A Simple Heuristic for Dynamic Order Sizing and Supplier Selection with Time-Varying Data Horst Tempelmeier Universit¨at zu K¨oln Seminar f¨ur Produktionswirtschaft Albertus-Magnus-Platz D-50923 K¨oln http://www.uni-koeln.de/wiso-fak/spw/
[email protected]
Working Paper Cologne, June 2000
Revised, December 2001 Abstract In this paper we consider the problem of supplier selection and purchase order sizing for a single item under dynamic demand conditions. Suppliers offer all-units and/or incremental quantity discounts which may vary over time. Although the problem refers to a typical planning task of a purchasing agent, which is often solved without algorithmic assistance, in an eBusiness (B2B) environment the need for the integration of an automatic performance of this planning task into a WEB-based procurement process becomes evident. A new model formulation for this problem is developed and a simple but easily extendible heuristic procedure is presented and tested. The heuristic is implemented as part of the Advanced Planner and Optimizer (APO) software of SAP AG, Walldorf, Germany.
2000, 2001 Prof. Dr. Horst Tempelmeier
1
1 Introduction The WEB-based automation of procurement processes for maintenance, repair, and operations (MRO) products aims at the realization of significant cost-savings through the streamlining of the traditionally labor-intensive and time-consuming tasks of procurement. This is particularly obvious for products with high volume and low value. However, the potential for cost-reduction will only be realized if the complete business process of purchasing can be automated, including the relevant operational purchasing decisions. Software vendors are now increasing their efforts for supporting Business-to-Business (B2B) procurement scenarios including planning tools either for automated application or as decision support for the purchasing agent. In a typical industrial purchasing environment the purchasing agent has the task of supplier selection and order sizing given several suppliers that often offer their customers lower unit prices on orders for larger quantities. In addition, base prices may vary over time due to marketing actions. In this paper we present a planning model for supporting short-term supplier selection and order sizing under dynamic demand conditions. We develop a heuristic solution procedure, named Iterative Supplier Selection and Order Sizing Procedure (ISSOS), that can be applied as a planning component in an automated procurement process.
2 Previous Work There is an increasing amount of literature that addresses purchasing problems, including supplier selection, within a long-term framework of supply-chain management. See e. g. Anupindi and Bassok (1999), Chaudry et al. (1993), Jayaraman et al. (1999), Munson and Rosenblatt (1997), Munson and Rosenblatt (2001), Tsay et al. (1999), Zeng (1998), and the references given therein. In the short-term planning environment focussing on the operational problems 2
of order sizing and supplier selection that we consider, however, it is assumed that the base of suppliers and the purchasing conditions, including discount structures, are already in place. Although research on production lot sizing and purchase order sizing dates back to the beginning of the 20th century and a large number of research papers have been written, the literature on order sizing under consideration of quantity discounts comprises relatively few papers. Among these, many papers focus on order sizing under stationary demand conditions, and therefore are not adequate for an operative MRP-like planning environment, which is dominated by dynamic conditions. Recent overviews over the literature related to the order sizing decision are provided by Benton and Park (1996) and Munson and Rosenblatt (1998). Relevant work that focusses on dynamic demand conditions includes Benton (1985, 1986), Benton and Whybark (1982), Bregman (1991), Bregman and Silver (1993), Callarman and Whybark (1981), Christoph and LaForge (1989), Chung, Chiang, and Lu (1987), Chung, Hum, and Kirca (1996), Chung, Hum, and Kirca (2000), Chyr, Huang, and de Lai (1999), LaForge and Patterson (1985), and Tersine and Toelle (1985). A paper by Prentis and Khumawala (1989) seems to be the only one where quantity discounts in the context of a multi-level MRP-type product structure are considered. Most of the work that has been done so far concentrates on isolated aspects of the complete problem. Reducing the complexity through setting several assumptions obviously increases the probability that a specialized (exact or heuristic) solution algorithm can be developed to solve the resulting model. For example, if only incremental quantity discounts are considered, it is possible to construct a shortest-path network where the cost associated to an arc are computed based on the cheapest supplier available in that period and for that given order quantity. See Zipkin (2000). However, this approach is not applicable if all-units discount structures are in
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effect, which is the case in the majority of practical situations. See Munson and Rosenblatt (1998). In addition, with increasing experience through the application of a decision support tool the planner’s aspiration level with respect to the problem characteristics that should be included in a solution procedure will also increase. This is an important point that must be kept in mind by an Advanced Planning Software vendor who intends to provide a solution procedure with wide-spread applicability to his or her customers. As a starting point, the most important problem aspects that we think, after several discussions with industrial purchasing agents, should be included in an automated purchasing decision support tool are: dynamic demands, several competitive suppliers, multiple price breaks, changing prices and costs over time, and supplier-specific delivery schedules.
3 A New Model Formulation In what follows we present a new model formulation for the problem of simultaneous supplier selection and order sizing. In particular we consider a single and independent item subject to the following assumptions: 1. Dynamic deterministic demands 2. Zero inventory at the beginning and the end of the planning horizon 3. Stockouts not permitted 4. Several suppliers, each one with (a) Time-varying deterministic prices dictated by time-varying (all-units or incremental) quantity discount structures with several discount levels. This includes rising or 4
falling prices starting with a specific period as well as special prices for limited time intervals. (b) Supplier-specific fixed ordering costs (transportation etc.) (c) Supplier-specific delivery periods; i. e. although the demand periods are days, a supplier may follow a delivery schedule including delivery only on, say, Mondays and Thursdays. (d) Supplier-specific delivery lead times (e) Supplier-specific minimum and/or maximum order sizes 5. Holding costs depending on the purchasing price 6. No capacity constraints These problem characteristics are relevant in many industrial purchasing environments. Depending on the type of discount structure (all-units or incremental), two versions of a planning model are formulated, with the following symbols used: Parameters: L Rlτ T
Number of discount levels for supplier l in period τ
alτ
dt glr τ plr τ h hlr τt
Net requirements for the item in period t
slτ
Number of suppliers
Number of demand periods 1 0
if supplier l can deliver in period τ else
Upper limit of discount level r for supplier l in period τ; g l0 τ 0 Unit price in discount level r for supplier l in period τ holding cost percentage per period Inventory holding costs for the complete demand of period t, if it is delivered by lr supplier l in period τ with discount level r ; h lr τ τt h⋅ pτ ⋅ dt ⋅ t fixed ordering costs for supplier l in period τ
5
Decision variables: δlr Proportion of demand in period t, that is delivered by supplier l in period τ with τt γτlr qlr τ
discount level r Binary variable for selecting discount level r in period τ for supplier l Quantity purchased from supplier l in period τ with discount level r
For the case of all-units discounts, the following mixed-integer linear optimization model (Uncapacitated Multi-Supplier Order Quantity Problem with Time-Varying All-units Discounts) can be formulated. Model UMSOQPVAD
Rlτ
lr Minimize Z ∑ ∑ ∑ ∑ hlr τt ⋅ δτt T
T
L
τ1 t τ l 1 r1
∑ T
Rlτ
lr ∑ ∑ slτ ⋅ γτlr qlr τ ⋅ pτ L
(1)
τ1 l 1 r1
s. t. Demand fulfillment: Rlτ
∑ ∑ ∑ δlr τt 1 t
L
t 1 2 T
τ1 l 1 r1
(2)
Definition of order sizes: T
∑ δlr τt ⋅ dt
t τ
qlrτ
l 1 2 L; τ 1 2 T ; r 1 2 Rlτ
(3)
l 1 2 L; τ 1 2 T ; r 1 2 Rlτ
(4)
l 1 2 L; τ 1 2 T ; r 2 3 Rlτ
(5)
Upper limit of a discount level: lr lr qlr τ ≤ g τ ⋅ γτ
Lower limit of a discount level:
l r 1
qlr τ ≥ gτ
1
⋅ γτlr
6
At most one active discount level per delivery period: Rlτ
∑ γτlr ≤ alτ
r1
l 1 2 L; τ 1 2 T
(6)
Define fixed ordering cost and discount level for the selected delivery period: l 1 2 L; τ 1 2 T ; t τ τ 1 T ; r 1 2 Rlτ; dt 0
(7)
qlr τ ≥0
l 1 2 L; τ 1 2 T ; r 1 2 Rlτ
(8)
γτlr ∈ 0 1
l 1 2 L; τ 1 2 T ; r 1 2 Rlτ
(9)
lr δlr τt ≤ γτ
δlr τt ≥ 0
τ 1 2 T ; t τ τ 1 T ; l 1 2 L; r 1 2 Rlτ
(10)
The model formulation is based on the well-known analogy between the plant location problem and the dynamic lot sizing problem (Krarup and Bilde, 1977). Supplier-specific lead-times are accounted for through the setting of the a lτ -values. If supplier l has lead time zl (based on the planning time 0), then we have a lτ 0, τ zl . Differences with respect to lead-times between suppliers will have an impact on the solution only in the first few periods of the planning horizon. For the case of incremental discounts, the following non-linear mixed-integer optimization model (Uncapacitated Multi-Supplier Order Quantity Problem with Time-Varying Incremental Discounts) is formulated.
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Model UMSOQPVID
Rlτ
Minimize Z ∑ ∑ ∑ ∑ h⋅ T
L
T
τ1 t τ l 1 r1
∑ T
L
Rlτ
∑ ∑
τ1 l 1 r1
s. t. l r lr vlr τ qτ gτ
1
lr
lr fτlr vlr τ ⋅ pτ ⋅ d ⋅ t τ ⋅ δτt t qlr τ
hlr τt
slτ fτlr ⋅ γτlr
plrτ ⋅ vlrτ
(11)
l 1 2 L; τ 1 2 T ; r 1 2 Rlτ
⋅ γτlr
(12)
(3) – (10) The variable vlr τ , which is used for correct inventory valuation, denotes the difference between the order quantity in period τ from supplier l and the base quantity of the associated discount level r. The coefficient fτlr denotes the variable purchasing costs associated with the portion of the order quantity up to the lower limit of the current discount level r, which is relevant for the current order size. For the case of incremental discounts this term evaluates to r 1
fτlr ∑ gliτ glτi i1
1
⋅ pliτ
(13)
Note that in the case of all-units discounts we would have f τlr plr τ ⋅ gτ
l r 1
for the remaining quantity vlr τ
qlrτ gτl r
1
. By adding the cost
we get the complete variable ordering costs for
ordering the quantity qlr τ . As far as the variable ordering costs are concerned, the difference between all-units and incremental discount structures can thus be reflected through the simple preparation of the input data. Table 1 shows the differences between the f τlr -values with all-units and incremental discount schemes for different values of the order size q.
8
r 0 1 2 3
glr τ 0 100 250 ∞
plr τ – 4.1 3.72 3.55
all-units – fτl1 q 50 0 fτl2 q 200 410 fτl3 q 400 8875
incremental – fτl1 q 50 0 fτl2 q 200 410 fτl3 q 400 410 558 968
Table 1: Comparison of all-units and incremental discount schemes
Through a combination of the models UMSOQP VID and UMSOQPVAD , situations where some suppliers offer all-units discounts and other suppliers offer incremental discounts may be captured. Unfortunately, as the holding cost coefficient h lr τt in the objective function (11) depends on the order size and the associated mean purchasing price for all units included in an order, for incremental discounts the objective function is non-linear. However, if only incremental discounts are in effect, then the above-mentioned shortest-path approach can be used to solve the problem exactly. The heuristic to be proposed is not aware of the type of discount structure, as it works with the f τlr vlr τ -data. Maximum and minimum order sizes – which may be specific to certain periods – and even forbidden ranges of order sizes can easily be accounted for with the help of dummy discount levels with associated prohibitively high prices. Note that the maximum order size for a supplier in a certain period can also be used to model a supplier-specific capacity constraint. It is well-known that – opposite to the undiscounted lotsizing problem – under the condition of all-units quantity discounts in the optimal solution a single demand may be fullfilled by two different orders with different prices. In the presence of multiple suppliers, a period demand may even be fullfilled by different suppliers. Therefore inventory may include units which have been bought at non-identical prices. As in the above formulation inventory is implicitly evaluated through the coefficients hlr τt denoting the holding costs for the complete demand of period t, if it is delivered by supplier l in period τ with discount level r, the inventory value can 9
be tracked back for each individual unit of inventory – even if the total inventory in a period comes from different orders. Therefore the above model formulation describes the holding costs exactly. This would not be possible in a standard MIP formulation of the order sizing problem. See Prentis and Khumawala (1989), who consider quantity discounts and holding costs that do not reflect the purchase price, an assumption that may not be reasonable from an economic point of view. Apart from the correct inventory evaluation the above model has the nice characteristic that it is much tighter than the standard ”Big-M”-based model formulation. In several computational tests we found that the solution time was dramatically reduced by an order of magnitude.
4 A New Heuristic In a situation where only a single problem instance of model UMSOQP VAD has to be solved, a standard MIP-solver may be the right tool. In our tests we used CPLEX 6.5 running on a Pentium-based Windows NT workstation with 200Mhz clock speed. Depending on the problem data, optimum solution times for problems with 40 periods, 5 discount levels and 3 suppliers ranged between a few dozen seconds and several hours. For a routine application of the model with many items in an automated B2B process, however, fast solution methods are required. The heuristic solution procedure that is presented in the sequel was developed according to several requirements dictated by the potential user: 1. It must consider all aspects included in model UMSOQP VAD,VID (i. e. solve the complete problem). 2. It must be fast. 3. It must be easily understandable by the user. 10
4. It must be easily extendible with respect to additional constraints without a major redesign of the complete solution procedure. With these requirements in mind, we propose a heuristic solution method that consists of two main phases. In phase one (construction phase), an initial solution is constructed. In phase two (improvement phase) local search procedures are iteratively applied to improve the solution.
4.1 Phase I: Construction Phase The construction phase is the adaption of the well-known LUC-rule to the current problem with special consideration of several alternative suppliers and quantity discounts. In this phase, the possibility of time-varying price schedules is not explicitly accounted for. This latter aspect of the problem is considered in the sequent improvement phase of the heuristic. An overview over phase one is given in Figure 1.
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While the current order period τ is less than the planning horizon T : Iteration τ: Construct an order for order period τ Iteration t t τ 1 τ 2 Compute unit costs for each supplier and all integer coverages between τ 1 and T : UCint l t . Stop increasing the coverage t, if the lowest price level has been reached and the unit costs increase again. Compute unit costs for each supplier and all fractional coverages UC f racl t associated with the lower limits of the discount levels provided by the suppliers. Select the order size/supplier combination with the minimum unit costs out of all UCint l t and UC f rac l t computed. Consider only coverages t, where at least one supplier is able to deliver in period t 1. Set τ t 1 and perform another iteration τ. Figure 1: Construction procedure
4.2 Phase II: Improvement Phase Typically the initial solution (order size/supplier combination) may be improved with respect to several aspects. An obvious improvement potential is located in the fact that the prices (discount structures) may change over time, which has been neglected in the construction phase of the heuristic. Therefore, under conditions of changing prices – the first improvement step attempts to shift orders into time periods with low prices. The subsequent improvement steps examine deleting orders and shifting orders between suppliers to search for further cost reduction.
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I MPROVEMENT
STEP
A: P RICE
CHANGES
When the price structure changes in period t leading to a shift of prices upwards or downwards, then a straightforward local improvement may be achieved through the shifting of order quantities to the ”nearest” point in time, when the lower price applies. If a price decrease happens in period t (period t must belong to the delivery schedule of the supplier considered), then it may be favourable to postpone part or all of the orders scheduled immediately before the price decrease. In this case, we try to find an order in period tv t (i. e. the order nearest to the price decrease). If such an order exists, the maximum number of units that may be postponed from period t v to period t is determined and the shift is evaluated. If a cost decrease is realized, the new solution is saved and the next iteration is performed. If an order is scheduled for supplier l in period tn following a price increase in period t, we try to shift this order from period t n backwards to period t 1. This testing is performed backwards from period T to period 1 with an inner loop including all suppliers before proceeding to the previous period. An upper bound for the quantity that may be shifted backwards is given by the on-hand inventory at the beginning of period t n. Further details of all improvement steps are presented in the appendix.
I MPROVEMENT
STEP
B: S PLITTING
OF ORDERS
Even with constant prices it may be possible to improve the quality of a order/supplier combination through local changes. Therefore the second type of improvement step is to evaluate the splitting and possibly deletion of orders. In particular, consider order q lτ . If this order could be completely deleted a reduction of the fixed ordering costs by the amount of s would occur. Order qlτ is now split into two parts. The first portion of this order is shifted backwards and
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added to the latest order scheduled before period τ. The remaining quantity is shifted into the future, i. e. the next possible ordering period for the same supplier l. In order to keep the increase of inventory costs through the backwards shifting of part of the order as small as possible, we consider as a candidate for increasing the order placed in the period tv nearest to period τ. This may be an order from the same supplier as order q lτ or from a different supplier [see (14)] tv max j qij 0; j 1 2 τ 1; i 1 2 L
(14)
In addition, we only shift backwards the minimum amount required to prevent backorders between period τ and the future period tn , where tn is the next available ordering period for the same supplier l. The second portion of q lτ is shifted forwards into the period t n . This improvement step is performed order by order working forward from the earliest to the latest order as long as cost reductions are achieved. For the last period of the planning horizon only a shifting backwards is performed.
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I MPROVEMENT
STEP
C: C OMBINATION
OF COMPLETE ORDERS
A further improvement step tries to combine complete orders. This may be favorable in cases with small orders, low holding costs and relatively high fixed ordering costs. This improvement step considers all orders in the sequence of the order period and tries to combine each order with its immediate successor.
I MPROVEMENT
STEP
D: P OSTPONEMENT
OF PARTIAL ORDER QUANTITIES
In time series with high variability it may happen that in several consecutive periods there are low demands followed by a very large demand in one of the next periods (see demand series #10 in Table 3). If the large demand quantity is included in an order in the low demand interval, it may be favorable to shift part of that order into the future with one additional order leading to an increase of fixed ordering costs but with a large reduction of inventory costs. This improvement step considers all possible shifts of (partial) orders into the future up to the period when the next order is placed.
OVERALL P ROCEDURE The improvement steps are integrated in an iterative procedure. The heuristic stops when a solution that cannot be further improved has been found. The complete structure of the proposed heuristic is shown in Figure 2. Through several exploratory computational tests we found that the sequence of improvements as given in Figure 2 performed best.
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Compute an order plan with the Construction heuristic PlanChanged := TRUE
WHILE PlanChanged = TRUE
Dynamic prices or costs present?
Improvement step A1: Consider price/cost decreases Improvement step A2: Consider price/cost increases Improvement step B: Splitting of orders Improvement step C: Combination of order Improvement step D: Postponement of partial orders
Has the order plan changed? PlanChanged := TRUE
PlanChanged := FALSE
Figure 2: Complete structure of the heuristic lr The heuristic works with the discount data given in the form f τlr vlr τ ⋅ pτ [see equation (11)],
which is valid for all-units as well as for incremental discounts. This has the slight disadvantage that special properties of incremental discount structures, e. g. no fractional coverages of orders, are not reflected in phase I of the heuristic, leading to a small increase of computational effort. The great advantage, however, is that the heuristic can also handle cases where one supplier offers all-units discounts and the others use incremental discounts. It is even possible that a
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supplier changes from all-units to incremental discounts and vice versa within the planning horizon.
5 Numerical Results In order to evaluate the performance of the heuristic, a numerical experiment was set up. The discount structures were taken from industrial practice. Three problem groups (items A, B and
40 periods and L 3 suppliers and one problem group (item D) with T 20 periods and L 2 suppliers were considered. For items A and B all-units discount structures C) with T
were assumed, whereas for items C and D incremental discounts were in effect.
I TEM A: A LL - UNITS
DISCOUNT STRUCTURE
For item A we used an empirically observed discount structure with all-units discounts provided by a given supplier named A1. In order to generate data for a multi-supplier problem setting, we created two additional hypothetical competitors (named suppliers A2 and A3), for which we modified the discount structure of supplier A1 such that the competitors offered slightly higher prices for lower quantities and slightly lower prices for higher quantity levels. With this price structure no supplier dominates the others. The prices are shown in Table 2. Supplier A1
Supplier A2
gtlr
gtlr
r 1 2 3 4 5
50 125 250 500 ∞
ptlr 4.10 3.72 3.55 3.39 3.16
r 1 2 3 4 5
50 125 250 500 ∞
ptlr 4.22 (+2%) 3.79 (+2%) 3.48 (–2%) 3.36 (–1%) 3.13 ( –1%)
Supplier A3 r 1 2 3 4 5
gtlr 50 125 250 500 ∞
ptlr 4.26 (+4%) 3.76 (+1%) 3.51 (–1%) 3.32 (–2%) 3.13 (–1%)
Table 2: Item A; All-units discount structure
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As main experimental factors we considered the variability of the demands and the holding cost percentage. In addition, a set of experiments was run with constant discount structures, whereas in a second set of experiments different time intervals with special prices were considered. The fixed ordering costs were set to s 10 for all periods and all suppliers. All suppliers had the same delivery schedule (a 1 in all periods). The demand series were generated based on a gamma-distribution with differing coefficients of variation (ranging between 0.2 and 4.05) and mean values between 37 and 64. Table 3 shows the realisations of the demands.
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Series #
1
2
3
4
5
6
7
8
9
10
µ
49.80
48.75
37.58
54.45
46.70
38.85
51.75
64.40
47.05
51.40
t CV
0.21
0.20
0.80
0.72
1.34
1.32
1.82
1.89
3.50
4.05
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 36 37 38 39 40
37 36 49 34 63 59 51 52 50 45 61 49 61 55 54 64 51 42 48 54 48 33 43 58 39 59 39 45 68 55 49 63 51 67 57 40 27 61 45 30
41 69 49 59 61 53 40 35 52 42 60 45 41 52 41 48 45 41 42 58 40 39 71 46 59 45 54 43 41 30 46 40 54 57 56 48 64 41 63 39
3 68 26 9 7 10 29 22 88 10 29 40 11 35 47 38 22 59 133 69 15 38 41 20 22 7 51 57 26 88 8 33 97 79 1 93 25 7 14 26
50 12 41 35 171 52 65 24 47 171 61 110 28 99 59 23 29 53 102 65 53 83 82 27 4 33 6 69 70 32 15 16 25 34 51 46 7 27 113 88
219 31 8 8 30 1 50 31 46 25 37 140 35 3 1 1 0 0 0 2 59 23 35 0 33 86 11 88 282 177 50 83 16 68 2 3 0 6 55 123
6 15 4 2 33 61 60 78 8 140 40 85 115 8 282 11 43 8 55 14 0 5 65 76 25 1 5 0 33 22 20 42 0 6 7 78 33 24 37 7
19 0 96 141 1 0 7 64 9 111 5 163 0 475 4 161 0 4 310 75 129 16 0 15 0 1 17 0 0 64 62 0 0 0 7 9 0 0 0 105
36 0 69 1 31 93 147 411 196 0 0 47 0 0 44 0 118 15 18 0 0 13 0 600 20 0 0 18 11 56 1 1 289 82 4 42 2 16 2 193
76 0 1 0 562 1 0 0 0 0 0 0 58 0 0 0 63 1 0 0 0 0 0 0 0 0 0 0 7 893 0 0 0 0 0 0 216 0 0 4
7 0 0 0 4 0 11 3 0 875 0 0 0 0 20 0 0 1029 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 105 0 1
Table 3: Demand series for items A, B, and C
For each problem instance the exact solution was computed with CPLEX 6.5 on an 200 Mhz NT-workstation. The CPU times for the exact solution varied between a few dozens of seconds
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and several hours. The results for the case of constant prices are presented in the Table 4. The numbers are the percentage deviations of the heuristic solutions from the exact solutions.
h
Demand series 1 2
3
4
5
6
7
8
9
10
Mean
0.01 0.02 0.03 0.04 0.05
0.13 0.82 1.56 1.68 1.92
0.00 0.06 0.87 1.29 1.70
0.00 3.54 0.00 1.53 1.04
1.29 1.77 2.63 3.18 3.64
0.10 2.98 1.62 3.35 4.87
0.66 2.52 3.60 3.34 3.22
0.08 2.94 4.79 4.91 2.90
1.15 2.68 5.68 6.26 1.84
4.94 0.09 0.35 3.50 4.41
0.06 0.07 0.99 1.29 3.08
0.84 1.75 2.21 3.03 2.86
Mean
1.22
0.78
1.22
2.50
2.58
2.67
3.12
3.52
2.66
1.10
2.14
Table 4: Results for item A, constant prices
Heuristic Exact ⋅ 100 Exact
In a second setting, we assumed that supplier A1 offered 20% lower prices in periods 26 to 35. The results are presented in the Table 5.
h
Demand series 1 2
3
4
5
6
7
8
9
10
Mean
0.01 0.02 0.03 0.04 0.05
0.02 1.06 1.75 1.87 1.59
0.00 0.76 1.28 1.59 2.33
0.75 1.83 0.00 1.65 0.96
0.24 1.75 2.25 1.80 3.10
1.07 2.94 1.51 1.64 2.63
0.28 2.32 3.85 3.60 3.33
0.61 2.46 3.28 4.89 2.48
1.83 3.54 6.39 7.08 1.32
0.06 0.10 0.10 3.96 4.97
0.06 0.07 1.00 1.30 3.10
0.49 1.68 2.14 2.94 2.58
Mean
1.26
1.19
1.04
1.83
1.96
2.68
2.74
4.03
1.84
1.11
1.97
Table 5: Results for item A, special prices for supplier A1
Heuristic Exact ⋅ 100 Exact
In the third setting, we additionally let supplier A2 reduce its prices in periods 11 to 20 by 5% and supplier A3 reduce his prices in periods 16 to 20 by 10%. The results are presented in Table 6.
20
h
Demand series 1 2
3
4
5
6
7
8
9
10
Mean
0.01 0.02 0.03 0.04 0.05
2.21 1.09 1.09 0.84 0.36
1.78 1.44 1.71 1.30 1.78
0.31 1.14 0.43 1.04 1.15
0.00 0.03 4.12 1.14 2.27
1.16 2.26 1.66 1.15 1.95
1.44 1.86 3.39 2.37 3.07
2.31 2.91 0.34 0.59 2.09
1.79 0.55 6.25 7.45 2.25
0.06 0.10 0.10 3.95 0.10
0.45 0.07 0.06 1.24 3.25
1.15 1.15 1.92 2.11 1.83
Mean
1.12
1.60
0.81
1.51
1.63
2.43
1.65
3.66
0.86
1.01
1.63
Table 6: Results for item A, special prices for all suppliers
I TEM B: A LL - UNITS
Heuristic Exact ⋅ 100 Exact
DISCOUNT STRUCTURE
Item B differs from item A only with respect to the discount breakpoints for the two hypothetical suppliers – all other data are the same as with item A. The discount structure is shown in Table 7. The results are given in Tables 8 – 10. Supplier B1
Supplier B2
gtlr
gtlr
r 1 2 3 4 5
50 125 250 500 ∞
ptlr 4.10 3.72 3.55 3.39 3.16
r 1 2 3 4 5
50 150 300 500 ∞
ptlr 4.22 (+2%) 3.79 (+2%) 3.48 (–2%) 3.36 (–1%) 3.13 ( –1%)
Supplier B3 r 1 2 3 4 5
gtlr 40 200 300 450 ∞
ptlr 4.26 (+4%) 3.76 (+1%) 3.51 (–1%) 3.32 (–2%) 3.13 (–1%)
Table 7: Item B; All-units discount structure
21
h
Demand series 1 2
3
4
5
6
7
8
9
10
Mean
0.01 0.02 0.03 0.04 0.05
0.09 0.62 0.89 1.59 2.15
0.03 1.19 1.53 1.94 1.87
0.08 1.16 2.59 3.06 2.96
0.52 0.40 1.22 1.56 3.62
0.33 1.18 1.84 2.39 5.29
1.49 2.54 5.52 2.15 3.78
0.23 1.12 2.19 3.49 1.58
0.64 3.13 5.38 8.69 7.53
0.05 0.06 0.05 3.37 4.18
0.06 0.07 0.07 0.06 0.06
0.35 1.15 2.13 2.83 3.30
Mean
1.07
1.31
1.97
1.46
2.21
3.10
1.72
5.07
1.54
0.07
1.95
Table 8: Results for item B, constant prices
h
Demand series 1 2
Heuristic Exact ⋅ 100 Exact
3
4
5
6
7
8
9
10
Mean
0.01 0.02 0.03 0.04 0.05
0.31 1.29 1.38 1.93 2.39
0.47 0.00 0.00 0.04 2.52
0.00 0.18 3.60 1.30 2.43
0.00 0.09 1.48 1.93 1.68
0.83 1.77 2.33 3.26 3.06
0.64 2.54 3.19 2.23 3.92
0.24 0.81 1.82 3.52 1.61
0.67 1.69 4.82 8.54 6.83
0.06 0.06 0.07 3.76 4.72
0.06 0.07 0.07 0.06 0.06
0.33 0.85 1.88 2.66 2.92
Mean
1.46
0.61
1.50
1.04
2.25
2.50
1.60
4.51
1.73
0.07
1.73
Table 9: Results for item B, special prices for supplier B1
h
Demand series 1 2
Heuristic Exact ⋅ 100 Exact
3
4
5
6
7
8
9
10
Mean
0.01 0.02 0.03 0.04 0.05
0.83 1.44 2.70 2.84 0.83
0.47 0.83 1.00 0.30 3.11
0.00 0.44 3.88 0.86 0.44
2.40 1.94 2.39 0.91 2.14
0.65 1.00 2.83 2.07 2.03
0.24 1.83 2.26 1.46 1.33
0.97 2.50 4.08 0.70 0.77
1.33 2.49 6.78 7.00 5.42
0.06 0.06 0.06 3.82 4.62
0.44 0.07 0.08 0.07 0.07
0.74 1.26 2.60 2.00 2.07
Mean
1.73
1.14
1.12
1.95
1.72
1.42
1.80
4.60
1.72
0.15
1.74
Table 10: Results for item B, special prices for all suppliers
I TEM C: I NCREMENTAL
Heuristic Exact ⋅ 100 Exact
DISCOUNT STRUCTURE
Item C is purchased from supplier C1 with an incremental discount scheme. In addition to this existing supplier C1 we created two hypothetical competitors along the lines described for item 22
A. The prices are given in Table 11. Supplier C1 r gtlr ptlr 1 10 40.00 2 100 20.00 3 250 15.00 4 500 12.00 5 ∞ 8.00
Supplier C2 r gtlr ptlr 1 10 40.40 (+1%) 2 100 20.40 (+2%) 3 250 14.85 (–1%) 4 500 11.52 (–4%) 5 ∞ 7.60 ( –5%)
Supplier C3 r gtlr ptlr 1 10 41.60 (+4%) 2 100 19.80 (–1%) 3 250 14.70 (–2%) 4 500 11.76 (–2%) 5 ∞ 8.16 (+2%)
Table 11: Item C; Incremental discount structure
In addition to the different pricing schemes the range of the holding cost factors was changed slightly. All other settings were the same as in the experiments for item A. Table 12 shows the percentage deviations of the solutions found with the proposed heuristic from the exact solutions, which were computed with the above-mentioned shortest-path approach.
h
Demand series 1 2
3
4
5
6
7
8
9
10
Mean
0.05 0.1 0.15 0.2
0.42% 1.62% 0.43% 1.11%
0.18% 0.14% 0.73% 0.85%
3.07% 1.96% 0.06% 0.14%
0.59% 2.12% 2.62% 0.12%
0.00% 1.23% 0.41% 0.35%
0.11% 0.29% 2.12% 0.45%
2.08% 2.84% 0.39% 1.21%
1.19% 0.36% 0.22% 0.21%
0.13% 0.12% 0.62% 0.97%
2.21% 0.04% 0.04% 0.04%
1.00% 1.07% 0.76% 0.55%
Mean
0.89%
0.48% 1.31%
1.36%
0.50%
0.74% 1.63%
0.49%
0.46% 0.58%
0.84%
Table 12: Results for item C, constant prices
h
Demand series 1 2
Heuristic Exact ⋅ 100 Exact
3
4
5
6
7
8
9
10
Mean
0.05 0.1 0.15 0.2
0.32% 1.60% 0.23% 0.91%
0.00% 0.03% 0.02% 0.89%
2.28% 1.78% 0.53% 0.11%
3.28% 1.55% 3.04% 0.07%
0.00% 0.19% 0.38% 0.58%
0.11% 0.30% 1.90% 0.46%
1.08% 3.78% 2.28% 1.36%
3.53% 0.15% 0.23% 0.22%
0.14% 0.13% 0.36% 0.36%
2.07% 0.04% 0.04% 0.04%
1.28% 0.95% 0.90% 0.50%
Mean
0.76%
0.24% 1.17%
1.99%
0.29%
0.69% 2.12%
1.03%
0.25% 0.55%
0.91%
Table 13: Results for item C, special prices for supplier C1
Heuristic Exact ⋅ 100 Exact
23
Demand series 1 2
h
3
4
5
6
7
8
9
10
Mean
0.05 0.1 0.15 0.2
0.05% 0.38% 0.34% 0.65%
0.13% 0.23% 0.48% 0.53%
1.20% 2.13% 0.53% 0.06%
0.38% 1.91% 2.06% 0.06%
0.00% 0.00% 0.20% 0.47%
0.89% 0.82% 1.78% 0.75%
0.03% 1.75% 2.31% 0.28%
5.75% 0.15% 0.23% 0.23%
0.14% 0.40% 0.13% 0.39%
0.05% 0.05% 0.04% 0.04%
0.86% 0.78% 0.81% 0.35%
Mean
0.36%
0.34% 0.98%
1.10%
0.17%
1.06% 1.09%
1.59%
0.27% 0.05%
0.70%
Table 14: Results for item C, special prices for all suppliers
I TEM D: I NCREMENTAL
DISCOUNT STRUCTURE , LOW- VOLUME DEMAND
The last problem class considered includes L 2 suppliers, T ∑ dt
Heuristic Exact ⋅ 100 Exact
20 periods, a total demand of
40 and the discount structures shown in Table 15. Supplier D1 r gtlr ptlr 1 1 41 2 5 33 3 10 28 4 15 24 5 20 22 6 25 18 7 ∞ 15
Supplier D2 r gtlr ptlr 1 1 43 2 7 32 3 11 26 4 20 23 5 30 16 6 ∞ 14
Table 15: Item D; Incremental discount structure
The demand series were again randomly generated based on a gamma-distribution with differing coefficients of variation (ranging between 0.16 and 1.08) and total demands between 35 and 40. Table 16 shows the realisations of the demands.
24
Series #
1
2
3
4
5
6
7
8
9
10
∑ dt
40
40
35
38
40
37
36
35
39
39
t CV
0.27
0.16
0.44
0.40
1.01
0.60
0.72
0.83
1.08
0.94
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 2 1 2 2 2 3 2 2 3 2 3 2 2 1 2 2 1 2 2
2 1 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2
2 3 3 2 1 1 2 1 3 2 1 2 1 2 2 1 1 3 1 1
2 3 2 2 1 1 1 2 1 2 3 3 2 3 1 3 1 2 1 2
1 6 1 2 0 5 8 0 2 2 1 0 2 1 2 1 1 1 1 3
2 5 1 4 2 1 1 2 2 3 1 2 2 1 1 3 1 1 1 1
2 3 3 2 0 1 1 1 1 3 2 0 0 4 4 0 1 2 3 3
3 1 4 2 1 2 5 2 0 1 5 2 0 1 1 2 1 0 1 1
1 1 0 1 1 5 3 0 1 4 0 3 2 1 1 3 1 0 9 2
2 0 3 1 0 2 6 0 4 2 3 0 1 0 5 1 2 5 0 2
Table 16: Demand series
Table 17 shows the results for constant prices. Table 18 refers to an experiment, where supplier D1 offered 5% lower prices in periods 11 to 15, while in Table 19 additionally supplier D2 reduced his prices in periods 13 to 20 by 6%.
h 0.1 0.2 0.3 0.4 0.5 Mean
Demand series 1 2 1.37 1.13 0.54 0.32 1.49 0.97
2.09 0.16 0.29 0.00 0.00 0.51
3
4
5
6
7
8
9
10
Mean
0.18 0.69 2.57 0.60 0.42 0.89
1.51 0.86 0.06 0.20 0.42 0.61
2.82 0.14 0.51 0.15 0.26 0.78
0.00 4.56 0.67 0.21 0.40 1.17
2.51 0.36 0.26 0.81 1.06 1.00
0.06 0.78 2.21 0.10 1.39 0.91
0.00 1.14 0.55 0.45 0.24 0.48
3.97 0.52 0.05 0.05 0.12 0.94
1.45 1.03 0.77 0.29 0.58 0.82
Table 17: Results for item D, constant prices
Heuristic Exact ⋅ 100 Exact
25
h 0.1 0.2 0.3 0.4 0.5 Mean
Demand series 1 2 1.37 0.21 1.11 0.32 1.66 0.93
0.18 1.18 0.49 0.02 0.00 0.38
3
4
5
6
7
8
9
10
Mean
1.67 1.28 2.77 0.45 0.37 1.31
0.00 0.47 0.00 0.44 0.43 0.27
1.99 0.14 0.43 0.15 0.26 0.59
0.48 2.93 0.06 0.15 1.03 0.93
0.09 0.37 0.26 0.64 0.87 0.45
0.10 0.79 2.24 0.07 0.95 0.83
0.00 0.90 0.18 0.06 0.06 0.24
4.03 0.00 0.05 0.05 0.51 0.93
0.99 0.83 0.76 0.24 0.61 0.69
Table 18: Results for item D, special prices for supplier D1
h 0.1 0.2 0.3 0.4 0.5 Mean
Demand series 1 2 1.37 1.11 0.78 0.32 1.17 0.95
0.18 0.86 0.30 0.00 0.00 0.27
Heuristic Exact ⋅ 100 Exact
3
4
5
6
7
8
9
10
Mean
0.91 1.16 2.79 0.45 0.39 1.14
0.00 0.90 0.00 0.45 0.43 0.36
1.75 0.19 0.49 0.15 0.20 0.56
0.48 4.42 0.06 0.00 0.84 1.16
0.52 0.44 0.26 0.57 0.83 0.52
0.10 0.79 2.25 0.00 1.18 0.86
0.00 1.44 0.12 0.00 0.00 0.31
4.05 0.00 0.06 0.05 0.60 0.95
0.94 1.13 0.71 0.20 0.56 0.71
Heuristic Exact ⋅ 100 Exact
Table 19: Results for item D, special prices for all suppliers
The results indicate that the heuristic – despite of its simplicity – performs quite well. This is especially true for the cases with higher holding costs. As the improvements are performed iteratively, we do not know as to which extent the solution quality is attributable to a specific improvement step. It appears that the overall quality is due to the iterative way of changing the solutions, with one improvement step leaving an intermediate solution that is a good starting point for the next improvement step. Table 20 illustrates the impact of the complete improvement phase on the overall solution quality of the heuristic – compared to the LUC solution. In the first column the item and the corresponding table number are given. Basically there are three conditions under which phase II improved the quality of the solutions: time-varying discount-structures, high variability of 26
demands and – to the largest extent – incremental discounts. Demand series Item–table 1 2
3
4
5
6
7
8
9
10
Mean
A–4 A–5 A–6
0.48 2.76 2.71
0.47 2.79 2.14
1.39 5.03 5.25
0.36 3.55 3.71
0.98 6.81 5.65
0.85 2.85 2.56
2.99 3.43 4.80
2.69 3.96 3.90
6.66 11.56 11.66
10.47 11.13 12.21
2.73 5.39 5.46
B–8 B–9 B–10
0.53 2.03 2.62
0.52 2.80 2.32
0.71 3.73 3.75
0.36 3.80 2.24
2.50 6.16 7.05
1.86 3.59 2.82
3.27 3.53 3.92
1.95 3.72 4.32
7.56 11.41 10.84
12.02 12.68 14.15
3.13 5.34 5.40
C–12 C–13 C–14
0.89 3.89 3.26
1.15 4.28 3.52
6.88 6.92 9.09
7.59 7.77 7.62
27.74 33.80 32.23
17.63 17.35 16.71
16.43 15.10 17.18
23.50 23.16 21.69
32.93 37.51 27.92
33.04 33.06 34.77
16.78 18.28 17.40
D–17 D–18 D–19
1.14 1.34 1.04
0.00 0.53 0.46
4.09 4.07 4.39
1.24 2.17 2.31
6.20 5.96 5.97
4.68 5.00 4.91
4.78 5.68 5.33
5.46 6.08 6.02
14.73 13.15 13.01
10.09 10.07 9.92
5.24 5.41 5.34
Table 20: Contribution of the phase II improvement steps to the overall solution quality Z (Phase I) Z (Phase II) ⋅ 100 Z (Phase I)
6 Concluding Remarks In this paper, we have developed new model formulations and a heuristic solution method for the dynamic order sizing and supplier selection problem under assumptions that are relevant for industrial purchasing practice. The heuristic has been implemented and tested in Visual Basic 6.0 which is an environment that facilitated the development of the heuristic. A problem instance of the size considered in the above experiments normally required less than one second. The maximum CPU time required was less than two seconds. The heuristic has been recoded and implemented as part of the Advanced Planner and Optimizer (APO) Software of SAP AG. It is designed in a modular form such that further improvement steps can easily be added. This will be a topic of future extensions. In a parallel project we currently concentrate on the development of more sophisticated heuristics for the considered problem. 27
Appendix Procedure Improvement Step A1 (price reduction) begin for l 1 to L for t 1 to T if cost reductionl t true then Find the earliest period τ≥ t with alτ 1. Find the latest order in period tv t from any supplier i. Determine the maximum postponable quantity qmax . If the objective value decreases, shift qmax from period tv and supplier i to period τ and supplier l end if next t next l end
Procedure Improvement Step A2 (price increase) begin for t T 1 to 2 step -1 for l 1 to L if atl 1 then if cost increasel t 1 true then Find the next order in period tn t from any supplier i. Determine the maximum shiftable quantity qmax . If the objective value decreases, shift qmax from period tn and supplier i to period t and supplier l. end if end if next t next l end
Procedure Improvement Step B (splitting of orders) begin for l 1 to L for t 1 to T if qtl 0 then Determine the next possible ordering period τ of supplier l and the latest order from any supplier i before period t in period tv . Determine the minimum quantity qmin to be shifted into period tv . If the objective value decreases, shift qmin from period t to tv and supplier i and shift qtl qmin to period τ and supplier l. end if next t
28
next l end Procedure Improvement Step C (combination of orders) begin for l 1 to L for t 1 to T if qtl 0 then Determine the latest order from any supplier i before period t in period tv . If the objective value decreases, shift qtl from period t to period tv and supplier i. end if next t next l end Procedure Improvement Step D (postponement of partial orders) begin for l 1 to L for t 1 to T 1 if qtl 0 then Determine the period tn of the next order for τ t to tn Determine the maximum quantity qmax postponable from period t to τ. If the objective value decreases then Shift qmax from period t to tn exit for end if next τ end if next t next l end
References Anupindi, R. and Y. Bassok (1999). Supply contracts with quantity commitments and stochastic demand. In S. Tayur, R. Ganeshan, and M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Boston, pp. 197–232. Kluwer. Benton, W. C. (1985). Multiple price breaks and alternative purchase lot-sizing procedures in material requirements planning systems. International Journal of Production Research 23, 1025–1047. Benton, W. C. (1986). Purchase lot sizing research for MRP systems. Operations & Productions Management 6(1), 5–14.
29
Benton, W. C. and S. Park (1996). A classification of literature on determining the lot size under quantity discounts. European Journal of Operational Research 92, 219–238. Benton, W. C. and D. C. Whybark (1982). Material requirements planning (MRP) and purchase discounts. Journal of Operations Management 2, 137–143. Bregman, R. L. (1991). An experimental comparison of MRP purchase discount methods. Journal of the Operational Research Society 42, 235–245. Bregman, R. L. and E. A. Silver (1993). A modification of the silver-meal heuristic to handle MRP purchase ciscount situations. Journal of the Operational Research Society 44, 717–723. Callarman, T. E. and D. C. Whybark (1981). Determining purchase quantities for MRP requirements. Journal of Purchasing and Materials Management 17(Fall), 25–30. Chaudry, S. S., F. G. Forst, and J. L. Zydiak (1993). Vendor selection with price breaks. European Journal of Operational Research 70, 52–66. Christoph, O. B. and R. L. LaForge (1989). The performance of MRP purchase lot-size procedures under actual multiple purchase discount conditions. Decision Sciences 20, 348–358. Chung, C.-S., D. T. Chiang, and C.-Y. Lu (1987). An optimal algorithm for the quantity discount problem. Journal of Operations Management 7, 165–177. Chung, C.-S., S.-H. Hum, and O. Kirca (1996). The coordinated replenishment dynamic lot-sizing problem with quantity discounts. European Journal of Operational Research 94, 122–133. Chung, C.-S., S.-H. Hum, and O. Kirca (2000). An optimal procedure for the coordinated replenishment dynmaic lot-sizing problem with quantity discounts. Naval Research Logistics Quarterly 47, 686–695. Chyr, F., S.-T. Huang, and S. de Lai (1999). A dynamic lot-sizing modell with quantity discount. Production Planning & Control 10, 67–75. Jayaraman, V., R. Srivastava, and W. C. Benton (1999). Supplier selection and order quantity allocation: A comprehensive model. The Journal of Supply Chain Management 35(2), 50–58. Krarup, J. and O. Bilde (1977). Plant location set covering and economic lot sizing: An O(nm) algorithm for structured problems. In L. Collatz (Ed.), Numerische Methoden bei Optimierungsaufgaben, Band 3, Optimierung bei graphentheoretischen and ganzzahligen Problemen. Basel: Birkh¨auser. LaForge, R. L. and J. W. Patterson (1985). Adjusting the part-period algorithm for purchase quantity discounts. Production and Inventory Management Journal (1), 138–150. Munson, C. L. and M. J. Rosenblatt (1997). The impact of local content rules on global sourcing decisions. Production and Operations Management 6, 277–290. Munson, C. L. and M. J. Rosenblatt (1998). Theories and realities of quantity discounts: An exploratory study. Production and Operations Management 7, 352–369. Munson, C. L. and M. J. Rosenblatt (2001). Coordinating a three-level supply-chain with quantity discounts. IIE Transactions 33, 371–384. Prentis, E. L. and B. M. Khumawala (1989). MRP lot sizing with variable production/purchasing costs: Formulation and solution. International Journal of Production Research 27, 965–984. Tersine, R. J. and R. A. Toelle (1985). Lot size determination with quantity discounts. Production and Inventory Management Journal (1), 1–23.
30
Tsay, A. A., S. Nahmias, and N. Agrawal (1999). Modeling supply chain contracts: A review. In S. Tayur, R. Ganeshan, and M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Boston, pp. 299–336. Kluwer. Zeng, A. Z. (1998). Single or multiple sourcing: An integrated optimization framework for sustaining time-based competitiveness. Journal of Marketing Theory and Practice 6(4), 10–25. Zipkin, P. H. (2000). Foundations of Inventory Management. Boston: McGraw-Hill.
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