SERVICE LEVEL MODELS AND INVENTORY OPTIMIZATION IN THE SUPPLY CHAIN

The great importance of the service level concept is universally recognized in the design of logistics operations, in communicating expectations regarding stock levels, in the relationship between customers and suppliers and in the segmentation of products and markets according to their importance or profitability. One of the main service level indicators adopted by companies is product availability. However, it turns out that this availability indicator can appear under different measurement models in different companies in the chain, which usually causes some confusion. Are all companies “speaking the same language”? When you hear the phrase “90% product availability,” exactly what is it referring to?

For example, among the different measurement or benchmarking models for the product availability indicator, it is worth highlighting the most common ones, as discussed below.

Probability of not running out of product, or the complement of the probability of running out of product: this indicator reflects the chances of having a shortage during resupply, regardless of the magnitude of the shortage. More specifically, an 85% probability of not running out of product indicates that, on average, out of every 100 resupplies there will be a shortage in 15 of them, no matter how much is missing (whether one unit or a thousand units, this indicator is the same).

In a typical supply chain, the probability of (not) missing a product tends to be the service level model most used in industrial relations, that is, between suppliers and manufacturers, and in contracts that have a strong bias towards penalizing shortages. product with fines, regardless of their magnitude, in order to guarantee the reliability of the supply. Some service level agreements (SLA- Service Level Agreement) between automobile assemblers and auto parts manufacturers contemplate fines in case the Just in Time supply is affected by breaches in the size of the supplier's batches.

Lost sales, or the basis for calculating the Fill Rate: this indicator reflects the average size of shortages during resupply, in situations where there is a shortage. Returning to the previous item, if 15 out of 100 resupply are missing, in the lost sales model it is possible to estimate, based on a probability distribution of the demand in the response time, the expected size of each shortage. For example, “20 units on average for each of the 15 resupply”.

It can be seen that, in a typical supply chain, lost sales models tend to be employed in the relationship between consumer goods manufacturers and retailers and between retailers and the final consumer. Great emphasis is placed on the demand service level or on the Fill Rate, that is, on the ratio between two estimates: the demand met and the total demand. More specifically, the Fill Rate can be approximated by:

FR = (Demand met)/(Total demand)
FR = 1 – (Lost sales)/(Total demand)

If the lot size is large enough, it can be used as an approximation for the total demand, in the form:

FR = 1 – (Lost sales)/(Batch size)

It can also be seen that, for companies that operate with reasonable inventory coverage, in percentage terms the Fill Rate indicator will always be greater than the probability indicator of not running out of product. Finally, by adopting lost sales as the basic service planning model, companies demonstrate their concern not with fines for breaching supply contracts, but rather with contribution margins eventually lost due to not having the product in stock for negotiation. It should be remembered that the contribution margin, or the price minus the variable cost (mc = p – cv), is another way of measuring the costs of lacking a certain product.

Backorders, or backorders: this indicator reflects the expected size of the total backorders for a given product, whenever there is a shortage in a given resupply and the situation cannot be characterized as a lost sale, that is, whenever the customer or consumer agrees to wait. It is a service level model that is quite common in wholesalers and some manufacturers of consumer goods, and is also seen in after-sales service, when providing technical assistance for spare parts, and in oligopolistic sectors, when the alternative for customers is imports. The existence of backlogs does not mean that there are no shortage costs: in this case, the shortage costs are associated with the time value of money of the contribution margin that the product generates. This is because what could be “invoiced today” and enter the company's cashier will only enter when the pending issue is resolved.

In this article, we indicate the analytical solutions for the first two models of service level (probability of not running out of product and lost sales) and we present, through two small case studies, how to optimize lot sizes and order points considering the logistical costs totals as a starting point. In each case, two basic situations are considered: one in which the response time demand is considered symmetric and modeled by the Normal distribution, and another in which the response time demand is considered asymmetric and modeled by the Exponential distribution. In a future article, the pending model will be discussed. The following section provides a brief description of these two important probability distributions.

A BRIEF REVIEW ON EXPONENTIAL AND NORMAL DISTRIBUTION

Table 1 presents the probability density function (f(x)) for the Normal and Exponential distributions, in its analytical form and in its form for implementation in MS-Excel. Also presented are the analytical form and the form in MS-Excel for the Probability of missing product during a given resupply (A(x)), which can be evaluated for a given order point (x = r). In the Table, μ and σ represent, respectively, the mean and standard deviation of demand in the response time, and, in the case of the Exponential distribution, μ = σ. Specifically, if r = μ (order point is the average demand in the response time), in the case of the Normal distribution A(r) = 50%, and A(r) = 36,79%, in the case of the Exponential distribution . It is noticed that the Exponential distribution is asymmetric, with demand bias in the response time being smaller than the order point.

Although the Normal distribution is symmetric around the mean and the Exponential distribution is not, both distributions represent diminishing returns in inventory management with respect to the three service level models under analysis. The sharp drop to the right of the charts in Chart 1 indicates an increasing effort in terms of lot sizes and order points to increase these indicators from higher plateaus. For example, it is possible to increase the probability of not running out of product by 10 percentage points, from 70% to 80%, for example, with much less investment in inventory than from 80% to 90% in both distributions. This property is important so that we can conceptualize higher service level policies as differentiated or premium policies. In other words, providing a 100% service level costs a lot and can be a difficult goal to achieve.

x = r : Evaluation at the point of order

Table 1 - Normal and Exponential Distributions

Next, we present the general solution for the total logistics costs of the probability model of not running out of product.

PROBABILITY MODEL OF NO MISSING PRODUCT

The Probability model of not running out of product is presented in Chart 2. In the upper left corner, the probability of occurrence of a shortage in each resupply plays an important role in each cycle or sawtooth of a policy (Q,r), where Q is the lot size (in units) and r is the reorder point (in units). Whenever there is a shortage during resupply (part of the sawtooth encompassed by the order point r), that is, whenever the demand verified in the response time is greater than r, the total costs will be increased by M*A( r), that is, the probability of occurrence of a fault multiplied by the fine applied to the company for each occurrence (as indicated in the Total Costs equation (TC) presented in the lower corner of Table 2).

Since there are two decision variables, lot size Q and order point r, the optimal solution for this service level model depends on deriving Total Costs as a function of Q and equating them to zero, so that it is set the minimum cost point. The result of this operation is shown in the lower corner of Table 2, where the optimal values ​​for the probability density function evaluated at the order point r (f(r)) and for the lot size Q are found.

Table 2 - General solution of the probability model of not running out of product

In general terms, the following can be stated, in qualitative terms, regarding the optimal solution:

• The higher the unit cost of acquiring the product (Caq), the capital opportunity rate (i) and the lot size Q, the greater the value of f(r). Observing the Normal and Exponential probability distributions, it is noticed that higher values ​​of f(r) tend to be associated with smaller order points (r). Thus, higher Caq, ie Q, lower order point. On the other hand, the higher D (annual demand) and M (fine incurred due to shortages), the higher the order point. It is interesting to note the trade-off between lot size (how much to ask) and order point (when to ask). In other words, larger lot sizes naturally decrease order placement frequency, ie, lead to operating with lower order points (resupply order takes longer to place).
• The optimal lot size (Q*) is basically a correction of the economic purchase lot (LEC) for the possibility of shortages (A(r)) and, consequently, a fine (M) at the end of a resupply. All other properties remain the same: the higher the fixed resupply costs (CTR) and the annual demand (D), the larger the lot; on the other hand, the higher the product acquisition cost (Caq) and the capital opportunity rate (i), the smaller the lot size.

Table 3 – Optimal Solutions for the Normal and Exponential Distributions in the Product Probability Model

Both the general solution for this model (Chart 2) and the solutions for the Normal and Exponential distributions (Chart 3) can be solved using the optimization algorithm suggested in the lower corner of Chart 3. This is because terms in r and Q appear in both formulas . There is no single optimization algorithm. Each author tends to develop his own and, nowadays, with the MS-Excel spreadsheet, it is very easy to enumerate values ​​for Q and r that simultaneously satisfy both equations. It should be noted that, depending on the combination of parameters, it may not be feasible to reach convergence of results: therefore, it is advisable to always list some possible values ​​of Q and r so that the decision maker acquires sensitivity to the results in terms of costs and level of service.

We present below a small case study to illustrate the main concepts and their operationalization in the probability model of not running out of product.

Mini-Case No 1: A large automobile manufacturer negotiated a Service Level Agreement (SLA) with its main clutch supplier, through which, in case of non-compliance with the delivery of the consumption requested in the JIT resupply, any amount of misconduct that occurs is punishable by a fine of $2.000.000. Considering the following parameters, what should be the optimal batch and ordering point adopted by the supplier in its clutch distribution and assembly operations? Assume that the consumption generated by JIT resupply is symmetrically (Normal) and asymmetrically (Exponential) distributed. Also assume that the standard deviation of demand in response time is equal to the mean of demand in response time.

In Table 4, the optimal solutions for the two probability distributions are presented in terms of Q, r, the probability of missing product (A(r)).

According to Table 4, in the case of the Normal distribution, the optimal solution is for the supplier to operate with a lot size of 30.210 units and an order point of 34.915 units, implying almost 23 occurrences of shortages in 100 resupply, which must be observed that in a typical year 6,62 replenishments (D/Q) are expected. The total cost of this operation (inventory maintenance + resupply + fines) totals $9.024.976,16/year.

It should be noted that the main effect of a skewed distribution, in the probability model of not running out of product, is that lots increase substantially when compared to order points. Oscillations around the mean, when there is an upward or downward bias in demand in the response time, are better accommodated with larger lots (order more) and not with larger order points (order earlier). In the case of the Exponential distribution, which has a downward bias in relation to the average demand in the response time (since A(µ) < 50%), the total costs are lower because of a lower frequency of shipments, due to larger batch sizes ( 5 shipments per year), more than compensates for a higher probability of missing product (40 occurrences out of 100).

LOST SALES MODEL

The lost sales model is shown in Table 5. In the upper left corner, the lost sales on each resupply play an important role in each cycle or sawtooth of a policy (Q,r). Whenever there are lost sales during resupply, total costs will be increased by B(r)*mc, that is, the expected value of lost sales (B(r)) by the mc contribution margin of each product. As in the previous model, it is necessary to derive the total cost equation as a function of the decision variables Q and equate them to zero.

Table 5 - General solution of the lost sales model

Table 6 - Optimal Solutions for the Normal and Exponential Distributions in the Lost Sales Model

A qualitative analysis of the optimal solutions presented in Tables 5 and 6 indicates that:

• The higher the average demand in response time (μ), the annual demand (D) and the contribution margin (mc) of the product, the higher the reorder point (r). On the other hand, lower order points are associated with higher unit acquisition costs (Caq), capital opportunity rates (i) and lot sizes (Q). As in the previous model, the lost sales model also verifies the trade-off between lot sizes and order points.
• The optimal lot size (Q*) is also a correction of the economic purchase lot (LEC) for lost sales (B(r)) over the course of resupply, and its previously described properties are maintained.

We present below another case study with the optimization of Q and solved with the aid of the algorithm presented in the lower corner of Table 6.

Mini-Case No. 2: A large retail chain wants to define the optimal Fill Rate levels for the final consumer for SKUs of a certain non-perishable food line, using information related to contribution margins and acquisition costs from the manufacturer. Considering the following operation parameters, what should be the optimal lot and reorder point adopted by the retailer in its resupply? Assume that consumption is symmetrically (Normal) and asymmetrically (Exponential) distributed. Also assume that the standard deviation of demand in response time is equal to the average of demand in response time.

In Table 7, the solutions for the two probability distributions are presented in terms of Q, r, A(r), B(r) and the approximate Fill Rate.

According to Table 7, in the case of the Normal distribution, the optimal solution is for the retailer to operate with a batch size of 42.909 units and an order point of 41.336 units, implying almost 14 occurrences of shortages in 100 resupply. in a typical year 4,66 replenishments (D/Q) are expected. The total cost of this operation (inventory maintenance + resupply + lost sales) totals $128.489,66, with the expected Fill Rate being 92,88% and the probability of not running out of product in each cycle is 85,7%.

It should be noted that the main effects of an asymmetric distribution with lower demand bias in response time, in the lost sales model, are lower total costs, resulting from lower shortages and slightly lower average inventory levels. There is a virtual tie between the Fill Rate indicators and the probability of missing product. Basically, because roughly speaking, in lost sales models, different probability distributions tend to present similar optimal solutions.

CONCLUSION

The service level can be defined in different ways in the supply chain and therefore it is essential to have a proper understanding of the main ways of measuring product availability indicators. In this article, the main characteristics of two important models were presented and discussed: probability of not running out of products and lost sales. In a future article, the pending model will be developed and discussed.

Among the main conclusions that can be drawn about each of the analyzed models, it is worth highlighting:

• Higher reorder points tend to result in higher demand and shortage costs. On the other hand, higher product acquisition costs, capital opportunity rate, and lot sizes tend to reduce order points.
• There is a clear trade-off between lot sizes and order points. Depending on the type of demand distribution in the response time, it may be more interesting to order earlier (increase the order point) than to order more (increase the lot size). This trade-off is stronger in the probability models of not running out of products than in the lost sales models.
• Ultimately, Optimal Lot Size is an Economic Lot correction for situations where fines or contribution margin losses occur during the resupply cycle. All its properties remain.
• Simple optimization algorithms are needed to determine optimal lot sizes and order points, since terms in Q and r appear simultaneously in both equations. With today's electronic spreadsheets, this is not a serious obstacle to determining optimal solutions.
• As it should be, although qualitatively the trade-offs involved in each formula are the same, depending on the probability distribution of the demand in the response time, the functional relations presented in Tables 4 and 6 vary a lot.

BIBLIOGRAPHY

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SILVER, E.; PYKE, D.; PETERSON, R. Inventory Management and Production Planning and Scheduling. New York: Wiley (2002).

WANKE, P.. Inventory Management in the Supply Chain: Decisions and Quantitative Models. São Paulo: Editora Atlas (2003).

ZIPKIN, P.. Foundations of Inventory Management. New York: McGraw-Hill (2000).