Journal Search Engine
Search Advanced Search Adode Reader(link)
Download PDF Export Citaion korean bibliography PMC previewer
ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.11 No.1 pp.11-17
DOI : https://doi.org/10.7232/iems.2012.11.1.011

Safer Zone Analysis for Multiple Investment Alternatives on the Total-Cost Unit-Cost Domain

Hirokazu Kono*
*Graduate School of Business Administration, Keio University
Received: November 12, 2011 / Revised: February 22, 2012 / Accepted: February 23, 2012

Abstract

Along with the recent trend toward increasing variety and shorter life of products in the market, evaluation of risk for economic investment alternatives is of practical importance in manufacturing companies. This paper assumes that each alternative is composed of demand volume and unit sales price as income factors, and unit variable cost and fixed cost as expense factors. The paper assumes that these four factors move worse from the originally expected values, toward the direction of decreasing profit. Values of these four factors are also assumed to fluctuate from year to year over the entire multi-period. By applying the analysis of the breakeven points to each of the four factors, safer area against these changes is represented on the two dimensional domain called normalized total-cost unit-cost domain. A practical numerical example is analyzed to verify the validity of the proposed method.

11-1-02.pdf292.7KB

1. INTRODUCTION

 Against the backdrop of increasing uncertainly in the recent economic conditions, a method of evaluating economic rigidity and safety of investment alternatives is of practical importance. Economic evaluation of rigid­ity and safety was developed in the area of engineering economy (Senjuet al., 1986; Senjuet al., 1994), as well as managerial decision making (Ruefliet al., 1999; Leung et al., 2006). Safety evaluation method was developed by Nakamura (1985, 2002) and Kono (2003, 2006). Kono and Mizumachi (2008) first proposed a method of safety evaluation applying the concept of breakeven point, but the evaluation procedure is restricted to a sin­gle investment alternative, leaving a method for compar­ing rigidity for multiple alternatives as a future research issue.
This paper examines a simple model of investment alternatives over multiple periods, composed of unit vari­able cost and fixed cost over each period. For the prod­uct, unit sales price and sales volume over each period is estimated. The paper analyzes a case of uncertainty in which one of the values for these four factors under con­sideration move worse toward the direction of decreas­ing profit. The paper first identifies the breakeven point for each of the four factors, and shows the values of these breakeven points on the normalized TC-UC do­main, whose horizontal axis represents total cost and vertical axis, unit-cost. Then the paper discusses an area on the normalized TC-UC domain where alternatives safer for all of the four factors are plotted. Thus, a prac­tical and visual method of rigidity analysis for invest­ment alternatives is formulated and proposed.

2. MODEL FORMULATION

This section summarizes assumptions and notations of the model for analysis.
1) The planning horizon covers n periods. For the j-th period, as sales conditions, unit sales price and sales volume for the product under consideration are estimated and denoted by by pj and Qj, j = 1, 2, , n, re­spectively
 
2) As production conditions, cost structure for producing the product is given by unit variable cost vj and fixed cost Fj  over the j-th
period, j = 1, 2, …, n. The sales volume is assumed to be equivalent to production volume in each period.
3) Capital cost, which is common over the entire planning horizon, is given by interest rate i.
4) Product is designated by A, B, C, … and represented by subscript when necessary. Based on these assumptions, the profit for the j-th
period, denoted by , j π is given by

Its discounted present value considering capital cost is obtained by 

 It follows that the total sum of profit over the entire planning horizon is represented by

3. BREAKEVEN POINTS

 This section analyzes breakeven points for each of the four factors, for the purpose of comparing rigidity against unexpected changes in each of the four factors under consideration.
It should be noted that the value of total profit  differs from product to product. For the purpose of com-paring rigidity among multiple products, the value of profit is converted into the ratio against total sales. At the same time, the weight of periodical sales should also be taken into account, considering that the period with high sales has greater impact on the ratio of profit over sales than that with lower sales.
Here, we denote the total sales for the j-th period, discounted to the present value, by Rj, where,

It follows,

 This paper investigates the following changes in each of the four factors, which is assumed to be inde­pendent of each other:

From statement (7), the following equations can be ob­tained. 

 Then, the breakeven point for each factor, denoted by α*, β* *, and δ respectively, can be derived as fol­lows:

On the TC-UC domain whose axes are normalized to the scale of (1, 1), the values of these breakeven points can be represented as shown in Figure 1. 

Fig. 1 Breakeven Points on the Normalized TC-UC Gomain.

4. SAFER AREA

 This section analyzes, on the normalized TC-UC domain, the area in which alternatives are safer against all the changes in values on the four factors. For sim­plicity of description, the paper defines following nota­tions:

 where subscript represents product identification.
We assume two alternatives A and B, as illustrated in Figure 2, where XA < XB and YA > YB are satisfied. For an alternative C, which is plotted to the upper right to the line segment AB as shown in Figure 2, The plot C satisfies

 This implies that plot C is located up right against AB , resulting that line segments connecting A, C, B shapes concaveto down left. Under these conditions, it is clear that either

  is satisfied against price decrease. Therefore, an alternative C can never be safer than ei­ther alternatives A and B against decrease in unit sales price.

Against decrease in sales volume, as shown in the case of    as in Figure 3(1), it is clear that

 is satisfied. On the other hand, for the case of

 as in Figure 3(2), it is clear that

   is satisfied. Then, it can be concluded that alternative C is less safe than alternatives A or B against decrease in sales volume.
Against increase in unit variable cost, where

 as in Figure 4(1), clearly

 is satisfied. While in the case of

 we can confirm that

   is satisfied as shown in Figure 4(2). Therefore, we can conclude that alternative C is not safer than alternatives A or B against increase in unit variable cost.
Lastly, against increase in fixed cost, two cases can be found as shown in Figure 5. Where of

 we can confirm that

is satisfied from Figure 5(1). While in the case of

 it is clear from Figure 5(2) that

 is satisfied. Then, it can be concluded that alternative C is not safer than alternatives A or B against increase in fixed cost.
As a result of the above discussion, a safer alternative than both A and B can never create concave to down left against line segment

Therefore, a safer area for a given set of alternatives is the area down left to the convex line segments given by connecting adja­cent plot of alternatives as described in Figure 6.
Further, for the convex line segments on the TC­UC domain, we shall investigate upper end and right end, as described in Figure 6. Let denote plots A and B as in the figure. At the upper end, setting a scale verti­callyon the verticalax is and turning it anti-clockwise centered on the point (0, 1), we can conclude that an alternative located in the upper right to the line segment connecting (0, 1) and plot A in Figure 6 is inferior in safety against changes in all of the four factors. In the same context, setting a scale horizontally on the horion­talax is and turning it clockwise centered on the point (1, 0) on the TC-UC domain, an alternative located in the upper to the line segment connecting (1, 0) and plot B on Figure 6 is also inferior in terms of safety against changes in the four factors.
Therefore, a safer area against changes in the four factors under consideration is represented as an area down left to the convexset of line segments as repre­sented in Figure 6.

Figure . 2 Consideration of These Alternatives.

Fig. 3 Decrease in Sales Volume.

Fig. 4 Increase in Unit Variable Cost.

Fig. 5 Increase in Fixed Cost.

Fig. 6 Safer Area on the Normalized TC-UC Domain.

 For simplicity, the paper takes a case of a single product. The paper assumes that the target life for the product is 3 years. For the sales conditions, the base case is given in Table 1. On the other hand, the base case of production is given in Table 2.

5. A NUMERICAL EXAMPLE

At this point, we consider several alternative cases for each of sales case and production case as follows;

For sales:
Base case I):
pjand Qjare given as in Table 1.
Scenario II):
pjis increased to 120% in each year from
the base case, and Qjis decreased to 80% in each year from the base case.
Scenario III): pjis increased to 150% in each year from the base case, and Qjis decreased to 60% in each year from the base case.

For production:
Base case a):
vjand Fjare given as in Table 2.
Scenario b):
vjis reduced to 85% in each year from
the base case, while Fjis increased 150% in each year from the base case.
Scenario c):    
vjis reduced to 80% in each year from the base case, while Fjis increased to 200% in each year from the base case.

 Similarly in the same manner, the x-y values of plots on the normalized TC-UC domain in each combination of sales case and production case are summarized in Table 3.
Then, we can plot all of nine combination cases on the normalized TC-UC domain, which is illustrated as in Figure 7. Clearly from this Figure, starting from plot (1, 0) and ending up at plot (0, 1), the convex line segments are organized by plots of a-III and b-III. It means that these two combinations are rigid against unexpected change in factors under consideration, but the other seven combinations can never be safer against changes in all factors under consideration. Thus, for this numerical example, applying the proposed procedure on the normalized TC-UC domain, we can point out production scenario C is disqualified in terms of economic rigidity against unexpected changes in factors, as well as sales scenarios I and II which are also disqualified. Thus, we can concentrate risk estimation on production scenario a and b, and sales scenario III, more specifically a-III and b-III. Thus, the proposed method can be a practical tool for eliminating disqualified alternatives in terms of risk and rigidity, and enable focusing on a limited number of alternatives for detailed risk and rigidity analysis. Thus,alternatives for detailed risk and rigidity analysis. Thus, the proposed method can support economic risk evaluation of investment alternatives.

Table 1 Base Case on Sales Conditions.

6. CONCLUDING REMARKS

This paper analyzed a problem of safety analysis for a set of alternatives over multiple periods. As a visual tool to represent a safer area, the paper proposed two-dimensional domain named normalized TC-UC domain.  
On the domain, breakeven points for each of the four factors, unit sales price, sales volume, unit variable cost, and fixed cost, are represented. Applying the result of breakeven point, the paper presented a safer area, against changes in the values of the relevant factors un­der consideration, on the normalized TC-UC domain. By plotting multiple alternatives on this domain, we can evaluate rigidity of given investment alternatives under uncertainties. Thus, the proposed TC-UC domain can work as a practical tool for economic rigidity evaluation.
The model can be extended to deal with such cases as simultaneous and dependent changes of relevant fac­tors, distinction of production volume and demand quantity, and varying change ratios over the multiple planning periods. Analysis of these cases is left as an area for future research.

Reference

1.Kono, H. (2003), Economic Safety Analysis for Mutually Exclusive Alternatives, Industrial Engineering and Management Systems, 2(2), 106-112.
2.Kono, H. (2006), Sensitivity Coefficient Index Analysis for an Investment Alternative under Uncertainties in Variables, Industrial Engineering and Manage- ment Systems, 5(2), 149-158.
3.Kono, H. and Mizumachi, T. (2008), Safety Analysis under Uncertainties for Investment Alternatives over Multiple Periods using the Total-Cost Unit- Cost Domain, Journal of Japan Industrial Man- agement Association, 58(6), 411-422.
4.Leung, S. C. H., Wu, Y., and Lai, K. K. (2006), A Stochastic Programming Approach for Multi-Site Ag-gregate Production Planning, The Journal of the Operational Research Society, 57(2), 123-132. Nakamura, Z. (1985), Economic Evaluation on the Variable-Cost Fixed-Cost Domain, Proceedings for the Spring Conference of the Japan Industrial Man- agement Association, Tokyo, 209-210.
5.Nakamura, Z. (2002), Safety Indices of Profit under Uncertainties, Proceedings for the Autumn Confer- ence of the Japan Industrial Management Associa- tion, Fukuoka, 54-55.
6.Ruefli, T. W., Collins, J. M., and Lacugna, J. R. (1999), Risk Measures in Strategic Management Research: Auld Lang Syne?, Strategic Management Journal,20(2), 167-194.
7.Senju, S., Fujita S., Fushimi T., Yamaguchi T. (1986), Engineering Economic Analysis, Nihon Kikaku- Kyokai, Tokyo.
8.Senju, S., Nakamura, Z., and Niwa, A. (1994), Exercises of Engineering Economy, Japan Management As- sociation Press, Tokyo.