multi-period supplier selection under price uncertainty yaman... · 2013-07-15 · this study i...

Multi-period Supplier Selection Under PriceUncertainty

Hande Yaman

Department of Industrial EngineeringBilkent University, Ankara, Turkey

July 8, 2013

joint work withAlper Sen (Bilkent), Kemal Guler and Evren Korpeoglu (HP Labs)

Hande Yaman Supplier Selection 1/28

Problem

I A manufacturer needs to procure large volumes of multipleitems over a planning horizon (quarter) that consists ofmultiple periods (months).

I A set of suppliers, each offering all or a selection of the items.

I Suppliers provide base prices and discounts.

I Sourcing: Which items should be purchased from whichsuppliers to minimize procurement and inventory holdingcosts?

I Common problem in many industries.


Supplier Cost Structure and Offers

I Supplier production costs exhibit economies of scale andscope: unit costs go down with more volume of a particularitem or a group of items.

I Suppliers express these by offering discounts to themanufacturer.


Discounts

I Discounts may take many forms:

I On one item, or a group of items

I Based on total available market (verifiable), volume or spend

I Conditions on items individually, or a set of items (these itemsmay differ from the items discount applies)

I Price discount applied for specific periods (these periods canbe different from which conditions are required)

I Disjunctions, i.e., discount not valid with other discounts

I Price discount applied incrementally or to all units

I Lump sum discount offers


Literature

I review supplier selection:Benton and Park (1996), Munson and Rosenblatt (1998) andAissaoui et al. (2006)

I single period:Bichler et al. (2011), Crama et al. (2004), Goossens et al.(2007), Manerba and Mansini (2012), Mansini et al. (2012),Murthy et al. (2004), Qin et al. (2012)

I multiple periods, dynamic demand, single item:Tempelmeier (2002), van de Klundert et al. (2005)


Literature

I Stadtler (2007): multiple items, discount rules involve a singleitem, fixed cost

I Xu et al. (2000): single supplier, business volume discountsand setup costs

I Rong et al. (2012): discounts are due to ordering in standardbatches

I This study: multiple items, periods and suppliers and generaldiscount rules and price uncertainty


This Study

I Most favored customer clauses and price uncertainty

I Most favored customer status: ”A contractual arrangementbetween a company and customer that guarantees thecustomer the best price the company gives to anyone.”

I The supplier’s price for other customers is uncertain. If theprice drops, most favored customers also benefit from thereduction in price.

I May require a minimum purchase volume on a set of itemsover a specified time period.


I Example 1: Agreement between Cisco Systems Inc. andFrontier Software Development Inc.

“Frontier represents and warrants to Cisco that the productprices/license fees offered to Cisco under this Agreement areno less favorable than the product prices/license fees offeredto any other party purchasing or licensing similar quantities.In the event Frontier offers more favorable productprices/license fees to any other party, Frontier will promptlynotify Cisco of such event and offer such more favorableproduct prices/license fees to Cisco commencing upon thedate such more favorable product prices/license fees wereoffered to the other party.” 1

1http://contracts.onecle.com/netscout/cisco.sales.1996.05.15.shtmlHande Yaman Supplier Selection 8/28

I Example 2: US Government purchasing through GeneralServices AdministrationA price reduction (MFC prices) clause in a contract betweenDoD and HP:“The prices under this blanket purchase agreement (BPA)shall be at least as low as the prices that the contractor hasunder any other contract instrument under like terms andconditions. If any time the prices under any other contractinstrument become lower than the prices in this BPA, thisBPA will be modified to include the lower prices”

Failure to do so leads to ...I Oracle to pay $199.5 million to settle charges, 2011I EMC reached a settlement to pay $87.5 million in 2011I NetApp paid $128 million in 2009I PeopleSoft (Oracle) paid $98.5 million in 2006


Problem Setting

I Supplier offers are collected prior to the start of the horizon.

I Suppliers may also have limits on the amount of capacity thatthey can allocate to the manufacturer.

I Inventory may be carried over from one period to the other.


Decisions

I Manufacturer’s decisions

I Usually, the manufacturer does not commit anything to thesuppliers (except perhaps that a list of items that he willprocure from each).

I Here–and–now decisions:I Volume to be purchased from each supplier for each item in

Period 1

I Wait–and–see decisions/recourses:I Volume to be purchased from each supplier for each period

contingent on new item prices


Motivating Example

I A company needs to procure an item over a two-periodhorizon

I Demand in the first and second period are δ1 and δ2

I Supplier a charges µa per unit and offers a MFC:I If the firm procures 100 m of the first period’s demand from

supplier a, it can benefit from a price reduction πa per unitwhich will happen with probability γ

I Supplier b charges µb per unit and offers a volume discount:I If the firm procures a total of ρ in two periods, the firm gets a

discount of πb per unit

I Assume that:I

δ1 + δ2 ≥ ρ > (1−m) δ1 + δ2,

I

µb < µa < µb + πa,


I Expected cost if firm procures enough from supplier a tobenefit from MFC:

Φa0 = µam δ1 + µb(1−m)δ1 + γ (µa − πa)δ2 + (1− γ)µbδ2.

I Expected cost if firm procures enough from supplier b tobenefit from volume discount:

Φb = (µb − πb)(δ1 + δ2).

I The firms will choose supplier a’s MFC status if (Φa0 < Φ b):

γ > γ0 =πb(δ1 + δ2) + (µa − µb)m δ1

(πa + µb − µa) δ2


I Certainty equivalent argument:I Supplier a’s second period price will be: µa − γπaI Firm’s cost if it chooses to use MFC:

Φa1 = µa m δ1 + µb (1−m) δ1 + (µa − γπa)δ2.

I The firms will choose supplier a’s MFC term if (Φa1 < Φ b):

γ > γ1 =πb(δ1 + δ2) + (µa − µb)m δ1 + (µa − µb) δ2

πa δ2.

I Result: γ0 < γ1


I For γ0 < γ ≤ γ1, the optimal decision is to purchase enoughto be eligible for supplier a’s MFC clause.

I However, certainty equivalent approach would lead to optingfor volume discount from supplier b

I In general, certainty equivalent approach may lead tosuboptimal decisions.

I Need to model uncertainty.


0

Month 1: Price µ

1

Month 2: Price µ

2

Month 2: Price µ− π

γ1

γ2 = 1− γ1

Example: A discount rule r is available at node 2 (if the supplier lowers itsprice to other customers) contingent on a condition c that the total volumepurchased in months Tc = {1} exceeds ρc . In this case, a discount k provides adiscount of π per unit for anything that is bought above θk = 0 in monthsTk = {2}.


Model: Inventory and Capacities

Iis = Iia(s) +∑j∈Ni

xijs − δiτ(s) ∀i ∈ I, s ∈ V

xijs ≤ κijτ(s) ∀i ∈ I, j ∈ Ni , s ∈ Vxijs ≥ 0 ∀i ∈ I, j ∈ Ni , s ∈ VIis ≥ 0 ∀i ∈ I, s ∈ V


Model: Discounts

∑i∈Ic

∑s∈Ps :τ(s)∈Tc

xijr s ≥ ρczsr ∀s ∈ V, r ∈ Rs , c ∈ Cr

y sk ≤∑i∈Ik

∑s∈Ps :τ(s)∈Tk

xijr s − θkzsr ∀s ∈ V, r ∈ Ds , k ∈ Kr

y sk ≤

∑i∈Ik

∑t∈Tk

κijr t − θk

zsr ∀s ∈ V, r ∈ Ds , k ∈ Kr

zsr + zsr ′ ≤ 1 ∀s ∈ V, {r , r ′} ∈ Es

zsr ∈ {0, 1} ∀s ∈ V, r ∈ Rs

y sk ≥ 0 ∀s ∈ V, r ∈ Ds , k ∈ Kr


Model: Objective

min∑s∈V

γs

(∑i∈I

∑j∈Ni

µijτ(s)xijs +∑i∈I

ηiτ(s)Iis

−∑r∈Ds

∑k∈Kr

πkysk −

∑r∈Ls

ωrzsr

)


What we can model

I All units discounts and incremental discountsI Discounts with multiple conditionsI Separation of items for which the conditions required and the

discounts are applied

I Supplier offers with most favored customer clauses (MFC)

I Supplier offers with meet the competition clauses (MCC)I Multiple periods

I Inventory decisionsI Separation of periods for which the conditions required and the

discounts are applied

I Offers with uncertain elementsI MFC, MCCI Spot price uncertaintyI Future trade deals


Impact of MFC

I three bidding events in two rounds

I 40-45 items, 3-5 suppliers

I 2 periods

I 1 MFC clause

EventsTest 1a 1b 2a 2b 3a 3b

Ignore MFC clause(regular discount offers only) 0.93 1.26 2.29 2.31 3.22 3.23

MFC clause when prob of drop islow 0.97 1.67 2.29 2.31 3.70 3.71

medium 1.48 2.29 2.29 2.31 4.42 4.42high 1.98 2.91 2.30 2.32 5.14 5.13


Certainty equivalent heuristics

I Static (SCE): solve at the beginning of the horizon andexecute the solution regardless of the actual realizations of therandom events

I Dynamic (DCE): solve at the beginning of each period


Numerical Example 1

I Prices drop with prob γ

I Prices remain the same

Discount Amount 5% Discount Amount 10%γ J∗ ∆SCE ∆DCE J∗ ∆SCE ∆DCE

0.1 0.00 0.00 0.00 0.00 0.00 0.000.2 0.00 0.00 0.00 3.81 100.00 100.000.3 0.00 0.00 0.00 21.99 84.12 0.000.4 0.00 0.00 0.00 40.17 46.79 0.000.5 3.36 100.00 100.00 58.35 32.72 0.000.6 10.54 66.84 0.00 76.53 23.35 0.000.7 17.72 29.82 0.00 94.71 14.15 0.000.8 24.90 14.15 0.00 112.89 7.91 0.000.9 32.08 5.49 0.00 131.06 3.41 0.00

Avg 17.72 43.26 20.00 67.44 39.06 12.50


Numerical Example 2

I Prices drop 5% with prob γ1

I Prices drop 10% with prob γ2


γ1 γ2 J∗ ∆SCE ∆DCE γ1 γ2 J∗ ∆SCE ∆DCE

0.1 0.1 0.00 0.00 0.00 0.3 0.3 43.53 30.36 0.000.2 0.1 0.00 0.00 0.00 0.4 0.3 50.71 22.59 0.000.3 0.1 7.18 100.00 100.00 0.5 0.3 57.89 16.75 0.000.4 0.1 14.36 75.66 0.00 0.1 0.4 47.35 35.98 0.000.5 0.1 21.54 42.26 0.00 0.2 0.4 54.53 28.01 0.000.1 0.2 10.99 100.00 100.00 0.3 0.4 61.71 21.90 0.000.2 0.2 18.17 80.78 0.00 0.4 0.4 68.89 14.85 0.000.3 0.2 25.35 50.96 0.00 0.5 0.4 76.07 8.00 0.000.4 0.2 32.53 34.30 0.00 0.1 0.5 65.53 26.45 0.000.5 0.2 39.71 23.67 0.00 0.2 0.5 72.71 19.32 0.000.1 0.3 29.17 57.38 0.00 0.3 0.5 79.89 12.40 0.000.2 0.3 36.35 41.20 0.00 0.4 0.5 87.07 6.62 0.00

Avg 45.51 38.61 9.09


Numerical Example 3

I Prices drop for item group 1 only with prob γ1

I Prices drop for item group 2 only with prob γ2

I Prices drop for both groups with prob γ3


γ1 γ2 γ3 J∗ ∆SCE ∆DCE γ1 γ2 γ3 J∗ ∆SCE ∆DCE0.1 0.1 0.0 0.00 0.00 0.00 0.1 0.2 0.1 7.99 100.00 100.000.2 0.1 0.0 0.00 0.00 0.00 0.1 0.3 0.1 12.16 100.00 100.000.3 0.1 0.0 13.65 98.98 0.00 0.1 0.4 0.1 16.33 100.00 100.000.4 0.1 0.0 27.66 44.03 0.00 0.2 0.2 0.1 21.99 84.12 0.000.1 0.2 0.0 0.00 0.00 0.00 0.2 0.3 0.1 26.17 76.94 0.000.2 0.2 0.0 3.81 100.00 100.00 0.3 0.3 0.1 40.17 46.79 0.000.3 0.2 0.0 17.82 94.65 0.00 0.4 0.4 0.1 58.35 32.72 0.000.4 0.2 0.0 31.83 48.81 0.00 0.0 0.0 0.2 3.81 100.00 100.000.1 0.3 0.0 0.00 0.00 0.00 0.1 0.1 0.2 21.99 84.12 0.000.2 0.3 0.0 7.99 100.00 100.00 0.2 0.2 0.2 40.17 46.79 0.000.3 0.3 0.0 21.99 84.12 0.00 0.3 0.3 0.2 58.35 32.72 0.000.4 0.3 0.0 36.00 47.68 0.00 0.0 0.0 0.3 21.99 84.12 0.000.1 0.4 0.0 0.00 0.00 0.00 0.1 0.1 0.3 40.17 46.79 0.000.2 0.4 0.0 12.16 100.00 100.00 0.2 0.2 0.3 58.35 32.72 0.000.3 0.4 0.0 26.17 76.94 0.00 0.3 0.3 0.3 76.53 23.35 0.000.4 0.4 0.0 40.17 46.79 0.00 0.0 0.0 0.4 40.17 46.79 0.000.1 0.1 0.1 3.81 100.00 100.00 0.1 0.1 0.4 58.35 32.72 0.000.2 0.1 0.1 17.82 94.65 0.00 0.2 0.2 0.4 76.53 23.35 0.000.3 0.1 0.1 31.83 48.81 0.00 0.0 0.0 0.5 58.35 32.72 0.000.4 0.1 0.1 45.83 30.98 0.00 0.1 0.1 0.5 76.53 23.35 0.00

Avg 32.94 64.76 22.22


Numerical Example 4

I 3 periods

I prices drop with prob γ1 in period 1

I prices drop with prob γ2 in period 2

γ1 γ2 J∗ ∆SCE ∆DCE

0.05 0.05 0.00 0.00 0.000.10 0.10 1.87 100.00 100.000.15 0.15 6.71 85.92 2.500.20 0.20 11.17 53.93 1.760.25 0.25 15.28 36.28 1.400.30 0.30 19.09 27.13 1.170.35 0.35 22.63 21.42 1.000.40 0.40 25.93 17.14 0.870.45 0.45 29.00 14.26 0.750.50 0.50 31.88 12.28 0.640.55 0.55 34.57 10.90 0.540.60 0.60 37.09 8.55 0.45

Avg 21.38 32.26 10.10


Summary of Experiments

SCE DCEExp. Description # Avg(∆) F S Avg(∆) F S

1a 2per: drop by 5% 5 43.26 1 0 20.00 1 41b 2per: drop by 10% 8 39.06 1 0 12.50 1 72 2per: drop by 5% or 10% 22 38.61 2 0 9.09 2 203 2per: drop for a group 35 64.76 8 0 22.22 8 274 3per: drop by 5% + 5% 11 32.26 1 0 10.10 1 0

All 81 49.38 13 0 15.91 13 58

I #: number of valid instances for which MFC leads to additional savings

I Avg(∆): average regret

I F: number of instances where the regret is 100%

I S: number of instances where the regret is 0%


Conclusions

I MFC clauses can lead to substantial savings

I Certainty equivalent heuristics may have large regrets

I Model uncertainty on other parameters

I Demand uncertainty - execution

I Polyhedral demand uncertainty - newsvendor

I Solve large instances: stronger formulations


multi-period supplier selection under price uncertainty yaman... · 2013-07-15 · this study i...

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