Washington Ski Company (WSC) is a small manufacturer of two types of popular all-terrain snow skis, the Axis and the Status models. The manufacturing process consists of two principal departments: fabricating and finishing. The fabrication department has 12 skilled workers, each of whom works 7 hours per day. The finishing department has 3 workers, who also work a 7-hour shift. Each pair of Axis skis requires 3.5 labor-hours in the fabricating department and 1 labor-hour in finishing. The Status model requires 4 labor-hours in fabrication and 1.5 labor-hours in finishing. The company operates 5 days per week. WSC makes a new profit of $50 on the Axis model and $65 on the Status model. In anticipation of the next ski-sale season, WSC must plan its production of these two models. Because of the popularity of its products and limited production capacity, its products are in high demand, and WSC can sell all it can produce each season. The company wants to determine how many of each model should be produced to maximize net profit. Below are the first three steps of the model-building process: Identifying the decision variables, objective function, and the constraints. Step 1 - Identify the decision variables. WSC makes two different models of skis. The decisions are stated clearly in the last sentence of the problem description: How many of each model of skis should be produced each day. Thus you can define two variables: Axis = number of pairs of Axis skis produced / day Status = number of pairs of Status skis produced / day It is important to be clear about the dimensions of the variables, for example, "pairs of skis produces / day" rather than "Axis skis". Step 2 - Identify the objective function (or objective). The problem states that WSC wants to maximize their net profit, and we are given the net profit figures for each type of ski. In some problems, the objective is not explicitly stated and we must use logic and business experience to identify the appropriate business objective. Step 3 - Identify the constraints. To identify constraints, look for clues in the problem statement that describe limited resources that are available, requirements that must be met, or other restrictions. In this example, we see that both the fabrication and finishing departments have limited numbers of workers, who work only 7 hours each day; this limits the amount of production time available in each department. Therefore, we have the following constraints: Fabrication: Total labor hours used in fabrication cannot exceed the total amount of labor hours available. Total available fabrication labor is 12 workers * 7 hours per day for 84 hours per week. Finishing: Total labor hours used in finishing cannot exceed the total amount of labor hours available. Total available finishing labor is 3 workers * 7 hours per day for 21 hours per week Finally, we must ensure that negative values for the decision variable cannot occur, Nonnegativity constraints are assumed in nearly all optimization models. # of pairs of Axis skis >= 0 # of pairs of Statis skis >=0 The complete optimization model for the WSC problem is Maximize Total Profit = 50 Axis + 65 Status (Axis and Status are your decision variables) Subject to the following constraints 3.5 Axis + 4 Status = 84 1 Axis + 1.5 Status <= 21 Axis >= 0 Status >= 0 An Excel spreadsheet that represents this problem has been included below. Your job is to use Solver to solve this problem and come up with the maximum possible profit given the revenue numbers and constraints.
Washington Ski Company (WSC) is a small manufacturer of two types of popular all-terrain snow skis, the Axis and the Status models. The manufacturing process consists of two principal departments: fabricating and finishing. The fabrication department has 12 skilled workers, each of whom works 7 hours per day. The finishing department has 3 workers, who also work a 7-hour shift. Each pair of Axis skis requires 3.5 labor-hours in the fabricating department and 1 labor-hour in finishing. The Status model requires 4 labor-hours in fabrication and 1.5 labor-hours in finishing. The company operates 5 days per week.
WSC makes a new profit of $50 on the Axis model and $65 on the Status model. In anticipation of the next ski-sale season, WSC must plan its production of these two models. Because of the popularity of its products and limited production capacity, its products are in high demand, and WSC can sell all it can produce each season. The company wants to determine how many of each model should be produced to maximize net profit.
Below are the first three steps of the model-building process: Identifying the decision variables, objective function, and the constraints.
Step 1 - Identify the decision variables. WSC makes two different models of skis. The decisions are stated clearly in the last sentence of the problem description: How many of each model of skis should be produced each day. Thus you can define two variables:
Axis = number of pairs of Axis skis produced / day
Status = number of pairs of Status skis produced / day
It is important to be clear about the dimensions of the variables, for example, "pairs of skis produces / day" rather than "Axis skis".
Step 2 - Identify the objective function (or objective). The problem states that WSC wants to maximize their net profit, and we are given the net profit figures for each type of ski. In some problems, the objective is not explicitly stated and we must use logic and business experience to identify the appropriate business objective.
Step 3 - Identify the constraints. To identify constraints, look for clues in the problem statement that describe limited resources that are available, requirements that must be met, or other restrictions. In this example, we see that both the fabrication and finishing departments have limited numbers of workers, who work only 7 hours each day; this limits the amount of production time available in each department. Therefore, we have the following constraints:
Fabrication: Total labor hours used in fabrication cannot exceed the total amount of labor hours available.
Total available fabrication labor is 12 workers * 7 hours per day for 84 hours per week.
Finishing: Total labor hours used in finishing cannot exceed the total amount of labor hours available.
Total available finishing labor is 3 workers * 7 hours per day for 21 hours per week
Finally, we must ensure that negative values for the decision variable cannot occur, Nonnegativity constraints are assumed in nearly all optimization models.
# of pairs of Axis skis >= 0
# of pairs of Statis skis >=0
The complete optimization model for the WSC problem is
Maximize Total Profit = 50 Axis + 65 Status (Axis and Status are your decision variables)
Subject to the following constraints
3.5 Axis + 4 Status = 84
1 Axis + 1.5 Status <= 21
Axis >= 0
Status >= 0
An Excel spreadsheet that represents this problem has been included below. Your job is to use Solver to solve this problem and come up with the maximum possible profit given the revenue numbers and constraints.
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