Jackson Hole Manufacturing is a small manufacturer of plastic products used in t
ID: 443367 • Letter: J
Question
Jackson Hole Manufacturing is a small manufacturer of plastic products used in the automotive and computer industries. One of its major contracts is with a large computer company and involves the production of plastic printer cases for the computer company's portable printers. The printer cases are produced on two injection molding machines. The M-100 machine has a production capacity of 25 printer cases per hour, and the M-200 machine has a production capacity of 40 cases per hour. Both machines use the same chemical material to produce the printer cases; the M-100 uses 40 pounds of the raw material per hour and the M-200 uses 50 pounds per hour. The computer company has asked Jackson Hole to produce as many of the cases during the upcoming week as possible and has said that it will pay $18 for each case Jackson Hole can deliver. However, next week is a regularly scheduled vacation period for most of Jackson Hole's production employees; during this time, annual maintenance is performed for all equipment in the plant. Because of the downtime for maintenance, the M-100 will be available for no more than 15 hours, and the M-200 will be available for no more than 10 hours. However, because of the high setup cost involved with both machines, management has a requirement that, if production is scheduled on either machine, the machine must be operated for at least 5 hours. The supplier of the chemical material used in the production process has informed Jackson Hole that a maximum of 1000 pounds of the chemical material will be available for next week's production; the cost for this raw material is $6 per pound. In addition to the raw material cost, Jackson Hole estimates that the hourly cost of operating the M-100 and the M-200 are $50 and $75, respectively.
Formulate a linear programming model that can be used to maximize the contribution to profit.
1- On excel do the sensitivity report
2- Provide the range of value for the decision variable coefficient
3- Identify constraints that have slack, and the range of the slack
Explanation / Answer
Formulation of linear programming problem
Decision Variables
The question is about determining quantities of Plastic printer cases to be produced using two machines M-100 and M-200. Let us denote X representing number of cases to be produced by M-100 and Y representing number of cases produced by M-200
Objective Function
As mentioned in the question is to Maximize Profit.
Given M-100 production rate is 25 per hour with consumption of 40 pounds of raw material at rate of $6/pound and hourly operating cost of $50, so per unit cost is $11.6 [(240+50)/25]
Similarly per unit cost for M-200 is $9.375 [(300+75)/40]
Given sales price as $18/case, contribution per unit by M-100 (X) is $6.4 (18-11.6)
& contribution per unit by M-200 (Y) is $8.625 (18-9.375)
Therefore Objective function is to Maximize Z = 6.4X + 8.625Y
Constraints:
Availability of manchine hours
Minimum of 5 hours for each and maximum of 15 hours of M-100 and maximum of 10 hours of M-200
therefore 5 <= X/25 <= 15 or X >= 125 and X <= 375 and similarly Y >= 200 and Y <= 400
Availability of raw material
40 * (X/25) + 50 * (Y/40) <= 1000 or 32X + 25Y <= 20,000
Lastly non-negativity constraint X, Y >= 0 rather we want X and Y as positive numbers
Sensitivity report on excel is as follows
Solution as per excel is as follows
Number of cases needs to be integer, second per unit contribution of M-200 is more (8.625 > 6.4) so firstly 10 hours of M-200 produces 400 units and consumes 500 pounds out of 1000 pounds available.
Remaining 500 pounds of raw material will be consumed by M-100 in 12.5 hours, hence maximum production by M-100 is 312 units only.
Microsoft Excel 12.0 Sensitivity Report Worksheet: [Book1]Sheet1 Report Created: 14-10-2015 12:23:01 Adjustable Cells Final Reduced Cell Name Value Gradient $C$2 Decision variables X 338.4627047 0 $D$2 Decision variables Y 366.767738 0 Constraints Final Lagrange Cell Name Value Multiplier $E$4 Constraints 20000 1Related Questions
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