In this section, the optimization model is described briefly in order to put the analysis in context. Readers interested in an in-depth description of the model are referred to the source publications of the development of the dynamic programming optimization algorithm.
Stationary Single-Stage Inventory Model
The demand process, the key equations of the safety stock model, and the optimization problem as described in the seminal publication are described next.
Demand Process. External demand is observed at demand stages and Is based on a stationary process with a daily average { H: The demand of the internal stages, which are all stages that are not demand stages, is the sum of the demand from each of the next-tier customers multiplied by the value of the corresponding
Orc.
The model does noi require any assumption about normality of demand. it is assumed though that managers are able to produce a meaningful upper bound of demand which will be used for inventory planning. One way to produce the vpper bound of demand is jo assume that demand is normally distributed and use the average demand, stanaara deviation of demand, and safety factor to praduce the upper bound as follows.
Dt) = tj, + kovr (1)
Where
D(t) Upper bound of demand over the days of exposure at stage |
J
1 Days of exposure k Safety factor
Ơ Standard deviation of demand
The number of days of exposure (t) depends on the stage’s processing time; the inbound service time, which is the maximum time it takes for the stage to get replenished from all its suppliers; and the quoted service time, which is the time promised to the next-tier customers. Specifically,
c=§| +T=§ (2)
Where
1 Days of exposure Sl Inbound service time
I Stage’s processing time
S, Service time quoted to next-tier customers
Safety Stock Model. The expected inventory level at a stage at the end of each period is the expected safety stock for the stage. At the end of each period, it is expected that the stage used all cycle stock. Thus, the expected inventory level at stage j at the end of time Tf Is:
E[l] =D(Sl+T-S)-(Sl+T-S)p | | | | | | | | |
=Dfrl-t L3)
The expected inventory at stage j at the end of period tf is equal to the Upper bound of demand minus the average demand over the exposure time.
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Optimization Problem. The optimization problem is designed to find the optimal service times; that is, the quoted service time to the next-tier customers.
P= min Yh,[D, (SI, +7,—S,)-(5%I,+T,— S,),]
subject to:
S-SIsT, forjJ= 1,2,...
SI, - S20 (for all (i, j) <A S, < s,for all demand nodes |
S, Sl, 20 and integer for} = 1, 2,... N Where:
D(t) Upper bound of demand over the exposure time h Annual holding cost rate
Sl Inbound service time I Stage’s processing time
S, Service time quoted to next-tier customers m Expected average demand
S, Maximum service time quoted to the next-tier customers Thus, the optimization problem is to minimize the summation of the holding cost percentage multiplied by the inventory level at all stages in the supply chain.
Mechanics of the Optimization Algorithm
The mechanics of the optimization algorithm are based on dynamic programming. Dynamic programming is used to solve sequential decision problems [3]. A dynamic program starts by solving the problem for the last stage and works backwards solving the problem for previous stages one ata time. In the context of the supply chain modeled here, the optimization algorithm determines the stock level for the demand generation stages (restaurants) and works backward to the warehousing stages (distribution centers of distributors and manufacturers) and on to the manufacturing and procurement stages (manufacturers).
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In a traditional setting, inventory levels are set based on the information available to each stage. This information is the stage’s processing time (stage time}, the time it takes for the stage to get repienisned from iis suppliers (the inbound service time}, and the time the stage has promised the nexi-tier customers if will take to have the product available to the next stage (the quoted service time}.
The stage time plus the inbound service time minus the quoted service time equals the time of exposure as shown in equation 2 {since in this research the unit of time was measured in days, itis called days of exposure hereafter).
Tables 3.2 and 3.3 shows an example for optimizing a supply chain formed by seven stages: procurement, manufacturing, interplant transportation, warehousing {at ihe manufacturer’s regional distribution center}, transportation to the distrioution center, and demand (at a restaurant}. Each stage is connecied to the stage to the right as shown in the tables. Table 3.2 shows the days optimal of exposure based on only the local information that was available to the stages.
Since each stage focuses on the next-tier customer, the quoted service time for al stages was zero in order fo mode! the fact that each stage promises immediate product availability. This forces alistages to hold safety stock. Allstages have days of exposure equal fo its stage time and all stages provide immediate product availability to the nexi-tier customer.
Table 3.3 snows the output of the optimization, which is the determination of the locations of safety siocks based on minimizing the holding costs. The optimization algorithm calculates the cost of possible solutions {safety stock levels and locations}
and outputs the least cost option. These calculations includes, for example, all risk pooling opportunities. Table 3.3 shows that safety stocks are located at the distrioution centers to provide product availability to the stores, and the furthest back in the supply chain as possible. Note that this solution is possible because some stages, those that do not hold safety stocks, quoted service times to the next-tier customers of at least one day. The stages that had the inbound service time higher than zero means that those stages will wait to receive their orders.
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These stages can wait because they hold enough safety stock to cope with uncertainty of their demand. itis also assumes that the management of those stages are willing to wait for their orders.
Since demand is considered independent across time, the algorithm considers the effect of pooling demand variability across time. Safety stocks are determined using equation {1} which includes the square root of the days of exposure at each stage. This square root represents the pooling of variance. Since the square root of the sum is smailer than the sum of the square roots, the fotal safety stock requirement across the supply chain is smaller with the solution shown in Table 3.3.