Concept explainers
a)
To solve: The linear programming problem and answer the questions.
Introduction:
Linear programming:
Linear programming is a mathematical modelling method where a linear function is maximized or minimized taking into consideration the various constraints present in the problem. It is useful in making quantitative decisions in business planning.
a)
Explanation of Solution
Given information:
Calculation of coordinates for each constraint and objective function:
Constraint 1:
Constraint 2:
Constraint 3:
Constraint 4:
Objective function:
The problem is solved with iso-cost line method.
Graph:
(1) Optimal value of the decision variables and Z:
The coordinates for the cost line is (45, 55). The cost line is moved towards the origin. The lowest at which the cost line intersects in the feasible region will be the optimum solution. The following equation are solved as simultaneous equation to find optimum solution.
Solving (1) and (2) we get,
The values are substituted in the objective function to find the objective function value.
Optimal solution:
(2)
None of the constraints are having slack. All ≤ constraints are binding.
(3)
Protein and T constraint have surplus.
Protein:
The surplus is 92 (272 – 180).
T:
The surplus is 10 (20– 10).
(4)
The protein constraint is redundant because, it does not intersect at any point in the feasible region.
b)
To solve: The linear programming problem and answer the questions.
Introduction:
Linear programming:
Linear programming is a mathematical modelling method where a linear function is maximized or minimized taking into consideration the various constraints present in the problem. It is useful in making quantitative decisions in business planning.
b)
Explanation of Solution
Given information:
Calculation of coordinates for each constraint and objective function:
Constraint 1:
Constraint 2:
Constraint 3:
Objective function:
The problem is solved with iso-cost line method.
Graph:
(1) Optimal value of the decision variables and Z:
The coordinates for the cost line is (12, 8). The cost line is moved towards the origin. The lowest at which the cost line intersects in the feasible region will be the optimum solution. The following equation are solved as simultaneous equation to find optimum solution.
Solving (1) and (2) we get,
The values are substituted in the objective function to find the objective function value.
Optimal solution:
(2)
Constraint F is having slack as shown below.
The slack is 4.6 (12 – 7.4).
(3)
There are no surplus. D and E constraints with ≥ are binding.
(4)
There are no redundant constraints
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Chapter 19 Solutions
Operations Management
- Play Things is developing a new Lady Gaga doll. The company has made the following assumptions: The doll will sell for a random number of years from 1 to 10. Each of these 10 possibilities is equally likely. At the beginning of year 1, the potential market for the doll is two million. The potential market grows by an average of 4% per year. The company is 95% sure that the growth in the potential market during any year will be between 2.5% and 5.5%. It uses a normal distribution to model this. The company believes its share of the potential market during year 1 will be at worst 30%, most likely 50%, and at best 60%. It uses a triangular distribution to model this. The variable cost of producing a doll during year 1 has a triangular distribution with parameters 15, 17, and 20. The current selling price is 45. Each year, the variable cost of producing the doll will increase by an amount that is triangularly distributed with parameters 2.5%, 3%, and 3.5%. You can assume that once this change is generated, it will be the same for each year. You can also assume that the company will change its selling price by the same percentage each year. The fixed cost of developing the doll (which is incurred right away, at time 0) has a triangular distribution with parameters 5 million, 7.5 million, and 12 million. Right now there is one competitor in the market. During each year that begins with four or fewer competitors, there is a 25% chance that a new competitor will enter the market. Year t sales (for t 1) are determined as follows. Suppose that at the end of year t 1, n competitors are present (including Play Things). Then during year t, a fraction 0.9 0.1n of the company's loyal customers (last year's purchasers) will buy a doll from Play Things this year, and a fraction 0.2 0.04n of customers currently in the market ho did not purchase a doll last year will purchase a doll from Play Things this year. Adding these two provides the mean sales for this year. Then the actual sales this year is normally distributed with this mean and standard deviation equal to 7.5% of the mean. a. Use @RISK to estimate the expected NPV of this project. b. Use the percentiles in @ RISKs output to find an interval such that you are 95% certain that the companys actual NPV will be within this interval.arrow_forwardWhen you use a RISKSIMTABLE function for a decision variable, such as the order quantity in the Walton model, explain how this provides a fair comparison across the different values tested.arrow_forwardAssume the demand for a companys drug Wozac during the current year is 50,000, and assume demand will grow at 5% a year. If the company builds a plant that can produce x units of Wozac per year, it will cost 16x. Each unit of Wozac is sold for 3. Each unit of Wozac produced incurs a variable production cost of 0.20. It costs 0.40 per year to operate a unit of capacity. Determine how large a Wozac plant the company should build to maximize its expected profit over the next 10 years.arrow_forward
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- Company is an automobile replacement parts dealer in large metropolitan community. B company is preparing its sales forecasts for the coming year. Data regarding both B company’s and industry sales of replacement parts as well as both the used and new B company sales in the community for the last 10 years have been accumulated. If B company wants to determine if there is a historical trend in the growth of its sales as well as the growth of industry sales of replacement parts, the company would employ A. Simulation techniques B. Queuing theory C. Statistical sampling D. Time series analysisarrow_forwardA farmer is considering planting five possible crop mixes. The outcomes depend on the weather (dry, average, or rainy). Assume the three possible weather states have probabilities: dry = 30%, average = 40%, rainy = 30%. Answer the following two questions. Mix A Mix B Mix C Mix D Mix E Dry -1 1 -1 2 01 5 Avera ប្តី ge T 1 -1 5 2 3 Rainy 9 54 13₂ 2arrow_forwardMartin owns an older home, which requires minor renovations. However, the neighborhood where Martin lives mostly includes newly constructed luxury homes. Why might Martin's home increase in value? Based on the principle of substitution, the value of Martin's house will equal the value of the newly constructed homes in the neighborhood. ○ The value of Martin's home will decrease due to the new competition in the neighborhood. Based on the principle of regression, the newly constructed homes in the neighborhood will increase the home values of the entire neighborhood. Based on the principle of progression, the newly constructed homes in the neighborhood will increase the home values of the entire neighborhood.arrow_forward
Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,