學習目標 1.2 獨特產品公司損益平衡分析 （1.2 節） 1.3 – 1.8 第 1 章 導論 補充教材 學習目標 1.2 獨特產品公司損益平衡分析 （1.2 節） 1.3 – 1.8 補充教材 廣告問題（華盛頓大學上課教材） 1.9 – 1.22 說明如何運用管理科學方法來解決問題。在華盛頓大學，此為 MBA 核心課程——管理科學，第一堂課的教材。 然而，其中也包括一些進階的主題（規劃求解以及非線性目標函數等），可以直覺且直接的方式在試算表上教授，這是一種非常有效介紹規劃求解（Solver）功能的方式。然後在接下來幾堂課程中，將會介紹線性規劃、模式化，以及規劃求解等更多基礎的內容。

學習目標 在讀完本章後，你應該能夠： 1. 清楚定義何謂「管理科學」。 2. 描述管理科學的本質。 3. 解釋什麼是數學模式。 1. 清楚定義何謂「管理科學」。 2. 描述管理科學的本質。 3. 解釋什麼是數學模式。 4. 運用數學模式來從事損益平衡分析。 5. 使用試算表模式來執行損益平衡分析。 6. 了解本書與眾不同的一些特點。

獨特產品損益平衡分析 獨特產品公司生產昂貴且不尋常的禮品 。 最近的新產品開發專案為限量版的老爺鐘。 資料： 若公司已決定生產此項產品，首先將衍生 50,000 美元的固定成本（fixed cost）以支付生產設備。 若生產每一座老爺鐘的變動成本為 400 美元。 每賣出一座老爺鐘可為公司創造 900 美元的營收。 將可以獲得銷售預測值。 問題：公司是否應該生產老爺鐘？若是的話，該生產多少數量？ Slides 1.2–1.6 cover the Special Products Break-Even Analysis covered in the text, and includes many of the figures from the text.

以數學形式表示問題 決策變數： 成本： 利潤： Q = 老爺鐘的生產數量 固定成本 = $50,000（若 Q > 0） 總成本 = 0，若 Q = 0 $50,000 + $400 Q，若 Q > 0 利潤： 利潤 = 總收益 – 總成本 利潤 = 0，若 Q = 0 利潤 = $900Q – ($50,000 + $400Q) = –$50,000 + $500Q，若 Q > 0

獨特產品公司試算表 Figure 1.1 A spreadsheet formulation of the Special Products Company problem.

問題的分析 Figure 1.2 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.

管理科學互動模組 可以藉由互動管理科學模組（為隨書所附 CD-ROM 中的 MS 教學 輔助軟體）中的損益平衡模組來執行敏感度分析。 Figure 1.3 A screen shot of the Break-Even Analysis module in the Interactive Management Science Modules after changing the fixed cost for the Special Products Company problem from $50,000 to $75,000. 可以藉由互動管理科學模組（為隨書所附 CD-ROM 中的 MS 教學 輔助軟體）中的損益平衡模組來執行敏感度分析。 從上圖可以看到將固定成本改變為 75,000 美元 時的影響。

獨特產品公司試算表 Figure 1.4 An expansion of the spreadsheet in Figure 1.1 that uses the solution for the mathematical model to calculate the break-even point.

廣告問題 Parker Mothers 是一家兒童玩具與遊戲製造商。其中最暢銷的玩具 是互動電子哈利波特娃娃。 相關資料： 單位變動成本： $48 單位售價： $65 固定成本： $42,000 Parker Mothers 分析哈利波特娃娃（以及其他類似玩具）過去相關 資料，並且決定出影響銷售量的幾項因素： 季節（例如： 聖誕節、哈利波特書籍或電影剛發行期間等） 專屬銷售人員數 廣告量 問題：哈利波特娃娃的廣告預算應該編列多少？（提議：$50,000） Slides 1.8–1.21 are based upon the first lecture in the core-MBA class on Management Science at the University of Washington (as taught by one of the authors). It represents an illustration of the management science approach to a problem. While it includes some advanced topics (Solver, nonlinear objectives, etc.) it is entirely spreadsheet-based and quite intuitive. It has proven to be a good introduction to the power of Solver. The next several lectures then would need to “back up” and cover more of the fundamentals of linear programming, modeling, the Solver, etc.

預測銷售量 在執行統計迴歸分析之後，他們估計每季的銷售量約略與季節及 廣告預算有關，關係如下： 季節因子（Seasonality Factors） Q1: 1.2 （哈利波特新書發表期間） Q2: 0.7 Q3: 0.8 Q4: 1.3 （聖誕節以及預期哈利波特新電影的發行期間） 廣告的影響：

第 1 季之試算表

試驗解（Trial Solutions） This slide shows the same column C as from the preceding slide, repeated many times in order to show the effect of different advertising levels. Alternatively, if this is done using a live spreadsheet in class, one can use the preceding spreadsheet and manually adjust the advertising levels to see the effect.

Excel Solver Here we first introduce the Solver. It is a simple model with no constraints. We’re simply letting the Solver (“magically” at this point) find the value of the changing cell (the advertising level) that maximizes the target cell (the profit).

最佳解

四季之試算表 Here, the spreadsheet is expanded to four quarters. The only difference between the four quarters is the seasonality factor. We begin by trying to guess the impact of the seasonality factor on the optimal advertising levels.

最佳解（四季）

殘差效應（Residual Effect） In previous spreadsheets it was assumed that advertising only affects the sales in the quarter. However, advertising can impact future quarters as well. This spreadsheet assumes that advertising is driven primarily by advertising in the current quarter, but are also impacted by advertising in the previous quarter (30% of the impact). This has the effect of making the 4 quarters no longer independent. It is no longer a simple task to find the optimal solution for each quarter, since they depend upon the other quarters. However, it is no problem for the Solver.

殘差效應（Solver最佳化後） The first attempt at solving the model results in negative advertising in quarter 2. While this “mathematically” maximizes the profit, it does not make sense. In practice it could not be implemented, nor would it be accurate. This leads into a discussion about constraints. The changing cells must be nonnegative.

Solver Options The “Assume Non-negative Option” in Solver Options forces all of the changing cells to remain nonnegative.

殘差效應（經Solver重新最佳化後） The model is “re-optimized” with the addition of the “Assume Non-Negative” option. While mathematically the profit is reduced ($327,823 rather than $328,688), this is now a solution that can be implemented.

殘差效應（預算最佳化後） We conclude with a discussion about how the proposed advertising budget in the previous slide ($327,823) is significantly higher than the original proposal ($50,000 per quarter). Perhaps this is beyond their ability to pay. Perhaps the relationship between units sold and advertising does not apply at these advertising levels. One possibility is to introduce a budget. This is implemented on this and the following slide.

在 Solver 中加入一條限制式 This slide shows how the budget constraint can be added in Solver.