.
Another set of constraints refers to the number of sheets sent to the laundry and to Mary’s friends. Specifically, it’s clear that we can’t send to the cleaners more sheets than the number of dirty sheets. So, just as before, let’s define a variable “Dirty” for the number of dirty sheets at the beginning of each week. The first week, the number of dirty sheets is zero, since we still have no sheets:[pic]. The number of dirty sheets at the beginning of week 2 will simply be 115; the number of sheets used during week 1: [pic]. The number of dirty sheets at the beginning of week 3 will be equal to the number of dirty sheets at the beginning of week 2, plus the number of sheets that became dirty in week 2 (210), minus the number that was sent for cleaning at the beginning of week 2:
The Term Paper on The intangible assets section of the balance sheet
E12-1 (Classification Issues—Intangibles) Presented below is a list of items that could be included in the intangible assets section of the balance sheet. Instructions (a)Indicate which items on the list above would generally be reported as intangible assets in the balance sheet. 13.Goodwill acquired in the purchase of a business. 15.Cost of purchasing a patent from an inventor 16.Legal costs ...
The next weeks follow the same logic:
And so on…
The variable “Dirty” is shown in the “Dirty sheets” column in Excel. So now we can define the constraints: the number of sheets sent each week for cleaning must be equal to or smaller than the number of dirty sheets:
Finally, we also set as constraints that all the decision variables (the x’s, y’s and z’s) must be non-negative, and that they must be integer values (it makes no sense to purchase “half a sheet”).
I entered all these data in Excel and ran the Solver. The optimal plan is shown in the orange cells:
Beginning of Week 1: Purchase 115 sheets
Beginning of Week 2: Purchase 210 sheets; send 115 to Mary’s friends Beginning of Week 3: Purchase 250 sheets; send 210 to Mary’s friends …
The total cost of this plan will be $11,586. That’s the minimum possible cost.
