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**Chapter 1- Introduction to Management Science**

The Management Science Approach to Problem Solving Model Construction: Production Management Science Modeling Techniques Breakeven Analysis and Excel Model Example Indifference Point Analysis

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**Applications of Management Science in Business: Opening a Fairwood Fast Food Restaurant**

Business questions: Is it worth doing? (Break-even analysis, Forecasting) What seating capacity should we build? (Decision analysis) How much food material to prepare? (Forecasting) 4. How to schedule the staff to ensure a certain customer service level? (Linear programming, Waiting line management)

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**The Management Science Approach**

Management science uses an objective (logical, scientific and mathematical) approach to solve management problems. It is used in a variety of organizations to solve many different types of problems. Investment, resource allocation, production mix, marketing, multi-period scheduling etc.

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**The Management Science Process**

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**Steps in the Management Science Process**

Observation - Identification of a problem that exists in a system or an organization. Definition of the Problem - Problem must be clearly and consistently defined showing its boundaries and interaction with the objectives of the organization. Model Construction - Development of the functional relationships that describe the decision variables, objective function and constraints of the problem. Model Solution - Model solved using management science techniques. Model Implementation - Actual use of the model or its solution.

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**Example of Model Construction Problem Definition**

Information and Data: - A bakery makes and sells birthday cakes (Cake A and Cake B) - Cake A costs $100 to produce, Cake B costs $120 - Cake A sells for $200 while Cake B sells for $250 - Cakes A and B require 0.5 and 0.8 pounds of double cream to make, respectively - The bakery has 20 pounds of double cream for each day Business problem: Assuming all cakes produced can be sold out, determine the numbers of different cakes to produce to make the most profit given the limited amount of double cream available.

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**Example of Model Construction Mathematical Model**

Decision Variable: x = number of Cake A to produce y = number of Cake B to produce Z = total profit Model: Z = $200x +$250y- $100x - $120y (objective function) 0.5x +0.8y <= 20 lb of double cream (resource constraint) Parameters: $200, $100, 0.5 lb, 20 lbs (known values) Formal specification of model: maximize Z = $200x + $250y - $100x - $120y subject to 0.5x +0.8y <= 20 x >= 0, y>= 0

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**Management Science Modeling Techniques**

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**Characteristics of Modeling Techniques**

Linear mathematical programming: clear objective; restrictions on resources and requirements; parameters known with certainty. Probabilistic techniques: results contain uncertainty. Network techniques: model often formulated as diagram; deterministic or probabilistic. Forecasting and inventory analysis techniques: probabilistic and deterministic methods in demand forecasting and inventory control. Other techniques: variety of deterministic and probabilistic methods for specific types of problems.

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Break-Even Analysis Break-even analysis is an important type of cost-volume analysis, which focuses on relationships between cost, revenue, and volume of output. One purpose of break-even analysis is to estimate the income of an organization under different operating conditions. It is a key component of most business plans and is especially important for starting-up companies seeking financing or investors. Performing a break-even analysis is a simple way to determine price levels and to estimate whether an expansion or cost saving project makes good business sense.

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Break-Even Analysis The goal of a break-even analysis is to determine when sales revenue equals total expenses; in simple terms, when a business or operation "breaks even." The real value lies in helping you determine the relationships between revenue, fixed costs, and variable costs. Changing one variable changes the results and allows you to model a variety of potential scenarios and make better business decisions. You can use a break-even analysis to: Make pricing decisions Determine the feasibility of selling new products Evaluate a project

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Break-Even Analysis Break-even analysis is used to determine the break-even point: the number of units of a product to sell or produce (i.e. volume) that will equate total revenue with total cost. The formula is simple: Total Revenue = Fixed Costs + Total Variable Costs

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**Break-Even Analysis Model Components**

Total cost (TC) - fixed cost plus total variable cost cf + Qcv Fixed cost (cf) - cost that remains constant regardless of number of units produced. For example, rent and salaries are fixed costs; a company will pay rent and salaries even if the company does not produce any product. Unit Variable cost (cv) - unit cost of product. For example, material cost, shipment cost and sales commission. Total variable cost (Qcv) - Cost that changes based on activity. Increase or decrease when the company produces more or less products. It is a function of production volume (Q) and unit variable cost.

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**Break-Even Analysis Model Components**

Total revenue (TR) - selling price per unit x sales volume pQ where p is the selling price per unit Profit(Z) - difference between total revenue Qp (p=price) and total cost: Z = pQ – (cf + Qcv)

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**Break-Even Analysis Computing the Break-Even Point**

The break-even point is the volume at which total revenue equals total cost and profit is zero: Qb/e = cf/(p-cv)

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**Example - Special Products Company**

The Special Products Company produces expensive and unusual gifts to be sold in stores that cater to affluent customers who already have everything. The latest new-product proposal to management from the company’s research department is a limited edition grandfather clock. Management needs to decide whether to introduce this new product and, if so, how many of these grandfather clocks to produce. Before making this decision, a sales forecast will be obtained to estimate how many clocks can be sold. Management wishes to make the decision that will maximize the company’s profit. If the company goes ahead with this product, a fixed cost of $50,000 would be incurred for setting up the production facilities to produce this product. (Note that this cost would not be incurred if management decided not to introduce the product since the setup then would not be done.) In addition to this fixed cost, there is a production cost that varies with the number of clocks produced. The unit variable (marginal) cost is $400 per clock produced. Each clock sold would generate a revenue of $900 for the company.

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**Special Products Company**

Q = Number of grandfather clocks to produce (Decision Variable) cf = $ if Q > 0 (cf = 0 if Q = 0) cv = $400 per unit p = $900 per unit Total variable cost = $400Q Total cost = $ $400Q Total revenue = $900Q Profit = Total revenue – Total cost = $900Q – ($ $400Q) Qb/e = 100 units, break-even point

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**Special Products Company**

To cover fixed and variable costs and break even the company will need to sell 100 clocks. Sell more clocks and as long as the fixed cost doesn't increase, each additional sale will generate an incremental gross profit of $500. On the other hand if the company sells less than 100 clocks the company will not cover its fixed cost and will operate at a loss.

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**Analysis of the Problem**

Figure 1.1 Break-even analysis for the Special Products Company shows that the cost line and revenue line intersect at Q = 100 clocks, so this is the break-even point for the proposed new product.

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**Special Products Company Spreadsheet**

Figure 1.4 An expansion of the spreadsheet in Figure 1.3 that uses the solution for the mathematical model to calculate the break-even point.

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Sensitivity Analysis A management science study usually devotes considerable time to investigating what happens to the recommendations of the model if any of the estimates turn out to considerably miss their targets.

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**Here is the impact on Qb/e of changing the fixed cost to $75,000.**

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**Here is the impact on Qb/e of changing the unit variable cost to $300.**

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**Here is the impact on Qb/e of changing the price to $1,500.**

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**Example - Great Threads Company**

Building an Excel Model to Find a Break-even Point

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**Background Information**

The Great Threads Company sells hand-knit sweaters. Great Threads is planning to print a brochure of its products and undertake a direct mail campaign. The cost of printing the brochure is $20,000 plus $0.10 a catalog. The cost of mailing each catalog is $0.15. In addition, the company will include direct reply envelopes in it’s mailings. It incurs $0.20 in extra cost for each direct mail envelope that is used by a respondent.

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**Background Information**

The sales revenue of a customer order is $40, and the company’s variable cost per order averages around 80% of the order’s value. The company plans to mail 100,000 catalogs. It wants to develop a spreadsheet model to answer the following questions:

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**Background Information**

How does a change in the response rate affect profit? For what response rate does a company break even? If the company estimates a response rate of 3%, should it proceed with the mailing? How does the presence of uncertainty affect the usefulness of the model?

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**Parameters Mailing: Fixed cost of printing = $20,000**

Variable costs: Printing = $0.1 Mailing, buying names = $0.15 Number of brochures mailed = 100,000 Order Unit revenue = $40 Variable cost (% of order) = 80% Variable cost of envelopes = $0.2

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**Excel Model of GREATTHREADS**

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**Excel Model of GREATTHREADS**

Note the clear layout of the model The input cells are outlined and shaded and separated from the outputs. There are boldfaced headings, several headings are indented. Numbers are formatted appropriately. Text boxes to the right spell out all the range names used.

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**Creating the Excel Model**

To create this model, proceed through the following steps. Enter heading and range names Obviously we have a lot of cells, more than you might want to enter, but you will see their value when we start entering formulas. Enter input values The values in the shaded cells are all given in the statement of the problem. Enter these values and format them appropriately.

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**Creating the Excel Model**

Model the responses We have not specified the response rate of the mailing, so enter any reasonable values such as 8% in the ResponseRate cell – we will perform sensitivity on this value later on – and enter the formula =NumMailed*ResponseRate in the NumResponse cell.

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**Creating the Excel Model**

Model the total revenue, costs and profit. Enter the formula=NumResponses*AvgOrder in the Total Revenue cell. Enter the formula=FCostPrinting, =SUM(VCostMailing)*NumMailed and =NumResponses*(AvgOrder*VCostOrderPct+ VcostEnvelopes) in the Cost cells (E10, E11, E12). Enter the formula=SUM(Costs) in the TotalCost cell,and enter the formula=Total Revenue-TotalCost in the profit cell.

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**Answering the Questions**

Now that a basic model has been created, we can answer the questions posed by the company. For question 1, we form a data table to show how profit varies with the response rate.

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Creating a Data Table First, enter a sequence of trial values of the response rate in column A, and enter a “link” to Total Revenue in cell B20 with the formula =E8 enter a “link” to Total cost in cell C20 with the formula =E13 enter a “link” to Profit in cell D20 with the formula =E14 Finally, highlight the entire table range, A20:D30, and select the Data/What-If Analysis Table menu item to bring up the dialog box shown here. Enter E4

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Creating a Data Table It should be filled in as shown to indicate that the only input ResponseRate, is listed along a column. When you click OK, Excel substitutes each response rate value in column A into the ResponseRate cell, recalculates the total revenue, total cost and profit, and reports them in the data table.

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Scatter Plot For a final touch, we have created a scatterplot (or in Excel’s terminology X-Y chart) of the values in the data table.

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**Answering the Questions**

Clearly, profit increases in a linear manner as response rate varies. More specifically, a 1% increase in the response rate always increases profit by $7800. Here is the reasoning. Each 1% in response rate results in 100,000*0.01=1000 more orders. Each order yields an average revenue of $40 but incurs a variable cost of $40*80%=$32 and a $0.20 envelope cost. The net gain is $7.80 per order.

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**Answering the Questions**

From the data table, we see that profit goes from negative to positive when the response rate is somewhere between 5% and 6%.

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**Answering the Questions**

Question 2 asks for the exact breakeven point. This could be found with trial and error but is easy with Excel’s Goal Seek tool. Goal Seek is useful for solving a single equation in a single unknown. Here the equation is Profit=0, and the single unknown is the response rate.

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**Answering the Questions**

In Excel terminology, the unknown is called the changing cell because we are allowed to change it to make the equation true. To implement Goal Seek, select Data/What-If Analysis/Goal Seek menu item and fill in the resulting dialog box as shown below.

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**Answering the Questions**

After clicking on OK, the ResponseRate and Profit cells have values 5.77% and $0. In words, if the response rate is 5.77% Great Threads breaks even. If the response rate is greater than 5.77%, the company makes money; otherwise, it loses money.

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**Answering the Questions**

Question 3 asks if the company should proceed with the mailing if the response rate is only 3%. From the data table, the apparent answer is “no” because profit is negative, a loss. However, like many business companies, we are taking the short term view with this reasoning.

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**Answering the Questions**

We should realize that many customers who respond to direct mail will reorder in the future. The company makes $7.80 per order. If each of the respondents ordered two or more times, say, the company would earn 3000*$7.80*2=$46,800 more than appears in the model, and profit would then be positive.

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**Answering the Questions**

The moral is that we must look at long-term impact of our decisions. However, if we want to incorporate the long term explicitly into the model, we must build a more complex model.

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**Answering the Questions**

Finally, question 4 asks about the impact of uncertainty in the model. We would be kidding ourselves to think that all model inputs are known with certainty. For example, the size of an order is not always $40 – it might be, say, from $10 to $100. When there is a high degree of uncertainty about model inputs, it makes little sense to talk about the profit level or the breakeven response rate.

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**Answering the Questions**

It makes more sense to talk about the probability that profit will have a certain value or the probability that the company will break even.

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**Indifference Point Analysis**

The indifference point analysis determines the point at which there is no difference in profit (cost) between two alternative methods. That is, at a particular point, the decision maker has no preference for one option over another, they are equally preferred. Example: Site Selection A firm will set-up a production line of a new product in Hong Kong or in Shenzhen. If the production line is set up in Hong Kong, the annual fixed cost and unit variable cost would be $10,000,000 and $300 respectively. If the production line is set up in Shenzhen, the annual fixed cost and unit variable cost would be $8,000,000 and $400 respectively. Determine the indifference annual demand volume (Q) of the product for which both alternatives are equally good.

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**Indifference Point Analysis**

TC(HK) = $10,000,000 + $300 *Q TC(SZ) = $8,000,000 + $400 * Q At the indifference annual demand quantity, TC(HK) = TC(SZ) Implies $10,000,000 + $300 *Q = $8,000,000 + $400 * Q Solving the equation, Q = 20,000 At this point, both options will give a total cost of $16,000,000

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General Conclusion If Q < 20000, set up the production plant in Shenzhen. If Q = 20000, it is indifferent to set up the production plant in either site. If Q > 20000, set up the production plant in Hong Kong

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**Please login to the English Window and download the file Breakeven_student.xls from Moodle.**

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