# Monetary Evaluation of Logistical Performance

Tammo Heuer, Vanessa Teuber, Peter Nyhuis

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Tammo Heuer, Vanessa Teuber, Peter Nyhuis. Monetary Evaluation of Logistical Performance. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2022, Gyeongju, South Korea. pp.494-501, 10.1007/978-3-031-16407-1_58 . hal-05386126

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# Monetary Evaluation of Logistical Performance

Tammo Heuer(*), Vanessa Teuber, and Peter Nyhuis

Institute of Production Systems and Logistics (IFA), Leibniz University Hannover,
An der Universität 2, 30823 Garbsen, Germany

heuer@ifa.uni-hannover.de

**Abstract.** Companies can differentiate themselves from competitors through higher logistical performance. It must be valued in monetary terms to realise the advantages of higher logistical performance. According to available information and diverse customer types, different analysis needs to be carried out. In this paper, the unit price potential of a delivery time reduction for customers with stock procurement as an exemplary customer type is modelled for the case of existing customer's process information and analysed based on individual customer-related revenue analysis and thus the logistic operating curves theory. Using storage and procurement models, unit price-delivery time combinations are derived with the same total costs of the customer to deduce the unit price potential of different supplier delivery times. Thus, this model enables supplier to quantify possible unit price increases for lower delivery times for customers with stock procurement.

**Keywords:** Logistics & distribution networks · Logistical modelling · Logistical performance · Delivery time · Monetary evaluation

## 1 Introduction

A company has to stand out from its competitors through competitive advantages to succeed in market competition. In addition to high quality and low prices, logistical performance is one way of achieving this [1]. The logistical performance towards the customer consists of the offered delivery time and delivery reliability [2]. In the following, only the delivery time is considered and the delivery reliability is assumed to be constant since the reliability can be regulated by the safety time [3].

Customers are willing to pay higher prices for shorter delivery times [2]. This paper, therefore, deals with the extent to which unit price increases can be achieved by shortening the delivery time. The topic is first placed in the context of the monetary evaluation of logistical performance by describing existing literature approaches in Sect. 2. Then, in Sect. 3, the influence of delivery time on the willingness of customers with stock procurement to pay a certain unit price is logistically modelled based on individual customer-related revenue analysis. The modelling approach and its application via the isocost curve are analysed in Sect. 4. The paper concludes in Sect. 5 with a summary and an outlook on further research activities.
## 2 Literature Review

When performing unit price analysis for individual customers, a distinction has to be made between customer types. Customers can be divided into two types: consumers and organisational customers [4]. Consumers procure the product for their personal use, so no post-procurement processes need to be considered. The organisational customers procure the products to process them in their own production. For this type of customer, no significant literature analysing the unit price in connection with the delivery time has been published yet.

Organisational customers can be further separated according to their procurement strategy. On the one hand, customers may procure only related to the orders they receive from their customers. In this case, the unit price analysis is complex because, in addition to the influences on the customer's supply chain, the impact on the supply chain of the downstream customer must also be considered. It is necessary because the unit price potential of an enterprise, in this case, also directly includes the unit price potential of the customer. On the other hand, the customer may use stock procurement. This procurement strategy decouples the procurement and the downstream processes. Therefore, a unit price analysis can be carried out detached from further influences of the downstream processes and customers through logistical modelling of the customer's procurement and storing processes.

Schönherr shows four different methods for the monetary evaluation of logistical performance. The choice of the appropriate method depends on the available information. A primary distinction is made between process information and potential information (see Fig. 1) [5].

<table><caption>**Fig. 1.** Classification of methods for the monetary evaluation of logistical performance [5]</caption><thead><tr><th>process information</th><th colspan="4">potential information</th></tr><tr><th></th><th>available</th><th colspan="2">not available</th><th></th></tr><tr><th></th><th></th><th>high effort</th><th colspan="2">low effort</th></tr><tr><th>individual customer-related revenue analysis</th><th>regression analysis</th><th>conjoint analysis</th><th colspan="2">expert interview</th></tr></thead></table>

Potential information is generally accessible information related to a market or a market segment. If this information is available, it can be analysed using regression analysis regarding the monetary value of the logistical performance. In this process, available data are mapped by a mathematical function and the quality of this mapping is evaluated [6]. Data can be collected through surveys or observations if no potential information is available. Conjoint analyses and expert interviews are used in this case.

The conjoint analysis is an individual analysis in which the market's reaction is concluded based on individual respondents. Product variants with different characteristics, such as unit price-delivery time combinations, are presented to rank them. The resulting total value can be used to conclude the partial value of the individual features. [7] In the expert interview, experts must be able to make well-founded statements about the
customers' needs based on experience and knowledge. Expert interviews are always subjective and can only be confirmed with difficulty by models. [8]

If process information is available, the logistical performance can be monetarily evaluated with the help of individual customer-related revenue analysis. As explained in the following paragraph, the customer's supply chain is modelled and thus the customer's cost reduction is quantified in case of a change in the supplier's logistical performance. From the cost reduction, it can be derived by how much the unit price can be increased that the customer has unchanged total costs. The analysis is based on the logistic operating curves theory [9], in which relevant elements of the customer's supply chain are modelled. The impact on the system is investigated through targeted variation of individual parameters. The procedure can be divided into four steps. [5]

In the first step, the primary conditions of the evaluation model are defined. For this purpose, the customer's supply chain is roughly structured and relevant elements are identified. These elements determine the logistic operating curve required for the analysis. In the second step, necessary data for the selected elements, such as the production data for the production operating curve, are selected. Ideally, the customer allows access to all data and missing values can be calculated. If this is not the case, data must be estimated from available industry sector averages or empirical values. The quality of the data directly impacts the validity of the analysis. The logistic operating curves for the current state are then modelled in step three (for further information, see [9]). In the final step, the parameters affected by the change in logistical performance are varied and their corresponding characteristic curves are modelled. By comparing these curve with the current curve, the effect of the change can be identified and the resulting unit price potential determined. [5] With the individual customer-related revenue analysis, precise statements about the unit price potential can be derived if a valid database is available.

# 3 Modelling Monetary Evaluation of Logistical Performance

The basic procedure of individual customer-related revenue analysis explained in Sect. 2 is used as the modelling basis. The first step is to determine which supply chain elements are relevant. Assuming a customer with stock procurement, only the logistical effects on procurement and storage must be modelled. Next, in Sect. 3.1, the influence of the delivery time on the unit price is modelled as a variable to be influenced by the manufacturing company. Following the possible unit price increases are concluded in Sect. 3.2, which can be used to deduct an isocost curve.

## 3.1 Modelling of Procurement Cost Components

The cumulative total costs incurred in procurement of one product within a reference period C are made up of purchasing costs CP, material costs CM and stock costs CS.

$$C = CP + CM + CS \quad (1)$$

In the following, it is assumed that fixed order costs CO are generated for each procurement transaction, which represents the administration and transport costs. The
purchasing costs CP are then the product of the order costs CO and the number of orders in the reference period. The number of orders itself results from the demand in the reference period DMD and the cost-optimal storage input quantity $QIN_{opt}$. [10]

$$CP = \frac{DMD}{QIN_{opt}} \cdot CO \qquad (2)$$

With help of lot-sizing models, cost-optimal storage input quantity $QIN_{opt}$ can be determined as a function of the order costs CO, the demand in the reference period DMD, the unit costs CU and the stock cost rate i. Order lot sizes can be calculated analogously to optimal production lot sizes, according to Andler and Harris [10–12].

$$QIN_{opt} = \sqrt{\frac{2*CO*DMD}{CU*i}} \qquad (3)$$

To calculate the material costs CM, the unit costs CU are multiplied by the demand in the reference period DMD.

$$CM = CU * DMD \qquad (4)$$

The stock costs CS, according to Andler and Harris, depend on the mean stock level $SL_m$, the unit costs CU and the stock cost rate i. The stock cost rate includes the interest for the fixed capital, the maintenance and administration costs of the storage, the risk and venture costs as well as the maintenance costs of the building and technology for storage [11, 12].

$$CS = SL_m*CU*i \qquad (5)$$

Stock operating curves help to position strategically between the mean stock level $SL_m$ and the service level SERL that describes the proportion of demand that can be met by the storage level over time. Input variables are the mean order quantity, which here corresponds to the optimal storage input quantity $QIN_{opt}$, the desired service level SERL, the minimum safety-stock level $SSL_{min}$ and the function parameter $c_L$ of the storage operating curve. The function parameter determines the deviation of the function from the ideal state [9, 13]. $c_L = 0, 31$ is used as a standard value [14].

$$SL_m = \frac{QIN_{opt}}{2} * SERL^2 + SSL_{min} * \sqrt[3]{1 - (1 - SERL)^{c_L}} \qquad (6)$$

The safety-stock level $SSL_{min}$ reflects process uncertainties and results from the maximum positive lateness $L_{max}^+$, the mean demand rate $RD_m$, the maximum demand rate $RD_{max}$ and the replenishment time [9]. It is assumed that delivery quantity deviations can be compensated as part of the following order. From the customer's point of view, the replenishment time is equivalent to the offered delivery time TD [15].

$$SSL_{min} = \sqrt{(L_{max}^+ * RD_m)^2 + ((RD_{max} - RD_m) * TD)^2} \qquad (7)$$
## 3.2 Deriving Enforceable Unit Price Increases

The cumulative total costs of the customer's procurement of one product within a reference period as a function of delivery time and unit price can thus be calculated by inserting Formula 2–7 into Formula 1:

$$ C(TD, CU) = \frac{DMD}{\sqrt{\frac{2*CO*DMD}{CU*i}}} * CO + CU * DMD + \frac{1}{2} * \sqrt{\frac{2*CO*DMD}{CU*i}} * SERL^2 * CU*i + \sqrt{(L_{max}^+ * RD_m)^2 + ((RD_{max} - RD_m) * TD)^2} * \sqrt{1 - (1 - SERL)^{c_L*i}} * CU*i \quad (8) $$

From Formula 8, a unit price CU can be determined that the producing company can enforce in the case of delivery time reductions without the customer experiencing increased total costs. Assuming that the customer's total costs remain the same, the unit price $CU_2$ is calculated as a function of the previous delivery time $TD_1$, the previous unit price $CU_1$ and the new delivery time $TD_2$ examined as follows. The mathematical derivation can be found in the appendix.

$$ CU_2(TD_2, TD_1, CU_1) = \left[ \frac{(SERL^2+1) * \sqrt{\frac{CO*DMD*i}{2}}}{\frac{1}{2} \sqrt{(L_{max}^+ * RD_m)^2 + ((RD_{max} - RD_m) * TD_2)^2 * \sqrt{1 - (1 - SERL)^{c_L*i}}}} + \left( \left( \frac{(SERL^2+1) * \sqrt{\frac{CO*DMD*i}{2}}}{\frac{1}{2} \sqrt{(L_{max}^+ * RD_m)^2 + ((RD_{max} - RD_m) * TD_2)^2 * \sqrt{1 - (1 - SERL)^{c_L*i}}}} \right)^2 + \frac{(SERL^2+1) * \sqrt{\frac{CO*DMD*i}{2}} * \sqrt{CU_1}}{\sqrt{(L_{max}^+ * RD_m)^2 + ((RD_{max} - RD_m) * TD_2)^2 * \sqrt{1 - (1 - SERL)^{c_L*i}}}} + \left( \frac{(\sqrt{(L_{max}^+ * RD_m)^2 + ((RD_{max} - RD_m) * TD_1)^2 * \sqrt{1 - (1 - SERL)^{c_L*i}} * DMD)}{\sqrt{(L_{max}^+ * RD_m)^2 + ((RD_{max} - RD_m) * TD_2)^2 * \sqrt{1 - (1 - SERL)^{c_L*i}} * DMD}} \right)^{\frac{1}{2}} \right] \quad (9) $$

## 4 Analysing the Monetary Evaluation of Logistical Performance

Users are to be supported in selecting attractive unit price-delivery time combinations with the available database and the developed modelling. For this purpose, isocost curves can also be derived from formula 8. For the discrete total cost levels of the customer, they represent the curves of unit price-delivery time combinations between which the customer is indifferent (see Fig. 2). From a unit price-delivery time combination ($CU_1, TD_1$), an isocost curve $ICC_A$ can be derived. This isocost curve can thus be used to easily determine other unit price-delivery time combinations that the customer would accept under the assumptions made since his total costs would remain unchanged. $ICC_B$ is an equivalent isocost curve with a higher total cost basis. In the example chosen here, with a fictitious dataset, a delivery time reduction from 13 to 7 days results in an increase in the unit price of 0.05€ that can be implemented.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Determination of unit price-delivery time combinations using isocost curves</figcaption></figure>

Parameters:

$L_{max}^{+}$ = 5 SCD

$RD_m$ = 10 units/SCD

$RD_{max}$ = 49 units/SCD

$c_L$ = 0,31

$DMD$ = 3.650 units

$SERL$ = 99 %

$i$ = 20 %

$CO$ = 140 €

$C(ICC_A)$ = 38.502 €

$C(ICC_B)$ = 39.904 €

A sensitivity analysis of the isocost curve was used to examine the influences of the parameters on the unit price potential. The ratio of unit price and the delivery time does not change, if there is a parallel shift of the isocost curve, as in the case of a change in the order cost CO or the maximum demand rate $RD_{max}$. An influence on the descent of the isocost curve is the relevant impact in this context, as a change in the relative unit price potential occurs. An increase in the stock cost rate i or the mean demand rate $RD_m$ also marginally increases the descent of the isocost curve, allowing minimal rising unit costs if the reduction in delivery time remains the same. An increase in demand in the reference period DMD leads to a decreasing descent of the isocost curve. The influence of the service level SERL is particularly noticeable. The service level reduction results in an apparent flattening of the isocost curve; at a service level below 35%, a unit price increase can no longer be achieved by reducing the delivery time.

## 5 Conclusion

According to the available information, different approaches can evaluate logistical performance monetarily. If potential information is available, the individual customer-related revenue analysis can be used through logistical modelling to quantify customers' cost savings with stock procurement concerning a reduction in delivery time. Using isocost curves, suppliers can determine which unit price growth would be enforceable at which delivery times for the customer in a cost-neutral manner. The resulting unit price potential depends on the available information, the influence of which was investigated by the sensitivity analysis. In particular, service level SERL has a strong influence on the potential. However, the supplier must consider that they may not be able to negotiate the total unit price growth as assumed in this paper. The cost saving would be split between the supplier and the customer within the negotiation.

Suppose companies can also determine the additional costs for reducing delivery times. In that case, the isocost curves can also be used for revenue-maximising positioning of the delivery time. Therefore, there is a need for research to quantify the costs
arising from delivery time reductions. Different order processing strategies can also be considered. For example, when choosing the order processing strategy, it must be considered that a make-to-stock production or a shift of the customer order decoupling point can also enforce a potential growth of the unit prices.

The modelled influence of delivery time on customers' willingness to pay a certain unit price for a reduced delivery time seems relatively low compared to descriptions in the literature (e.g. [2]). However, the high willingness to pay is due to a differing customer classification. In these cases, the high willingness to pay results from situations in which time-critical demands occur in the event of incorrect planning rejects or short-term customer orders. Since these demands are not decoupled, companies in these situations are forced to pay a high price premium to their suppliers to prioritise and fast-track their orders. Further research is therefore needed into the model-based quantification of unit price potential when the customer's production is not decoupled by stock.

# Appendix

Using the Power Rule for Exponents, Formula 8 can be simplified as follows:

$$C = \left( \sqrt{(L_{max}^{+} + RD_{m})^2 + ((RD_{max} - RD_{m})*TD)^2} * \sqrt[4]{1 - (1 - SERL)^{c_L}} \right) * i + DMD) * CU + (SERL^2 + 1) * \sqrt{\frac{CO*DMD*i}{2}} * \sqrt{CU} \quad (10)$$

With the following substitutions A and B Formula 10 can be modified to Formula 11:

$$C(CU, TD) = A*\sqrt{CU} + B(TD)*CU \quad (11)$$

$$A = (SERL^2 + 1) * \sqrt{\frac{CO*DMD*i}{2}} \quad (12)$$

$$B(TD) = \sqrt{(L_{max}^{+} + RD_{m})^2 + ((RD_{max} - RD_{m})*TD)^2} * \sqrt[4]{1 - (1 - SERL)^{c_L}*i + DMD} \quad (13)$$

Equating $C_1(TD_1, CU_1)$ and $C_2(TD_2, CU_2)$ according to Formula 8 using Formula 11 with the substitutions A and B results in:

$$A*\sqrt{CU_1} + B(TD_1)*CU_1 = A*\sqrt{CU_2} + B(TD_2)*CU_2 \quad (14)$$

Zeroing Formula 14 in Formula 15 offers the basis for the Quadratic Formula usage that results in Formula 16. The subtraction in Formula 16 can be mathematically excluded. Therefore, Formula 17 quantifies $CU_2$ as follows:

$$CU_2 + \frac{A}{B(TD_2)} * \sqrt{CU_2} - \frac{A * \sqrt{CU_1} + B(TD_1) * CU_1}{B(TD_2)} = 0 \quad (15)$$

$$\sqrt{CU_2} = \frac{-1}{2} \frac{A}{B(TD_2)} \mp \sqrt{\left(\frac{1}{2} \frac{A}{B(TD_2)}\right)^2 + \frac{A * \sqrt{CU_1} + B(TD_1) * CU_1}{B(TD_2)}} \quad (16)$$
$$CU_2(TD_2, TD_1, CU_1) = \left[ -\frac{1}{2} \frac{A}{B(TD_2)} + \sqrt{ \left( \frac{1}{2} \frac{A}{B(TD_2)} \right)^2 + \frac{A*\sqrt{CU_1} + B(TD_1)*CU_1}{B(TD_2)} } \right]^2 \quad (17)$$

By substituting A and B, $CU_2(TD_2, TD_1, CU_1)$ follows according to Formula 9.

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