Cost Considerations
7.1 Cost Models: Profit Maximization
Usually, in cost objective functions there is a need to consider the time value of cash flows. The usual approach is through a discounted cash flow in which cash flow in the future is discounted downwards to the present day. Although the concept of this cash flow is well understood and is analytically tractable, nevertheless a seri- ous technical/managerial issue arises in regard to what interest rate (I%, sometimes called the hurdle rate) to assume in the discounted cash flow calculations. The nature of the different elements in the objective function may be such that some parts of these elements are integer valued only, whereas others may be continuous. This gives rise to the usual computational difficulties associated with such objective functions.
Because of the different considerations mentioned in this section, there are a number of different possible formulations of production line design problems involving cost considerations. A number of profit objective functions will be considered below in increasing order of sophistication.
A relatively simple profit objective function would be as follows:
max F1= (R−C)XK∗−ChW IP, (7.1) where
R is the deterministic selling price of a unit of the product.
C is the direct cost associated with each unit produced. This direct cost should include the operative wages cost per unit produced, the cost of material used per unit and the recurrent machine costs per unit produced. In some situations, the latter costs may not be included.
XK is the normalized throughput of the specified production line consisting of K work-stations and K−1 intermediate buffers, whereas XK∗is the normalized through- put multiplied by the maximum physical throughput of the system, i.e., the XK∗is the actual physical output per unit time.
W IP is the average work-in-process (WIP) summed over all the K−1 buffers.
Chis the inventory holding cost, reminiscent of the same factor, which appears in inventory models, and is a measure of the cost of holding an item in inventory for the same time period as is in the throughput.
Clearly, this rather simple objective function has a number of associated difficult measurement/estimation problems. As indicated above, C, the direct cost, may not be very inclusive because some of the direct costs associated with the production of one unit of product are fuzzy. For example, it may be quite difficult to estimate the per unit product cost of machine repair. Needless to remark, there is no specific assignment of overhead costs and F1could not be considered to be a normal profit function as understood by accountants. Sometimes, the term “contribution to profit and overhead” is used to indicate the term(R−C)above. Moreover, it is necessary to specify the system to which objective function (7.1) is to be applied. What is fixed and what may be considered decision variables?
To illustrate the use of cost objective functions in general, the following exper- iment was undertaken. Perfectly reliable production lines with K =3,4,5 and 6 stations were considered with the same average processing time at each station (bal- anced lines). The number of total buffer slots to be allocated among the K−1 buffers varied from 1 up to 65, depending on the size of the production line.
Consider the output of a production line with K stations in which the finished product is sold at a value of 50 financial units (FU) and costs 40 FU to produce. The contribution margin per unit produced is therefore 10 FU. The assessment of Ch, the average holding cost per unit held in work-in-process (WIP), may be approximated as follows:
Ch=αCI,
where 0<α<1, C is the cost of production, defined above, and I is the relevant or assigned interest rate per annum. The function ofα is to take into account the fact that the value of an item in WIP increases as the item progresses down the line. In the example here, ifα is assumed to be 0.5 and I=10% per annum, then Ch=0.5C(0.1) =0.05C. In general, the value ofα would be estimated based on material, labor and machine content of the item in inventory.
In this case, define the ratio, r, of the marginal contribution divided by the average unit per annum holding cost as
r= f1
f2 =R−C
Ch = 50−40 (0.05)(40)=5.
To continue the illustration, assume the isolated throughput of all the stations is 1 unit per minute (balanced line) and that the facility operates 2 shifts of 8 hours per day for 250 working days per annum. The total maximum output is therefore
60∗2∗8∗250=240,000 units/annum.
The objective function now becomes
max F1=f1XK∗−f2W IP=f1(240,000)XK−f2W IP.
400000 500000 600000 700000 800000 900000 1000000 1100000 1200000
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 F1 F1
Ntotal
Fig. 7.3. Value of F1as a function of N for a 5-station production line with R=50 FU, C=40 FU, I=10%,α=0.5 (r=5)
This is equivalent to maximizing
(240,000)rXK−W IP=1,200,000XK−W IP.
This latter equation indicates that in any conceivable practical situation (where W IP<1,200,000XK), an objective function of type (7.1), given above, leads to the same optimal operating strategy as the more simple objective function of maximizing throughput for the given number of buffer slots, N. Of course, equation (7.1) assumes that R and C are constants independent of the value of XK∗and so, for example, there is no overtime premium and all product produced is sold. In Figure 7.3, the value of the objective function for each N for these parameter values is given.
A modification of the profit objective function given in (7.1) would be
max F2=max F1−bN= (R−C)XK∗−ChW IP−bN (7.2) where bN represents the cost on an annual basis of the buffer space used, i.e., each buffer slot costs b financial units (FU) per annum, where physical output, XK∗, and inventory costs, Ch in FU, are on a per annum basis. The term bN gives the plan- ner some scope for financial justification of a proposed design of a production line in which the number of stations is fixed, the cost of the proposed machines at the stations is given but the decision in relation to the total number of buffer slots, N, is still open. In a practical situation, where the effect of the W IP term is small and the buffer allocation associated with maximum throughput for any N is known, it is only necessary to determine the lowest value of N at which the marginal contribution to the first term of F2is lower than the cost of providing an extra buffer slot. A slightly
400000 500000 600000 700000 800000 900000 1000000 1100000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 F2 F2
Ntotal
Fig. 7.4. Value of F2as a function of N for a 5-station production line with R=50 FU, C=40 FU, I=10%,α=0.5, b=1000 FU (r=5)
more appropriate objective function might replace the bN term with∑Ki=2biNiwhere biis the cost per annum of providing a buffer slot of type i for each of the Nislots, i=2,3,...,K.
It is clear from expression (7.2) that in a practical system, a situation will arise where the marginal advantage of increasing the throughput by the optimal allocation of an additional buffer slot will result in a reduction in the objective function, F2. This is illustrated in Figures 7.4 and 7.5. Figure 7.4 refers to the following system with parameters: K=5 stations, R=50 FU, C=40 FU, I=10%,α=0.5, b=1000 FU, whereas, Figure 7.5 refers to a system with parameters as follows: K=5 stations, R=50 FU, C=40 FU, I=10%,α=0.5, b=5000 FU.
As indicated in Figure 7.4, F2increases monotonically as N increases rather sim- ilarly as F1, shown in Figure 7.3. On the other hand, Figure 7.5 indicates that an optimal value of F2 is achieved and that increasing the total number of slots, N, above a certain level leads to a reduction in the value of the objective function F2
(from the value in this case of N=27 onwards).
Extending the profit objective function further, it is desirable to bring into con- sideration discounted cash flows or the time value of money. If one considers a production line to be a generator of cash flows, there are three types of cash flows involved:
• Initial investment in production line facility, i.e., machines, stations, buffer slots, all at time t=0. These flows are considered to be negative.
• Cash flows during the useful life of the production line. These flows would nor- mally be considered to occur with regular frequency and consist of such flows
F2
600000 650000 700000 750000 800000 850000 900000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 F2 MAX
Ntotal
Fig. 7.5. Value of F2as a function of N for a 5-station production line with R=50 FU, C=40 FU, I=10%,α=0.5, b=5000 FU (r=5)
as revenue from sales, wages paid, energy used, materials purchased and used, maintainance/repair costs, etc. In a net sense, these flows would be expected to be positive.
• End of life flows, such as salvage value of machines and buffers, human resources consequences and final disposal of remaining work-in-process (WIP). Normally, these flows could be positive or negative.
In discounted cash flow analysis, the three different types of flows listed above must be treated separately. The basic concept is to develop a present value of all the flows at a chosen time t, usually, t=0. If t=0, the initial investment in the production line facility does not have to be discounted but it is considered negative and is here denoted byΩ1. On the other hand, the other two cash flows have to be discounted.
Taking the end of life flows first and assume that they occur at time t=T , each of these flows must be discounted using the following Present Worth Factor, P.W.F.∗, of a cash flow received at time T from now (t=0):
P.W.F.∗= 1 (1+I)T
where I is the interest rate per unit time (usually, per annum).
The above P.W.F.∗is difficult to use in analysis and as a result the concept of continuous discounting is introduced. The idea is basically as follows.