The dynamics of material flows in supply chains Dr Stephen Disney Logistics Systems Dynamics Group Cardiff Business School

The bullwhip effect in supply chains Methodological approaches to solving the bullwhip problem Supply chain strategies for taming the bullwhip effect The golden replenishment rule Solutions to the bullwhip problem Implementing a smoothing rule in Tesco Economics of the bullwhip effect Square root law for bullwhip The future of bullwhip

The bullwhip effect in supply chains

Measures of the bullwhip effect Stochastic measures Deterministic measures

The bullwhip effect is important because it causes Up to 30% of costs are due to the bullwhip effect! Poor customer service due to unavailable products Runaway transportation and warehousing costs Excessive labour and learning costs Unstable production schedules Insufficient or excessive capacities Increased lead-times

How the bullwhip effect creates unnecessary costs Demand Variance Overtime / Agency work / Subcontracting + Stock-outs + Costs Lead-time Obsolescence + Capacity + Utilisation - - Stock

Methodological approaches to solving the bullwhip problem

Representations of time Discrete Time Inventory positions are assessed and orders are placed at discrete moments in time -At the end of every day, or the end of every week, for example -May be suitable of the way a supermarket operates, or a distribution company -In between the discrete moments of time nothing is known about the system Inventory positions are assessed and order rates are adjusted at all moments of time -May be suitable for a petrol refinery or in a chemical plant -The system states are known at every moment of time v Continuous Time

Continuous time approaches Lambert W functions Johann Heinrich Lambert 1728 – 1777 Leonhard Euler Laplace transforms Pierre-Simon Laplace Differential equations Aleksandr Mikhailovich Lyapunov

Discrete time approaches Stochastic processes / ARIMA George Box

The ARMA(1,1) demand process for 16 P&G products in their Homecare range

Discrete time approaches State space methods Joseph Leo Doob Martingales Rudolfl Kalman Stochastic processes / ARIMA George Box z-transforms Yakov Zalmanovitch Tsypkin

Table of transforms and their properties

Other useful approaches Jean Baptiste Joseph Fourier ( ) Fourier transforms System dynamics / simulation Jay Forrester (1918-) The beer game John Sterman

Supply chain strategies for taming the bullwhip effect

Traditional supply chains Definition: ‘Traditional’ means that each level in the supply chain issues production orders and replenishes stock without considering the situation at either up- or downstream tiers of the supply chain. This is how most supply chains still operate; no formal collaboration between the retailer and supplier. Bullwhip increases geometrically in a traditional supply chain

Supply chains with information sharing Definition: Information exchange (or information sharing) means that retailer and supplier still order independently, yet exchange demand information in order to align their replenishment orders and forecasts for capacity and long-term planning. Bullwhip increases linearly in supply chains with information sharing

Synchronised Supply (VMI) Definition: Synchronized supply eliminates one decision point and merges the replenishment decision with the production and materials planning of the supplier. Here, the supplier takes charge of the customer’s inventory replenishment on the operational level, and uses this visibility in planning his own supply operations. Bullwhip may not increase at all in VMI supply chains

Integrating internal and external decision in supply chains with long lead-times RFID technologies now allow us to monitor the distribution leg

Solutions to the bullwhip problem

Replenishment rules and the bullwhip problem Replenishment decisions influence both inventory levels & production rates. A common replenishment decision is the “Order-Up-To” (OUT) policy…. Set via the newsboy approach to achieve the critical fractile Forecasts

Generating forecasts inside the OUT policy Exponential smoothing Moving average Conditional expectation We will assume normally distributed i.i.d. demand & exponential smoothing forecasting from now on

The inventory and WIP balance equations The replenishment lead-time, T p

The influence of the replenishment policy The inventory balance equation…. ….shows us that the replenishment policy influences both the orders and the net stock. Therefore, when studying bullwhip we should also consider

The impact of forecasting on net stock variance amplification As then NSAmp approaches 1+T p. Minimising the Mean Squared Error between the forecast of demand over the lead-time and review period and its realisation will result in the minimum inventory variance. This holds in a single echelon (Vassian 1954) and across a complete supply chain (Hosoda and Disney, 2006) when the traditional OUT policy is used

The impact of forecasting on bullwhip As then bullwhip approaches unity. Thus, we can see that as we make more accurate forecasts the bullwhip problem is reduced (but is not eliminated in this scenario)

Reducing lead-times Reducing lead-times usually (but not always) reduces bullwhip However, reducing lead-times will always reduce the inventory variance

The OUT policy through the eyes of a control engineer… Unity feedback gains! A control engineer would not be at all surprised that the OUT policy generates bullwhip as there are unit gains in the two feedback loops Let’s add in a couple of proportional feedback controllers…. Inventory feedback gain ( T i ) WIP feedback gain ( T w )

The first proportional controller: The Maxwell Governor

The golden replenishment rule

Matched feedback controllers When T w =T i the maths becomes very much simpler With MMSE forecasting ( ) we have…

The golden ratio in supply chains For i.i.d. demand, matched feedback controllers, MMSE forecasting

The golden ratio …

Economics of the bullwhip effect

Economics of inventory Inventory costs are governed by the safety stock (TNS) The Target Net Stock (TNS*) is an investment decision to be optimized In each period, when the inventory is positive a holding cost is incurred of £H per unit. In each period, if a backlog occurs (inventory is negative), a backlog cost of £B per unit is incurred

The economics of capacity Capacity per period = Average demand +/- slack capacity Production above capacity results in some over-time working (or sub-contracting). The cost of this type of capacity is £P per unit of over-time. Production below capacity results in some lost capacity cost of £N per unit lost. The amount of slack capacity (S*) is an investment decision to be optimized

Costs are a linear function of the standard deviation Setting the amount of safety stock we need via the newsboy… … and the amount of capacity to invest in… …for a given set of costs ( H, B, N, P ) and lead-time, ( T p ) Inventory costs Bullwhip costs Constants Total costs are thus linearly related to the standard deviations

Sample designs for the 4 different scenarios Assuming the costs are; Holding cost, H=£1, Backlog cost, B=£9 Lost capacity cost, N=£4, Over-time cost, P=£6

Square root law for bullwhip

… One manufacturer Distribution Network Design: Bullwhip costs 12 customers n DC’s Each customer produces an i.i.d. demand, normally distributed with a mean of 5 and unit variance All lead-times in the system are one period long

Number of DC’s (n) Demand process faced by each DC Factory demand …and it all depends on how many distribution centres we have… Each customer’s demand = N(5,1) DC demand= Factory demand=

Number of DC’s (n) Demand process faced by each DC Factory demand … for 2 DC’s… Each customer’s demand = N(5,1) DC demand= Factory demand= DC demand=

Number of DC’s (n) Demand process faced by each DC Factory demand … for 3 DC’s… Each customer’s demand = N(5,1) Each DC’s demand= Factory demand=

Number of DC’s (n) Demand process faced by each DC Factory demand … for 4 DC’s… Each customer’s demand = N(5,1) Each DC’s demand= Factory demand=

Number of DC’s, n Inventory cost £8.59£12.15£14.89£17.20£21.06£29.78 £8.59 £8.60 The Square Root Law Number of DC’s, n Capacity cost £13.38£18.93£23.18£26.77£32.78£46.36 £13.38£13.39£13.38£13.39£13.38 “If the inventories of a single product (or stock keeping unit) are originally maintained at a number (n) of field locations (refereed to as the decentralised system) but are then consolidated into one central inventory then the ratio exists”, Maister, (1976).

Proof of “the Square Root Law for bullwhip” The bullwhip (capacity) costs are given by In the decentralised supply chain the standard deviation of the orders is, In the centralised supply chain the standard deviation of the orders is Thus, which is the “Square Root Law for Bullwhip”.

Implementing a smoothing rule in Tesco

Tesco project brief Tesco’s store replenishment algorithms were generating a variable workload on the physical delivery process –this generated unnecessary costs The purpose of the project was to; –investigate the store replenishment rules to evaluate their dynamic performance –to identify if they generated bullwhip –offer solutions to any bullwhip problems

Inventory replenishment approaches

High volume products Account for 65% of sales volume and 35% of product lines Deliveries occur up to 3 times a day

The simulation approach

Weekly workload profile: Before and after Peak weekly workload amplified by existing system Peak weekly workload smoothed by modified system

Summary Tesco’s replenishment system was found to increase the daily variability of workload by 185% in the distribution centres A small change to the replenishment algorithms was recommended that smoothed daily variability to approximately 75% of the sales variability The solution was applied to 3 of the 7 order calculations. This accounted for 65% of the total sales value of Tesco UK This created a stable working environment in the distribution system. The Tesco case study will be discussed in more detail this afternoon in the President’s Medal presentation

The future of bullwhip

Multiple products with interacting demand Demand for product 2 at time t Demand for product 1 at time t Random processes Auto-regressive process with itself Auto-regressive interaction with the other product

The Inventory Routing and Joint Replenishment Problem In a multiple product or multiple customer scenario Place an order to bring inventory up-to S, –if inventory is below a reorder point –OR if inventory is below a “can-deliver” level AND another product (or retailer) has reached its reorder point Consolidation of orders/ deliveries can generate significant savings

The interaction between bullwhip inventory variance & lead-times Manufacturer Consumer Demand Production Lead time Retailer orders Replenishment orders Retailer uses an OUT policy and places orders onto the manufacturer Manufacturer is represented by a queuing model. Operates on a make to order principle Processes orders on a first come first served basis If the retailer smoothes his orders (with a proportional controller) then the manufacturer can replenish the retailers orders quicker. Thus there is an interaction effect between bullwhip and lead- times that allows supply chains to break the inventory / order variance trade-off!

Multi-echelon supply chain policies

The impact of errors Demand parameter mis-identification Demand model mis-identification Lead-time mis-identification Information delays Random errors in information Non-linear, time-varying systems …

Thank you The dynamics of material flows in supply chains Dr Stephen Disney Logistics Systems Dynamics Group Cardiff Business School

The IOBPCS family

Stability issues (Tp=1)

Stability issues (Tp=2)