Chemcal Supply

Available online at www.sciencedirect.com

Computers and Chemical Engineering 32 (2008) 2481–2504

Chemical supply chain network optimization

Jeff Ferrio ∗, John Wassick

The Dow Chemical Company, 1776 Building, Midland, MI 48667, USA

Received 22 May 2007; received in revised form 5 September 2007; accepted 6 September 2007Available online 11 September 2007

Abstract

Chemical supply chain networks provide large opportunities for cost reductions through the redesign of the flow of material from producer tocustomer. In this paper we present a mixed-integer linear program (MILP) capable of optimizing a multi-product supply chain network made upof production sites, an arbitrary number of echelons of distribution centers, and customer sites. The emphasis of our approach is on the redesignof existing supply chain networks. The model does not lump customer demand into zones, but rather deals with individual customer demand todirectly address customer preferred mode of transport at each location. Historical records can be used to fix decision variables in the model so thata base case can be computed to validate the model and contrast it against the optimized network. The details inherent in this approach allow theoptimization results to be partitioned and prioritized for implementation. The model results are processed to assign cost components to individualcustomer records. A simple case study is presented to illustrate the method and actual industrial results are reviewed.© 2007 Elsevier Ltd. All rights reserved.

Keywords: Supply chain; Optimization; MILP

1. Introduction

The cost pressures in the chemical industry require cost reductions for manufacturers to remain competitive. Chemical supplychains are a fruitful area of cost reduction opportunities because (1) they represent a significant portion of the total cost to servecustomers, (2) they constantly change, and (3) their complexity often hides a lowest cost option.

The decisions involved in managing a supply chain range from the tactical, such as detailed production scheduling, to the strategic,such as the number and location of production facilities. There are a large number of variables that affect these short- and long-termdecisions. Likewise, there are myriad costs that are encountered in supply chains including transportation costs, inventory costs, rawmaterial costs and terminal operating costs just to name a few.

The scope of a particular supply chain optimization effort is defined by business considerations and the need to balance effortversus value returned. In this paper, we confine our attention to the movement and handling of manufactured materials from theproduction sites to customer locations in a multi-echelon supply chain. Our goal is to improve the design of an existing supply chainnetwork made up of production sites and shipping terminals serving customer locations.

The importance of supply chain network optimization is attested by an immense body of literature on modeling approaches andsolution techniques. A complete review of the literature is beyond the scope of this paper. The interested reader can refer to surveypapers by Meixell and Gargeta (2005) and Vidal and Goetschalckx (1997). We briefly review previous work that is most closelyrelated to the mixed-integer linear programming (MILP) approach we present.

Geoffrion and Graves (1974) presented some of the earliest work on approaching the design problem as a MILP. They focused onthe design of a distribution system with a single echelon of DCs connecting production plants to customer zones. The objective was tominimize the sum of transportation costs and DC operating costs. Their model included constraints for production and DC capacitiesand a variety of other constraints related to the allowable configuration of the network. Fixed and variable costs at DCs were included.

∗ Corresponding author.E-mail addresses: [email protected] (J. Ferrio), [email protected] (J. Wassick).

0098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.compchemeng.2007.09.002

mailto:[email protected]

mailto:[email protected]

dx.doi.org/10.1016/j.compchemeng.2007.09.002

2482 J. Ferrio, J. Wassick / Computers and Chemical Engineering 32 (2008) 2481–2504

Nomenclature

SetsAcust(K, R, P, M) a viable arc to the customer on record r from facility k of product p via mode mFoutplant(K) keys that are outplantsFplant(K) keys that are production facilities or plantsF

MapP (K, P) facility k can handle or produce product p

Gi(K) group of facilities to be considered for site selection constraint iI site selection constraintsK, K′, K′′ keys, or facilitiesLcust(K, R, M) “Legal” to ship from facility k to record r via mode mLkey(K, K′, M) “Legal” to replenish facility k′ from facility k via mode mM, M′ modes of transportP productsP

MapM (P, M) product p can be shipped using mode m

R customer recordsR* preferred customer records where demand must be satisfiedR

MapM (R, M) customer record r receives shipments via mode m

RMapP (R, P) customer record r receives product p

Rate(M) modes of transport to customers that use a rate-based (per mass) freight correlationShipment(M) modes of transport to customers that use a shipment-based (per shipment) freight correlationV in

key(K, M) facility k can receive inbound replenishments via mode m

V outcust(K, M) customer can receive shipments via mode m from facility k

V outkey(K, M) facility k can ship outbound to other facilities via mode m

Z facility throughput zones

Indices into setsi user defined site selection constraintk, k′, k′′ a facility or keym, m′ a mode of transportp a productr a customer recordz, z′ a throughput zone for a facility

ParametersBk,p,z breakpoint: first point at which throughput quantity for product p at facility k enters zone zBk,z breakpoint: first point at which throughput quantity for total volume at facility k enters zone z

clog,ink,m logistical costs associated with an outplant k receiving inbound shipments via mode m

clog,outk,m,cust logistical costs associated with facility k shipping to customers via mode m

clog,outk,m,replen logistical costs associated with facility k replenishing other facilities via mode m

Cmaxk maximum total throughput for facility k

Cmaxk,p maximum total product p throughput for facility k

CMaxBigM a large cost

Cmink minimum total throughput for facility k

Cmink,p minimum total product p throughput for facility k

Dr demand on customer record rDmax upper bound on total volume of all products through any siteDmax

p maximum total demand of a single product p

fcust,rater,k,m rate-based freight rate to customer record r from facility k via mode m

fcust,shipmentr,k,m shipment-based freight rate to customer record r from facility k via mode m

freplenk,k′,m replenishment freight rate for shipping between facility k and k′ via mode m

FCk,p,z fixed cost (y-intercept) for linear cost segment for product p at facility k in zone z

J. Ferrio, J. Wassick / Computers and Chemical Engineering 32 (2008) 2481–2504 2483

FCk,z fixed cost (y-intercept) for linear cost segment for total volume through facility k in zone zNr number of shipments in customer record rrk,p production rate in mass per time for product p at plant kSk,k′,p,m historical quantity shipped between facilities k and k′ of product p via mode mSk,r,p,m historical quantity shipped from facility k to customer record r of product p via mode mTK total time available for production at plant KVCk,p,z variable cost (slope) for linear cost segment for product p at facility k in zone z

VCk,z variable cost (slope) for linear cost segment for total volume through facility k in zone zzmaxk,p maximum number of throughput zones defined for product p at facility k

zmaxk maximum number of throughput zones defined for total volume through facility k

Θmaxi maximum number of facilities used in a user-defined constraint i

Θmini minimum number of facilities used in a user-defined constraint i

ρk,k′,m,p probability that product p should ship between facilities k and k′ via mode mρk,r,m,p probability that product p should ship from facility k to record r via mode mφp,k,k′,m fraction of total product p sent to facility k′ that came from facility k via mode mωk,k′,m,p number of times product p was sent between facilities k and k′ via mode mωk,r,m,p number of times product p was sent from facility k to record r via mode m

Semi-continuous (positive) variablesCk,p throughput cost for product p at facility kCK→K′ total replenishment freightCK→R total customer freightCk site cost for facility kC

logP total logistics cost at plants

Clog,inK total logistics costs for outplants associated with inbound shipments

Clog,outK total logistics costs for outplants associated with outbound shipments

Clog,OutPlantIBk,k′,p total inbound logistics costs for outplant k′ for product p received from facility k

Clog,OutPlantIBr total outplant inbound logistics costs allocated to customer record r

Clog,Plantr total plant logistics costs allocated to customer record r

Clog,PlantDSk,r total logistics cost for plant k direct shipping (DS) to customer record r

Clog,PlantIDSk,k′,p total logistics costs for plant k indirect shipping (IDS) product p to outplant k′

Coutplantr total outplant costs allocated to customer record r

Cout siter total outplant site costs (based on total volume through the outplant) allocated to customer record r

COB Log Custk,r total outbound logistics costs for outplant k shipping to customer record r

COB Log replenk,k′,p total outbound logistics costs for shipping product p from outplant k to outplant k′

COutplant OB Logr total outplant outbound logistics costs allocated to customer record r

Cprodr total production cost allocated to customer record r

Cprodn siter total production site costs (based on total production of the plant) allocated to customer record r

Creplenr total replenishment freight costs allocated to customer record r

freplenk,p total freight cost to replenish product p at facility k

J total network cost: the objective function valueQtot

k,p total flow of product through facility kSk,r,p,m amount shipped from facility k to customer record r of product p via mode mSk,k′,p,m amount shipped from facility k to another facility k′ of product p via mode mStot

k,k′,p total amount of product p transferred from facility k to facility k′

Stotk,r,p total amount of product p transferred from facility k to customer record r

Binary variablesNk,p,z defined as 1 if the volume of product p through facility k exceeds breakpoint Bk,p,zNk,z defined as 1 if the total volume through facility k exceeds breakpoint Bk,z

N ′k,p,z defined as 1 if the volume of product p through facility k is in zone z


N ′k,z defined as 1 if the total volume through facility k is in zone z

Uk defined as 1 if facility k is being usedUk,p defined as 1 if a facility k is handling product pδ a binary variable

Special ordered set variables of type II (SOS2)qk,p,z non-zero if quantity of product p through facility k is near zone zqk,z non-zero if total volume through facility k is near zone z

Transportation costs and mode of transport selection were not specifically modeled, so the main cost opportunities identified werethose associated with DC selection and single sourcing customers from a DC. They presented practical considerations such as modelvalidation and sensitivity analysis to parameter changes. They also discussed the selection of changes to be implemented, but theirapproach is limited to scenario testing. Geoffrion and Graves also demonstrated a special Benders decomposition approach to solvingthe MILP.

In papers published after Geoffrion and Graves, researchers proposed models that quantify other aspects of supply chain networks.For instance, Cohen and Moon (1991) added a raw material vendor echelon and a piecewise linear cost model to capture economiesof scale in the manufacturing echelon. The DC locations were fixed and transportation to customers were ignored. The objectivewas to minimize the combination of fixed and variable production costs, raw material transportation costs and transportation costs tothe distribution centers subject to raw material and finished product capacity constraints, material balances in the network, demandsatisfaction, and assorted network configuration constraints. The model did not include details of logistics and did not explore howthe costs in the supply chain network can be examined to identify the changes that have the highest benefit-to-cost ratio. The authorsintroduced a variant of Benders decomposition to solve the problem.

Arntzen, Brown, Harrison, and Trafton (1995) used a multi-period model for a global supply chain because it not only includedproduction costs, distribution costs, and transportation costs, but it also dealt with duty costs, inventory costs, and import restrictions.The objective function was a weighted combination of cost, duty drawback credits, and time required for processing and transportation.Constraints were used to enforce demand satisfaction, material balances, throughput limits, production limits, inventory limits, anda variety of network configuration limitations. The paper does not deal with the analysis of the results of an optimization, but itreports multi-million cost savings from the use of supply chain network redesigns.

Tsiakis, Shah, and Pantelides (2001) modeled a supply chain composed of manufacturing plants, a warehouse and distributioncenter echelons, and customer zones. The intent of the model was to choose warehouse locations and assign production and networkflow to minimize an objective function comprised of fixed and variable infrastructure costs, production costs, material handlingcosts and transportation costs which are modeled as piecewise linear functions of volume. Constraints in the model include capacitylimits, material balances, flow limits, demand satisfaction, and network structure constraints. The model also included a constrainton the use of shared resources between products at the manufacturing sites. The authors presented the use of a scenario planningapproach to deal with uncertainty in the parameters of the model. However, they did not deal with the issue of network redesignand the need to prioritize proposed changes identified from the optimum design. A commercial solver, rather than decompositionmethods, was used to solve the model.

In this paper we discuss a single period network design MILP model for multi-product supply chains with an arbitrary number ofechelons for distribution centers. Cost as a function of volume is modeled in a piecewise linear fashion for both production sites anddistribution centers. Transportation costs are assigned by mode of transport. Logistics costs are also explicitly modeled. Constraintsin the model enforce demand satisfaction, facility capacity limits, material balances, and network structure constraints. The numberand geographic locations of manufacturing sites and customers is fixed but the model provides the means to choose distributioncenters from a predefined pool of candidates through the use of site selection constraints. The model can be used to minimize thetotal network cost which is a sum of production costs, DC throughput costs, freight costs, logistic costs and fixed costs for selectinga distribution center. The model can also be used to maximize the value of the supply chain network by considering the value ofdemand satisfaction as a component of the objective function.

Our model differs from those reported in the literature in two significant ways. First, we do not lump customer demand intozones but rather deal with individual customer demand. This allows the model to deal with the details of the logistical limitationsrelated to mode of transport at each customer location. These limitations play an important role in determining permissible modes oftransportation which have a significant impact on freight costs. The model is also structured to assign cost components to individualcustomer records. Historical records can be used to fix decision variables in the model so that a base case can be computed to validatethe model and contrast it against the optimized network. The details inherent in this approach also allow the optimization results tobe partitioned and prioritized for implementation.


The remainder of the paper is organized as follows. Section 2 presents the conceptual supply chain model that guided thedevelopment of mathematical model. Section 3 covers the mathematical model we use to determine an optimal design. In Sections4 and 5 we discuss the additional modeling components used to analyze the sources of cost savings attained by implementing theoptimum network. Dealing with uncertainty is covered in Section 6, while Section 7 illustrates the use of our approach on a smallexample problem. Finally in Section 8 we review how the model is being used deliver value to The Dow Chemical Company.

2. Supply chain network overview

The supply chain network considered here is described as a collection of nodes comprising customers, distribution centers (DCs)and production sources connected by transportation lanes for shipping material by truck, rail, vessel or any other defined transportmode (see Fig. 1). The optimization problem is to determine how to satisfy the customer demand using the supply chain networksuch that either costs are minimized or the total value is maximized.

A customer is defined as an entity that has a fixed demand of one single product and resides at a geographic location. A customer isgiven a status of either preferred or non-preferred. A preferred customer’s demand must be met, even if cost of serving that customeris non-optimal. The long-term value for a 100% service level for preferred customers is a business decision. The total supply to anon-preferred customer is allowed to be less than or equal to the customer demand, including zero which would indicate dropping thecustomer altogether. Customers with multiple product demands are broken down into separate entities with single product demand.

Distribution centers (DCs) are storage locations that are geographically separate from the production plants and the customers.Bulk liquid terminals, rail-to-truck transloaders and warehouses are common examples of distribution centers. Each DC has

• a set of viable inbound transport modes (e.g., rail, truck, vessel);• a set of viable outbound transport modes (e.g., rail, truck, vessel);• a piecewise linear cost structure with both fixed and variable components;• maximum volume limitations.

The production sources supply product to the system to satisfy the demand of the customers. Each source has a maximumproduction capacity, but not all may produce every product. Each production source has

• the ability to ship product via various outbound transport modes;• maximum capacity to produce each product in its portfolio;• a linear cost structure with both fixed and variable components.

One measure of the complexity of a supply chain network is the number of echelons or inventory holding places that exist tomove product from the producers to the customers. The general supply chain network of Fig. 1 is capable of representing networksof mixed complexity since it provides for direct shipping from production source to customer, as well as indirect shipping from theDC to customer. It is also capable of representing an arbitrary number of echelons since it provides for creating a chain of DCs tiedtogether by shipping lanes.

A decision support application based on the general supply chain network in Fig. 1 can be used to study a multitude of design issues:

• Amount of demand satisfied for non-preferred customers.• Size and capacity of DCs.• Number of DCs.• Mode of transport for each transportation leg.• Production requirements at sources.• Number of echelons used to serve a customer.

Fig. 1. General supply chain network problem representation.


In practice these decisions are interdependent so that an optimized design includes all of them.While arriving at an optimized network design is essential to reducing costs in a supply chain, it is equally important to be able to

break down the cost savings into its component parts. This information allows decision makers to make more informed choices aboutthe changes that are identified. Given the uncertainty associated with future demand or future costs, not all supply chain modificationsare worth the risk or effort. Or the changes might be staged so that the most beneficial are done first, thereby capturing value earlywhile limiting the downside risk of a complete implementation. The deferred changes can be implemented if the economics holdin the future. The assessment of individual changes has not been explored significantly in the literature. In this paper we presentthe means to assess individual modifications to a supply chain network. Specifically, we provide the incremental savings associatedwith each proposed change to the supply chain network.

3. Basic optimization model

This section describes the translation of the supply chain structure described in Section 2 into a complete mathematical program.Nomenclature is provided at the end.

3.1. Network entities

3.1.1. Customer recordUnderpinning our model is the concept of a customer record which contains the information on finished product demand. The

customer demand record can be uniquely identified by five attributes:

• Customer Name,• Customer Location,• Demand Quantity,• Product Name or ID,• Mode of Transport.

These five attributes represent a unique “customer record” or simply a “record”. This nomenclature allows one to consolidatemultiple pieces of information concerning the customer requirements into one unique container. In this way, the model output issimplified and can be easily referenced in a database. For example, by satisfying “Record 42”, we mean that customer XYZ at theirChicago site received 20 tonnes of product A by truck. Customer records are contained in the set R.

3.1.2. Distribution centerAs stated earlier, bulk liquid terminals, rail-to-truck transloaders and warehouses are common examples of distribu-

tion centers. Distribution centers will be referred to as keys, which is consistent with the enterprise resource planningsystem.

For each product that is processed at a distribution center, a piecewise linear cost structure represents the net cost as a functionof the product throughput. Each line segment is described using a fixed cost (FC) intercept and a variable cost (VC) slope. Fig. 2illustrates a cost function with three separate zones demarked by four breakpoints Bi.

Piecewise linear functions are often used to model nonlinear cost functions, such as the effect of economy of scale: as volumeincreases past a breakpoint, a discount is received. Note that at zero throughput (breakpoint B1) the function exhibits a step and thecost is actually zero.

Fig. 2. General piecewise linear cost function.


3.1.3. Production sourcesThe production sources supply product demanded by the customers. Each source has a maximum capacity for each product it

produces. Not all sources may produce each product. A piecewise linear cost structure identical in form to that described for thedistribution centers and shown in Fig. 2 is used to describe the cost of producing a product. Production sources are synonymous withplants. Plants are also referred to as keys within the enterprise resource planning system. Distribution centers and plants as keys arecontained in the set K.

3.2. Key model variables

The model contains the following sets for enumerating the entities in the supply chain network:

• K: keys, distribution centers or production sources• R: customer records• R*: preferred customer records where demand must be satisfied• P: products• M: modes of transport

The primary decision variables in the model are Sk,r,p,m which specifies the flow of product p from key k to customer record r usingtransportation mode m and Sk,k′,p,m which specifies the flow of product p from key k to another key k′ using transportation mode m.

3.3. Specifying valid transportation lanes

Not all transportation modes are available at all network entities. For instance, a distribution center called a rail-to-truck transloaderis only capable of receiving material contained in a rail car and only capable of sending material out in a truck. Customer records andproduction sources may also have specific limitations. Therefore not all transportation lanes distinguished by mode of transportationwill exist between all network entities. To handle this situation we use a set Acust that defines whether a link (i.e., arc or shipmentleg) is available between a key k and a customer record r:

Acust(K, R, P, M) = (Foutplant(K) ∩ V outcust(K, M) ∩ Lcust(K, R, M) ∩ R

MapM (R, M) ∩ R

MapM (R, P) ∩ P

MapM (P, M))

∪ (Fplant(K) ∩ V outcust(K, M) ∩ Lcust(K, R, M) ∩ F

MapP (K, P) ∩ R

MapP (R, P) ∩ P

MapM (P, M)

∩ RMapM (R, M)) (1)

The multidimensional set Acust is used throughout as a shorthand notation to indicate whether a shipment can occur between afacility and a customer. For a DC k to achieve membership in set Acust requires that:

• Facility k is a member of the outplants Foutplant(K).• DC k can ship via some mode of transport that the customer could receive, dictated by V out

cust and RMapM . R

MapM defines all modes of

transport that the customer could take, and V outcust defines all viable outbound modes that DC k can ship.

• Product p shipped to the customer record is the product required by the customer (dictated by RMapP ).

• Product p can be shipped by mode of transport m (dictated by PMapM ).

• The route from DC k to the customer record via mode m is permitted, dictated by set Lcust(K, R, M).

Plant k will be a member of Acust if

• Facility k is a member of the plants Fplant(K).• The plant k can ship outbound by the customer requested mode of transport m (dictated by V out

cust and RMapM ).

• The plant can produce the customer requested product p (dictated by FMapP and R

MapP ).

• The product can be shipped by the mode of transport (dictated by PMapM ).

• The route from plant k to the customer record via mode m is permitted, dictated by set Lcust(K, R, M).

Set Akey defines whether one facility can ship to another facility:

Akey(K, K′, P, M) = (Foutplant(K) ∩ V outkey(K, M) ∩ V in

key(K′, M) ∩ PMapM (P, M) ∩ Lkey(K, K′, M))

∪ (Fplant(K) ∩ FMapP (K, P) ∩ V out

key(K, M) ∩ V inkey(K′, M) ∩ P

MapM (P, M) ∩ Lkey(K, K′, M)) (2)


Two DCs k and k′ can ship a product to each other if they share a common mode of transport that can be used to ship and receivethe product p. A plant k could replenish a product p to an DC k′ if the plant and DC share a common mode of transport betweenthem, and if the plant can produce the product. Set Lkey specifies all viable movements that should be considered.

3.4. Material balance constraints

3.4.1. Preferred customer material balanceThe demand on every preferred customer record must be satisfied, either by a shipment from a DC, or by a direct shipment from

a production source:∑(k,p,m) ∈ Acust(K,r,P,M)

Sk,r,p,m = Dr ∀r ∈ R∗, (r, p) ∈ RMapP (R, P) (3)

Dr is the demand quantity for the customer record. RMapP (R, P) is a set that maps the product against a given customer record and

ensures that all network flows to this record are written over only this product.

3.4.2. Non-preferred customer material balanceTotal supply to a non-preferred customer must be less than or equal to the customer’s demand:∑

(k,p,m) ∈ Acust(K,r,P,M)

Sk,r,p,m ≤ Dr ∀r ∈ R, r /∈ R∗, (r, p) ∈ RMapP (R, P) (4)

3.4.3. DC material balanceDCs are assumed to operate at constant inventory, which requires that the mass into the DC must equal the mass leaving the DC.

Mass into the DC can arrive from plants as well as from other DCs. Mass leaving the DC can go to customers, as well as to otherDCs: ∑

k′, m ∈ Akey(k, K′, p, M)k /= k′

Sk′,k,p,m =∑

r,m ∈ Acust(k,R,p,M)

Sk,r,p,m +∑

k′, m ∈ Akey(k, K′, p, M)k /= k′

Sk,k′,p,m

∀k ∈ Foutput(K), p ∈ P (5)

3.5. Facility use constraints

3.5.1. Indicating a facility is usedThe big-M formulation defines binary variable Uk to be true if any product flows through the facility with the big-M constant set

equal to the maximum demand of the product:∑k′,m,p ∈ Akey(k,K′,P,M)

Skeyk,k′,p,m +

∑r,m,p ∈ Acust(k,R,P,M)

Scustk,r,p,m ≤

∑p ∈ P

Dmaxp Uk ∀k ∈ K, Uk ∈ [0, 1] (6)

Uk is forced to be false if less than one unit of material is flowing through the facility:∑k′,m,p ∈ Akey(k,K′,P,M)

Skeyk,k′,p,m +

∑r,m,p ∈ Acust(k,R,P,M)

Scustk,r,p,m ≥ Uk ∀K, Uk ∈ [0, 1] (7)

In addition to facility use, it is useful to have another set of indicator variables Uk,p that define whether a facility is handling acertain product. Uk,p is set false if less than one unit of product p flows through facility k and true otherwise:∑

k′,m ∈ Akey(k,K′,p,M)

Skeyk,k′,p,m +

∑r,m ∈ Acust(k,R,p,M)

Scustk,r,p,m ≤ Dmax

p Uk,p ∀k ∈ K, p ∈ P, Uk,p ∈ [0, 1] (8)

∑k′,m ∈ Akey(k,K′,p,M)

Skeyk,k′,p,m +


Scustk,r,p,m ≥ Uk,p ∀k ∈ K, p ∈ P, Uk,p ∈ [0, 1] (9)


3.6. Facility constraints

3.6.1. Facility time limitsThe total time used at a production facility must be constrained by the time available for actual production. This constraint is

useful when multiple plants can produce a wide range of products, or in cases where a plant might have expected downtime andsourcing must be redistributed to other production facilities. The actual production time is calculated using an average productionrate rk,p for product p at plant k:∑

r,m,p ∈ Acust(k,R,P,M)

Sk,r,p,m

rk,p+

∑k′,m,p ∈ Akey(k,K′,P,M)

Sk,k′,p,m

rk,p≤ Tk ∀k ∈ Fplant(K) (10)

The first summation covers all direct shipments from plants to customers, and the second summation covers replenishments to DCs.The total time available for production is given by Tk.

3.6.2. Facility capacity limitsThe total throughput of all products moving through a DC is be constrained between a minimum and maximum limit:

Cmink Uk ≤

∑r,m,p ∈ Acust(K,R,P,M)

Sk,r,p,m +∑

k′,p,m ∈ Akey(k,K′,P,M)∩Foutplant(K′)

Sk,k′,p,m ≤ Cmaxk ∀k ∈ Foutplant(K) (11)

Eq. (11) is also written for each production plant to ensure that the model does not violate the plant’s maximum production capacity.The binary variable Uk defined by Eqs. (6) and (7) is applied to the minimum throughput value Cmin

k so that a facility k can haveeither Cmin

k throughput or greater, or zero throughput.The amount of an individual product flow through the facility can be similarly constrained:

Cmink,p Uk,p ≤


Sk,r,p,m +∑


Sk,k′,p,m ≤ Cmaxk,p ∀k ∈ Foutplant(K), p ∈ P (12)

3.6.3. Site selectionThe number of facilities used in the supply chain can be constrained in general by

Θmini ≤

∑k ∈ Gi(K)

Uk ≤ Θmaxi ∀i ∈ I (13)

where index i is a constraint number to represent multiple constraints, with set Gi(K) being the ith group of facilities whose totalcount is to be constrained, and Θmin

i and Θmaxi are lower and upper limits on the facility count.

For example, consider a network with 12 DCs labeled T1 through T12. Eq. (13) could represent a wide variety of constraints suchas:

• Use at least two of the terminals T1, T2, T3, T4: 2 ≤ UT1 + UT2 + UT3 + UT4 .• Use between two and four of terminals T1, T3, T6, T7: 2 ≤ UT1 + UT3 + UT6 + UT7 ≤ 4.• Use any two of T1, T2 and T3 AND either T8 or T12: 2 ≤ UT1 + UT2 + UT3 ≤ 2, 1 ≤ UT8 + UT12 ≤ 1.

3.7. Network costs

3.7.1. Piecewise linear functionPiecewise linear functions can be modeled conveniently using Special ordered sets of type II (SOS2), which is a set where at

most two adjacent members can be greater than zero. Most commercial solvers internally implement the necessary constraints tosatisfy this property.

An additional zone, called “Zone 0” is added to the piecewise linear function that was previously shown in Fig. 2 to model afixed cost when using SOS2 variables. As shown in Fig. 3, “Zone 0” starts at the origin and extends to breakpoint #1 (B1) at a smallthroughput value of 1 unit. This zone models a rapid jump in cost for a small increase in throughput and simulates a fixed cost. Fig. 3is not to scale: the slope of the first segment in Zone 0 is much larger in reality.

3.7.2. DC throughputThe total throughput through a DC k is defined as∑

z≤zmaxk,p

qk,p,zBk,p,z =∑


Sk,r,p,m +∑


Sk,k′,p,m ∀k ∈ Foutplant(K), p ∈ P (14)


Fig. 3. Cost function structure when using special ordered set formulation.

where qk,p,z is a SOS2 across zone index z, which is multiplied by each user defined breakpoint for each plant–product pair (k, p).The structure of the SOS2 set will force at most two values of qk,p,z to be non-zero for each plant and product pair. The total volumeof product through the plant is given on the RHS of Eq. (14), which is the total of all product going to customers and all productreplenishing other DCs.

3.7.3. Production plant throughputSimilar to the DC throughput, the same equation is written for every product p that a plant k can produce:∑

z≤zmaxk,p

qk,p,zBk,p,z =∑


Sk,r,p,m +∑


Sk,k′,p,m ∀(k, p) ∈ Fplant(K) ∩ FMapP (K, P) (15)

3.7.4. ConvexityThere is no restriction on the magnitude of the SOS2 variables qk,p,z, other than they must be positive. The following convexity

constraint imposes the restriction that the facility throughput expressed in Eqs. (14) and (15) must be equal to a weighted averageof up to two adjacent breakpoints:∑

z≤zmaxk,p

qk,p,z = 1 ∀(k, p) ∈ FMapP (K, P) (16)

3.7.5. Facility costThe SOS2 variables q provide a convenient means for computing the facility cost by taking the weighted sum of the cost function

values at each breakpoint:

Ck,p =∑

z≤zmaxk,p

qk,p,z{FCk,p,z + VCk,p,zBk,p,z} ∀(k, p) ∈ FMapP (K, P) (17)

3.7.6. Replenishment freight costsThe replenishment freight cost for each arc or leg in the network connecting two facilities (either plant to DC or DC to DC) is

calculated as the product of the quantity shipped between two facilities and the replenishment freight rate. The replenishment freightrates are a function of (1) the source facility, (2) the destination facility, and (3) the mode of transport. They could also be set asa function of product as well. The total replenishment freight for the entire network, CK→K′ , is the summation over all potentialreplenishment moves:

CK→K′ =∑

k,k′,p,m ∈ Akey(K,K′,P,M)

Skeyk,k′,p,mf

replenk,k′,m (18)

3.7.7. Customer freight costsThe customer freight cost is the product of the quantity shipped to the customer and the freight rate. For customer demand, freight

rates are calculated either on a rate basis or a shipment basis. Rate-based freight costs are calculated using a $/mass freight rate,while shipment-based costs use a $/shipment rate. The type of freight calculation for customer shipments is based on the mode oftransport (sets Rate(M) and Shipment(M)); for example, all rail freight to the customer might be calculated on a per shipment basis,while truck freight to customers might use a rate basis.


The total customer freight is

CK→R =∑

k, r, p, m ∈ Acust(K, R, P, M)m ∈ Rate(M)

Scustk,r,p,mf

cust,rater,k,m +

∑k, r, p, m ∈ Acust(K, R, P, M)

m ∈ Shipment(M)

(Scust

k,r,p,m

Dr

)Nrf

cust,shipmentr,k,m (19)

For shipment-based freight calculations, the shipment amount to the customer Scustk,r,p,m is a continuous variable and is normalized

using the customer demand on the record. If a facility k fully satisfies customer record r, the second term in Eq. (19) will calculatethe freight using the historical number of shipments to that record (Nr) since the ratio S/D will go to unity.

3.7.8. Plant logistics costsSite logistics cost for outbound shipments at the production facility is broken into two parts: logistical costs associated with

replenishments sent from plant k via mode m to DCs (clog,outk,m,replen) and costs associated with direct shipping from plant k via mode m

to customers (clog,outk,m,cust):

ClogP =

∑k,k′,p,m ∈ Akey(K,K′,P,M)∩k ∈ Fplant(K)∩k′ ∈ Foutplant(K)

Skeyk,k′,p,mc

log,outk,m,replen

+∑

k,r,p,m ∈ Acust(K,R,P,M)∩k ∈ Fplant(K)∩m ∈ Rate(M)

Scustk,r,p,mc

log,outk,m,cust

+∑

k,r,p,m ∈ Acust(K,R,P,M)∩k ∈ Fplant(K)∩m ∈ Shipment(M)

(Scust

k,r,p,m

Dr

)Nrc

log,outk,m,cust (20)

For direct shipping, the structure of the cost equation depends on whether the cost for shipment is determined by shipment or byrate. The last term on the RHS of Eq. (20) accounts for shipment-based costing by taking the total quantity sent to the customer andnormalizing it with respect to the customer demand and applying this fraction to the number of shipments to the customer record r,Nr.

3.7.9. DC inbound logistics costsLogistical cost associated with receiving shipments at DCs is taken as the cost of receiving an inbound shipment at a DC k via

mode m (clog,ink,m ) multiplied by the total amount inbound from all plants and other DCs:

Clog,inK =

⎛⎝ ∑k,k′,p,m ∈ Akey(K,K′,P,M)∩k ∈ Fplant(K)∩k′ ∈ Foutplant(K)

Skeyk,k′,p,m +

∑k,k′,p,m ∈ Akey(K,K′,P,M)∩k,k′ ∈ Foutplant(K),k /= k′

Skeyk,k′,p,m

⎞⎠× c

log,ink,m (21)

3.7.10. DC outbound logistics costsLogistics cost for shipping outbound from a DC k via mode m is separated into costs associated with replenishing other facilities

(clog,outk,m,replen) and shipping to the customer (clog,out

k,m,cust):

Clog,outK =

∑k,k′,p,m ∈ Akey(K,K′,P,M)∩k,k′ ∈ Foutplant(K),k /= k′

Skeyk,k′,p,mc

log,outk,m,replen

+∑

k,r,p,m ∈ Acust(K,R,P,M)∩k ∈ Foutplant(K)∩m ∈ Rate(m)

Scustk,r,p,mc

log,outk,m,cust

+∑

k,r,p,m ∈ Acust(K,R,P,M)∩k ∈ Foutplant(K)∩m ∈ Shipment(m)

Scustk,r,p,m

Dr

Nrclog,outk,m,cust (22)

Similar to the plant logistics costs (see Section 3.7.8) the structure of the logistics cost in Eq. (22) for shipping to the customerdepends on whether the costing method is by shipment or by rate.


3.7.11. Total network costThe total network cost for the supply chain is the sum of the facility product throughput costs, replenishment and customer freight

costs, the production and facility logistics costs, and finally the site cost:

Jcost =∑

(k,p) ∈ Foutplant(K)

Ck,p +∑

(k,p) ∈ Fplant(K)∩FMapP

(K,P)

Ck,p + CK→K′ + CK→R + ClogP + C

log,inK + C

log,outK +

∑k

Ck (23)

One optimization objective is to minimize the total network cost Jcost.

3.7.12. Total network valueThe total value of the network, Jvalue, is the difference between the amount shipped to the customers with each customer record

having a value vr and the total network cost:

Jvalue =∑

(k,r,p,m) ∈ Acust(K,R,P,M)

vrSk,r,p,m − Jcos t (24)

Another optimization objective is to maximize the total network value Jvalue.

4. Modifying the model for historical baseline

The network solution of the MILP model provides the “optimized” supply chain network. This solution is not often analyzedby itself, but instead contrasted with the “historical” network. The most convenient means of evaluating the performance of thehistorical network is to use the same cost structures that are already defined within the model. This evaluation can be accomplishedby imposing the historical network flows into the model structure.

4.1. Using historical data

The historical customer record can be imposed, or fixed, within the existing model structure by setting the customer shipmentsequal to that shipped historically, Sk,r,p,m:

Sk,r,p,m = Sk,r,p,m ∀k ∈ K, r ∈ R, p ∈ P, m ∈ M (25)

In addition, the plant to DC replenishments and also shipments between DCs can also be forced:

Sk,k′,p,m = Sk,k′,p,m ∀p ∈ P, m ∈ M, k ∈ K, k′ ∈ Foutplant(K′) (26)

The actual replenishment data Sk,k′,p,m in Eq. (26) is usually available, but direct application of this constraint would likely resultin an infeasible solution since the model assumes constant inventory, which will not be reflected in the historical data.

Rather than adjust the actual historical replenishments to meet a steady state condition, we instead define a replenishment fractionφp,k,k′,m which is the ratio of the amount of product p shipped between facility k and k′ via mode m to the total amount of product pshipped to facility k′:

φp,k,k′,m = Sk,k′,p,m∑k′′,m′ Sk′′,k′,p,m′

∀p ∈ P, k ∈ K, (k′′, m′) ∈ V outkey(K′′, M ′), (k′, m′) ∈ V in

key(K′, M ′) (27)

Note that φp,k,k′,m in Eq. (27) is calculated using historical data. This constraint is imposed in the model as

φp,k,k′,m ∗∑k′′,m′

Sk′′,k′,p,m′ = Sk,k′,p,m ∀p ∈ P, k ∈ K, (k′′, m′) ∈ V outkey(K′′, M ′), (k′, m′) ∈ V in

key(K′, M ′) (28)

Thus by forcing the amount shipped to the customers, along with the DC replenishment fractions and the steady state constraint, theentire network will be forced to duplicate the historical supply chain.

4.2. Base case model

The model is run under a more limited set of constraints when attempting to duplicate history. Material balance constraints, Eqs.(3)–(5), and freight and logistics, Eqs. (18)–(22), impose sufficient constraints to emulate the historical, or “base case” network.Omitted from the equation list are the facility capacity limits, facility time limits, facility use, and site selection constraints. Theobjective function used for base case runs is usually cost minimization (Eq. (23)).


Since the shipment variables S to the customer are fixed, the optimization is doing little more than calculating the total cost of thenetwork. This output, however, provides a valuable baseline reference since the freight and DC cost structures are applied in exactlythe same way for both a baseline run and an optimization run.

5. Customer record cost allocation

This section provides the equations used to allocate various aggregate network costs to a single customer record. This costinformation allows one to make record by record comparisons between the base and optimized network. For example, the costopportunity in the optimized network can be directly contrasted with the base case costs by individual cost components such asproduction cost, replenishment costs, facility throughput costs, logistics costs and overall site costs. The equations provided in thissection are solved after the optimization is complete.

The equations involved can become cumbersome in very complex network arrangements. It is assumed in all sub-sequent equations that the most complex arc in the network will consist of no more than three DCs between theproduction facility and the customer. In other words, a product can move at most through a supply chain suchas

plant → DC1 → DC2 → DC3 → customer

Movements using four or more DCs are considered rare and are neglected here, although the equations can be extendedto accommodate these scenarios if necessary. Although the optimal solutions do not normally include this many trans-actions, the historical case might and thus requires inclusion of those costs. In addition, note that the equations thatfollow are really for reporting purposes only, and the actual constraints in the model are not limited at all to the abovestructure.

5.1. Intermediate totals

Various intermediate sums used in subsequent allocation equations are defined below:

5.1.1. Total product leaving a facilityQtot

k,p =∑k′,m

Sk,k′,p,m +∑r,m

Sk,r,p,m (29)

The total product leaving a facility is equal to the total quantity sent to other DCs as replenishments and the total quantity goingto customers.

5.1.2. Total product to a customer from facility kThe total product a customer receives from a facility is the sum of all products by each viable transport mode:

Stotk,r,p =

∑m

Sk,r,p,m (30)

Typically, the customer will receive product in only one mode, but the model is general enough to handle multiple modes eitherin the optimized solution or the historical data.

5.1.3. Total product replenishment between facilitiesThe total product moving between two facilities (plant to DC, or DC to DC) is given by summing over all transport modes between

the facilities:

Stotk,k′,p =

∑m

Sk,k′,p,m (31)

5.1.4. Total product replenishment freight costs for a DCThe total product replenishment freight for a facility is the sum of material inbound times the replenishment rate:

freplenk,p =

∑k′

∑m

Sk′,k,p,mfreplenk,k′,m (32)


5.1.5. Plant logistics cost for direct shipment to customerThe logistics costs for direct shipment (DS) from plant k to customer record r is

Clog,PlantDSk,r =

∑p,m ∈ Acust(K,R,P,M)∩k ∈ Fplant(K)∩m ∈ Rate(M)

Sk,r,p,mclog,outk,m,cust

+∑

p,m ∈ Acust(K,R,P,M)∩k ∈ Fplant(K)∩m ∈ Shipment(M)

(Sk,r,p,m

Dr

)Nrc

log,outk,m,cust (33)

The first term in the RHS is used when the transport mode is rate based, and the second term is used when the transport mode isshipment based.

5.1.6. Plant logistics cost for indirect shipment to customerThe logistics costs for indirect shipping (IDS) from plant k to an DC k′ is found by summing over all transport modes from the

plant to the DC:

Clog,PlantIDSk,k′,p =

∑m ∈ Akey(K,K′,P,M)

Sk,k′,p,mclog,outk,m,replen ∀k ∈ Fplant(K), k′ ∈ Foutplant(K), p ∈ F

MapP (K, P) (34)

5.1.7. DC inbound product logistics cost from one facility to anotherThe logistics cost for each DC k′ receiving inbound product volume from either other plants or other DCs k is found by summing

overall all viable transport modes from the upstream facility:

Clog,OutPlantIBk,k′,p =

∑m ∈ Akey(K,K′,P,M)∩(k,k′) ∈ K

Sk,k′,p,mclog,ink′,m (35)

5.1.8. DC outbound product logistics costs to customersDC logistics cost from a DC k to a customer record r is calculated by summing over all transport modes to the customer:

COB Log Custk,r =

∑p,m ∈ Acust(K,R,P,M)∩k ∈ Foutplant(K)∩m ∈ Rate(m)

Scustk,r,p,mc

log,outk,m,cust

+∑

p,m ∈ Acust(K,R,P,M)∩k ∈ Foutplant(K)∩m ∈ Shipment(m)

Scustk,r,p,m

Dr

Nrclog,outk,m,cust (36)

The first term is used when the transport mode is rate based, and the second term is used when the transport mode is based pershipment.

5.1.9. DC outbound product logistics cost for replenishmentsOutbound logistics cost is calculated by summing over all transport modes between two DCs:

COB Log replenk,k′,p =

∑m ∈ Akey(K,K′,P,M)

Skeyk,k′,p,mc

log,outk,m,replen ∀(k, k′) ∈ K, p ∈ P (37)

5.1.10. Production cost on the customer recordThe production cost on the customer record for a direct shipment from a plant k is computed by taking the fraction of the total

plant production that is sent to the customer record and applying that to the net production cost. In the case of indirect shipments(plant to DC to customer), the production cost is diluted further by taking the fraction of the DC throughput to the customer, thenthe fraction of the plant to the DC and applying both of these to the net production cost. Replenishments using more than one DCfollow the same approach.

Eq. (38) below consists of four terms. The first term represents the production cost for a direct shipment from plants to a customerrecord. The second term represents a plant to DC to customer movement, the third term represents a plant to DC to DC to customer,


and the fourth a plant to DC to DC to DC to customer:

Cprodr =

∑k ∈ Fplant(K)∩F

MapP

(K,p)

Stotk,r,p

Qtotk,p

Ck,p +∑

k ∈ Fplant(K)∩FMapP

(K,p)

∑k′ ∈ Foutplant(K)

Stotk,k′,pStot

k′,r,pQtot

k,pQtotk′,p

Ck,p

+∑


(K,p)


∑k′′ ∈ Foutplant(K)

Stotk,k′,pStot

k′,k′′,pStotk′′,r,p

Qtotk,pQtot

k′,pQtotk′′,p

Ck,p

+∑


(K,p)



∑k′′′ ∈ Foutplant(K)

×Stot

k,k′,pStotk′,k′′,pStot

k′′,k′′′,pStotk′′′,r,p

Qtotk,pQtot

k′,pQtotk′′,pQtot

k′′′,pCk,p ∀(r, p) ∈ R

MapP (R, P) (38)

5.1.11. DC costs on the customer recordThe aggregate cost of routing product through terminals is calculated by taking the fraction of DC throughput that goes to the

customer and applying it to the DC cost. For more complex networks where an upstream replenishing DC “A” replenishes a DC “B”serving the customer, the fraction of the replenishment quantity from “A” to “B” is applied to incorporate the “B” DC cost.

Eq. (39) gives the total DC cost (for up to three DCs in the chain) for supplying record r with product p:

Coutplantr =

∑k ∈ Foutplant(K)

Stotk,r,p

Qtotk,p

Ck,p +∑

k ∈ Foutplant(K)


Stotk,k′,pStot

k′,r,pQtot

k,pQtotk′,p

Ck′,p

+∑

k ∈ Foutplant(K)



Stotk,k′,pStot

k′,k′′,pStotk′′,r,p

Qtotk,pQtot

k′,pQtotk′′,p

Ck′′,p ∀(r, p) ∈ RMapP (R, P) (39)

5.1.12. Replenishment freight on a customer recordThe customer replenishment freight is the total cost to replenish each DC in the network that provides product to the customer.

The mass fraction of product leaving the DC to the total product throughput is applied to the total replenishment cost at each DC:

Creplenr =


Stotk,r,p

Qtotk,p

freplenk,p +



Stotk′,k,pStot

k,r,p

Qtotk′,pQtot

k,p

freplenk′,p

+∑

k ∈ Foutplant(K)



Stotk′′,k′,pStot

k′,k,pStotk,r,p

Qtotk′′,pQtot

k′,pQtotk,p

freplenk′′,p ∀(r, p) ∈ R

MapP (R, P) (40)

The three terms in Eq. (40) are the replenishment costs for a single, double, and triple DC replenishment chains.

5.1.13. DC site costs on the customer recordThe DC site cost is allocated to the customer record by taking the fraction of product in the DC total volume that goes to the

customer multiplied by the total site cost of facility k. Replenishment chains involving two DCs allocate the furthest upstream DCsite cost by applying first the fraction of that DC’s total volume that replenishes the downstream DC and then the fraction of DCproduct volume that went to the customer:

Cout siter =


Stotk,r,p∑pQtot

k,p

Ck +∑

k ∈ Foutplant(K)


Stotk,r,pStot

k′,k,pQtot

k,p′∑

pQtotk′,p

Ck′

+∑

k ∈ Foutplant(K)



Stotk,r,pStot

k′,k,pStotk′′,k′,p

Qtotk,p′Qtot

k′,p′∑

pQtotk′′,p

Ck′′ ∀(r, p′) ∈ RMapP (R, P), p ∈ P (41)

The three terms in Eq. (41) correspond to the single, double and triple DC replenishment schemes.

5.1.14. Production site costs on the customer recordThe production site costs for a direct ship customer is simply the fraction of the total production that is shipped to the customer

multiplied by the plant site cost. For situations where a single DC is used, the site cost is allocated by applying the fraction of site


production that was used to replenish the DC and the fraction of the total DC product volume that went to the customer. Double andtriple DC replenishment schemes follow analogously:

Cprodn siter =

∑k ∈ Fplant(K)

Stotk,r,p′∑

p ∈ FMapP

(K,P)

Qtotk,p

Ck +∑

k ∈ Fplant(K)


Stotk,k′,pStot

k′,r,p′

Qtotk′,p′∑

p ∈ FMapP

(K,P)Qtot

k,p

Ck

+∑

k ∈ Fplant(K)



Stotk,k′,p′Stot

k′,k′′,p′Stotk′′,r,p′

Qtotk′,p′Qtot

k′′,p′∑

p ∈ FMapP

(K,P)Qtot

k,p

Ck

+∑

k ∈ Fplant(K)




Stotk,k′,p′Stot

k′,k′′,p′Stotk′′,k′′′,pStot

k′′′,r,p′

Qtotk′,p′Qtot

k′′,p′Qtotk′′′,p′

∑p ∈ F

MapP

(K,P)Qtot

k,p

Ck

∀(r, p′) ∈ RMapP (R, P), p ∈ P (42)

5.1.15. Plant outbound logistics costs on the customer recordThe plant logistics cost is separated into direct (plant to customer) and indirect (plant to DCs and then to customer). The direct

cost is a summation across all plants that service the customer record, while the indirect costs must be allocated using variousnetwork flows. For a single DC between the plant and customer, the plant logistics cost is allocated by multiplying with the ratio ofthe customer demand to the total product throughput at the DC. For multiple DCs, the ratio of each DC to DC product flow to thefirst DC’s total product volume is used to allocate the costs through the network:

Clog,Plantr =

∑k ∈ Fplant(K)

Clog,PlantDSk,r +

∑k ∈ Fplant(K)


Stotk′,r,p

Qtotk′,p

Clog,PlantIDSk,k′,p

+∑

k ∈ Fplant(K)



Stotk′,k′′,pStot

k′′,r,pQtot

k′,pQtotk′′,p


+∑

k ∈ Fplant(K)






Qtotk′,pQtot

k′′,pQtotk′′′,p


∀(r, p) ∈ RMapP (R, P) ∩ F

MapP (K, P) (43)

The first term on the RHS in Eq. (43) is for direct shipments, and the remaining three terms account for single, double and tripleDC replenishment chains.

5.1.16. DC inbound logistics costs on the customer recordThe inbound logistics costs for a single DC replenishment chain is allocated to the customer record by multiplying the DC inbound

cost with the ratio of the customer demand to the total inbound product flow of the DC. For multiple DC chains, the same principleapplies for the first plant to DC leg, and subsequent legs in the chain are calculated by taking the ratio of downstream demand tototal product inflow and multiplying with the corresponding DC’s total product inbound cost:

Clog,OutPlantIBr =

∑k ∈ Fplant(K)

∑k′∈Foutplant(K)

Stotk′,r,p

Qtotk′,p

Clog,OutPlantIBk,k′,p +

∑k ∈ Fplant(K)




k′′,r,pQtot

k′,pQtotk′′,p

Clog,OutPlantIBk,k′,p

+∑

k ∈ Fplant(K)






Qtotk′,pQtot

k′′,pQtotk′′′,p


+∑

k ∈ Foutplant(K)


Stotk′,r,p

Qtotk′,p


+∑

k ∈ Foutplant(K)




k′′,r,pQtot

k′,pQtotk′′,p

Clog,OutPlantIBk,k′,p ∀(r, p) ∈ R

MapP (R, P) (44)

The first three terms in Eq. (44) account for the inbound logistics costs for plant to DC replenishments for single, double and tripleDC replenishment chains. The fourth term accounts for DCs one level upstream from the customer, and the final term accounts for


DCs two levels upstream from the customer.

5.1.17. DC outbound logistics cost on the customer recordThe DC outbound logistics cost is allocated to each customer by

COutplant OB Logr =


COB Log Custk,r +



Stotk′,r,p

Qtotk′,p

COB Log replenk,k′,p

+∑

k ∈ Foutplant(K)




k′′,r,pQtot

k′,pQtotk′′,p

COB Log replenk,k′,p (45)

The first term in Eq. (45) is the outbound logistics cost for the last leg to the customer. The second term accounts for logistics cost indouble DC replenishment chains by taking the outbound cost between the DCs in the chain and applying the ratio of the customerdemand to the total product flow from the second DC. The third term allocates the outbound costs between the first and the secondDCs in triple DC replenishment chains to the customer.

6. Dealing with uncertainty

Uncertainty is encountered regularly in the application of the MILP model to practical situations. Because of changing marketconditions, future demand always has a level of uncertainty associated with its predicted value. Cost parameters of the network arealso uncertain due to varying contract terms and the inability of accounting systems to track some details. For these reasons, customerdemand, freight rates and facility costs are usually uncertain. Brute force techniques like Monte Carlo can be used to ascertain theeffects of parameter uncertainties for many models that are not too large. The technique involves repeatedly selecting a value of theparameter from a prescribed distribution, solving the model to an acceptable integrality gap, and reporting the results with supportingstatistics. Although this method can take time, the computational capabilities available today are often sufficient to tackle manypractical problems. Neglecting the uncertainty in supply chain models can lead to erroneous conclusions as demonstrated in theexample presented later in this paper.

The output of the Monte Carlo optimization runs provides two pieces of useful information. First, probability estimates that agiven transportation lane is actually used can be defined as

ρk,k′,m,p ≈ ωk,k′,m,p

Nor ρk,r,m,p ≈ ωk,r,m,p

N(46)

where ωk,k′,m,p and ωk,r,m,p are the number of times the decision variables Sk,k′,p,m and Sk,r,p,m are non-zero and N is the totalnumber of Monte Carlo runs. This probability reflects the robustness of a sourcing change under the uncertainties in the model.Higher probabilities offset the expected burden of making the change with the expectation of long-term success.

Second, when a transportation lane is utilized, Monte Carlo simulations provide estimates of the underlying distributions. Thesedistributions, along with the lane usage probability, can help decision makers identify “sure bets” that should be implemented, othersupply chain changes that may need further investigation, or those changes that offer little certainty of a return and can be discarded.

7. Example supply chain

Fig. 4 illustrates a simple, contrived supply chain on a peninsula on a deep lake capable of supporting vessel traffic. There are sixcustomer locations, denoted as customer records. Each customer record is a point demand of a product at a location. There are twoproducts, products A and B, which are manufactured by only two plants, the Calumet plant and the Dollar Bay plant. The landlockedCalumet plant is limited in that it can only ship outbound via rail and truck. The Dollar Bay plant is positioned at a port, and canship by rail, truck and vessel.

There are three terminal facilities that can be used to store and distribute products A and B: the Allouez terminal, the Eagle Riverterminal and the Keystone terminal. The Allouez terminal is landlocked and can receive and ship products in rail and truck. TheKeystone and Eagle River terminals are on the lake, and can receive and ship products in rail, truck and vessel.

7.1. Customer demand

The customer demand is given in Table 1. Each row in the table constitutes a customer record, representing an aggregate ofshipments that occurred over a year. For example, Record #1 represents a total of 100 units of truck shipments of product A fromthe Keystone terminal while Record #2 represents 500 units of direct truck shipments of product B right from the Calumet plant.


Fig. 4. Supply chain example on a Peninsula.

Table 1Customer product demand with historical (base case) source

Record # Customer ID Historical shipping source Product Customer preferred modeof transport (MOT)

Year’s demand

1 Smith’s Fisheries Keystone terminal A Truck 1002 CliffDyne Calumet plant B Truck 5003 ThimbleBerry Tech Eagle River terminal A Truck 3004 Phoenix Ltd. Allouez terminal B Truck 5005 Copper Haven Calumet plant A Rail 6006 Linden Industries Dollar Bay plant B Truck 400

7.2. Base case replenishment fractions

As discussed in Section 4, the historical replenishments into the network can be specified as fractions. In this example, wewill assume that the Dollar Bay plant has historically utilized the terminal network, shipping product A exclusively by rail toKeystone and product B by rail to Allouez, while the Calumet plant always ships directly to customers. In addition, there have beensome rail replenishments of product A from Keystone Terminal to Eagle River. The historical data can be formulated for use inEq. (27) as

φA,Dollar Bay,Keystone,rail = 1, φA,Keystone,Eagle River,rail = 1, φB,Dollar Bay,Allouez,rail = 1

7.3. Customer freight costs

Currently all customers prefer to receive product by truck, with the exception of Record #5 which prefers rail. This example willnot consider changing the customer’s preferred mode of transport. Both truck and rail freight rates (f cust,rate

r,k,m ) from each potentialsource to each potential customer are given in Table 2.

Table 2Truck and rail freight rates ($/unit of mass) to the Customer

Source Customer record

1 (trucka) 2 (trucka) 3 (trucka) 4 (trucka) 5 (raila) 6 (trucka)

Calumet plant 0.93 0.58 0.65 0.81 0.20 0.69Dollar Bay plant 0.89 0.67 0.97 0.65 0.18 0.51Allouez terminal 0.75 0.35 0.55 0.78 0.13 0.40Eagle River terminal 0.25 0.39 0.86 0.23 0.05 0.33Keystone terminal 0.40 0.87 0.95 0.83 0.08 0.48

a Preferred MOT.


Table 3Rail and vessel replenishment rates

Destination Source

Calumet plant Dollar Bay plant Allouez terminal Eagle River terminal Keystone terminal

Rail replenishment rates ($/unit mass)Allouez terminal 0.25 0.15 Exclude 0.18 0.35Eagle River terminal 0.33 0.25 0.18 Exclude 0.90Keystone terminal 0.50 0.35 0.35 0.90 Exclude

Vessel replenishment rates ($/unit mass)Allouez terminal Exclude Exclude Exclude Exclude ExcludeEagle River terminal Exclude 0.10 Exclude Exclude 0.07Keystone terminal Exclude 0.08 Exclude 0.07 Exclude

7.4. Replenishment freight costs

The replenishment rates (f replenk,k′,m) for both rail and vessel movements are given in Table 3. Note that rates are also provided for

movements that did not happen historically, such as replenishing the terminals from the Calumet plant. The rates apply for bothproducts A and B; if there was a difference, then different MOTs can be assigned (e.g., Rail A, Rail B, etc.).

Some movements are marked as “Exclude”, which prevents any shipment between the source and destination by that mode. Inthe example given, we exclude movements from the terminal back to itself, and also movements by vessel to the landlocked Allouezterminal. These exclusions are reflected by omitting them from the set Lkey.

7.5. Production and distribution center costs

The production and DC costs are modeled using a piecewise linear cost function as shown in Fig. 5. Each facility has a fixed cost,so the y-intercepts of the cost functions are non-zero. If the facility is not used at all, then the fixed cost is not applied.

The maximum production of A at both the Calumet and Dollar Bay plants is 1000, and for B it is 1400 units. The maximumthroughput of A and B in any terminal is also set to 1000 and 1400 units, respectively.

7.6. Optimization results and analysis

The model was used to optimize this supply chain network to achieve minimum total cost which is the sum of all freight costs,production costs and DC costs. Logistical costs and costs based on total facility throughputs were not used. The overall cost summarycomparison between the historical network and the optimized network is given in Table 4.

The demand and freight rates given in Tables 1–3 were used in “Scenario 0”. An additional 2000 scenarios were runusing uncertain demand and freight rates. The demand can increase or decrease in the future, so that was varied byup to ±20% for every record using a uniform distribution. In addition, it is possible that freight rates to the customers(Table 2) might increase in the near future due to rising fuel costs, so these were varied by up to +10% using a uni-form distribution. The vessel replenishment rate from Dollar Bay to Eagle River (see Table 3) may also increase from

Table 4Supply chain cost comparison: historical vs. optimal

Cost item Historical Optimal (Scenario 0,deterministic case)

Optimal (Scenarios 1–2000)with 95% confidence interval

�(optimal Scenario0 − historical)

Plant costs $515 $348 $375 ± 1 $(166)Freight cost, DC to customer ($) $461 $215 $272 ± 2 $(246)Freight cost, plant to DC ($) $215 $90 $101 ± 3 $(125)Freight cost, DC to DC ($) $270 – – $(270)Freight cost, plant to customer ($) $881 $653 $655 ± 3 $(228)Site logistics, plant outbound ($) – – – –Site logistics, DC inbound ($) – – – –Site logistics, DC outbound ($) – – – –Plant site costs ($) – – – –DC costs ($) $146 $195 $178 ± 1 $49DC site cost ($) – – – –

Total costs ($) $2488 $1501 $1581 ± 5 $(987)


Fig. 5. Terminal and production cost functions.

anywhere to $0.10 up to $0.15. The average cost values for these scenarios are reported in Table 4 as 95% confidence inter-vals.

Table 4 shows that overall cost for the optimal network in Scenario 0 is $987 less than the current supply chain network. If theaverages from Scenarios 1 to 2000 are used, the overall savings is $907 which is quite close to the deterministic case. The greatestreductions in cost occurred by reducing the direct shipments (plant to customer, $228), the indirect shipments (DC to customer,$246), the inter-DC movements (DC to DC, $270) and the production costs ($166). The DC costs increased only slightly comparedto greatly decreased freight costs.

Each of these cost elements will now be analyzed in greater detail to determine how the network was changed to achieve thesesavings. The analysis will contrast how each customer was sourced historically against the optimal sourcing strategy with supportingeconomics. This approach can be used to identify the key sourcing changes that will yield the greatest savings.

7.6.1. Deterministic case: Scenario 0A record report for the Peninsular Supply Chain Problem is provided in Table 5 for the deterministic case, Scenario 0. Each row

in the table indicates how a customer record was sourced historically, and how it should be sourced to minimize total supply chaincost. The records are sorted from the largest opportunity to the smallest.

For example, the first row in Table 5 indicates that Record #3 was historically sourced by using the route “DKE”,where each single letter is the first letter of a facility, meaning “Dollar Bay plant to Keystone terminal to Eagle River ter-minal”. The first letter of the route will always be a production facility, followed by zero or more possible distributioncenters. Although not shown in the table, the customer in Record #3 receives material only by truck, which the optimizerwas not allowed to change in this example. The modes of transport for replenishment are also reported in Table 5, sincethese can be changed by the optimizer. In the case of Record #3, both distribution centers were historically replenished byrail.

In the optimal network, Record #3 is proposed to be direct sourced from the Dollar Bay plant (D). The last five columns on theright of the table are the individual changes in the economic components of the supply chain costs that would be incurred if thissourcing change is implemented. The cost data is derived using the equations from Section 5. An economic opportunity is defined

J.Ferrio,J.Wassick

/Com

putersand

Chem

icalEngineering

32(2008)

2481–25042501

Table 5Record report for the Peninsular Supply Chain Problem: deterministic Scenario 0

Record no. Product Historicalroute

Historical source wasreplenished using anMOT of

Optimal route Optimal sourcewas replenishedusing an MOT of

Change in lastleg freight(optimal-base)

Change inreplenishmentfreight(optimal-base)

Change in DC costs(optimal-base)

Change inproduction costs

Sum of netopportunity

3 A DKE Rail D Ship Direct $54 ($375) ($85) ($23) ($429)5 A C Ship Direct D Ship Direct ($12) $0 $0 ($136) ($148)6 B D Ship Direct DE Vessel ($256) $40 $87 $0 ($129)2 B C Ship Direct DE Vessel ($290) $50 $108 $7 ($125)4 B DA Rail C Ship Direct $50 ($75) ($49) ($7) ($81)1 A DK Rail D Ship Direct ($20) ($35) ($13) ($8) ($75)

Grand total ($474) ($395) $48 ($166) ($987)


Table 6Comparison of cost savings between deterministic and stochastic optimization cases

Record no. Historical route Optimal route Net cost savings ($)

Deterministic Stochastic Frequency of occurrence (%) Deterministic Stochastic (mean)

3 DKE D D 100 429 420

5 C DD 86

148145

D/C 14 102

6 D DEDE 51

129127

DK 49 110

2 C DE DE 100 125 105

4 DA DC 99

8163

DA 1 43

1 DK D

DA 31

75

71D 21 77C/D 19 25D/C 15 30C 12 33DA/C 2 25

as that found in the optimal case minus the value in the historical case, so a net negative economic change is desirable since thisrepresents a reduction in supply chain cost.

Continuing with the row for Record #3, the change in last leg freight is $54, meaning it would actually cost more if we sourcedirect from plant. However, there is a large savings in replenishment freight of $375, since direct sourcing means no longer havingto rail from the plant to Keystone Terminal and then rail again to Eagle River Terminal. The elimination of this product at the DCsreduce the costs by $85, and the cost of producing this material is decreased by $23. The net opportunity is the sum of the freightand DC cost components, or $429.

One might question why the production cost decreased for Record #3 since the same amount of material is sent to the cus-tomer, and it still originates from the same source, Dollar Bay. The reason is that other changes have been proposed by theoptimizer that are not clearly visible in the record report. Historically, Dollar Bay produced a total of 400 units of A, of which300 went to satisfy Record #3. Producing 300 units of A at Dollar Bay costs a total of $70, so 3/4 of that, or $53, is historicallyattributed to Record #3. In the optimal case, Dollar Bay now produces 1000 units of A, at a total cost of $100. The productioncost attributed to Record #3 is now 300/1000, or 3/10 of $100, or $30, giving a net decrease of $53–30 or $23 in savings. Theproduction cost reduction is therefore attributed to diluting the cost by producing more of material A at Dollar Bay in the optimalnetwork.

The record report rigorously attributes costs throughout the supply chain to each individual customer. As a result, the net sum ofall the cost components in Table 5 is exactly equal to the total aggregate savings as shown in Table 4, or $987.

7.6.2. Stochastic case: Scenarios 1–2000Table 6 summarizes the differences between the stochastic and deterministic optimizations. Each record in the table compares

the historical route against the optimal route found in Scenario 0 (the deterministic case) and all the routes found from Scenarios 1to 2000.

Table 6 indicates that direct sourcing Record #3 from Dollar Bay is by far the greatest opportunity with a $429 savings in thedeterministic case. With uncertainty in demand, customer freight rates and a possibly higher vessel replenishment rate from DollarBay to Eagle River, all 2000 scenarios still showed direct sourcing as the best option. The mean of all these scenarios was $420,which is very close to that found in the deterministic case.

The source of Record #5 was changed to Dollar Bay in the deterministic case, and was also favored 86% of the time in thestochastic case. Fourteen percent of the cases suggested that both Dollar Bay and Calumet (shown as D/C in Table 6) source tothis record. The complexity and low frequency of this dual-source arrangement precludes this route. Both the deterministic andstochastic cases indicate nearly identical savings of $148 and $145, respectively.

Record #6 indicates moving this customer from a direct sourcing arrangement from Dollar Bay to indirectly sourcing throughEagle River terminal. Under uncertainty, this is favorable 51% of the time, but 49% of the time Keystone terminal is favored. Theeconomic savings is $110 using Keystone, only slightly less than $125 using Eagle River. This is the only record where Keystoneterminal is ever proposed in the optimal network, and is a result of the uncertainty in the vessel rate replenishment rate from DollarBay to Eagle River. As Keystone is so infrequently used, it may be warranted to close it or to conduct more detailed analysis. Inaddition, the optimal network also proposes to source Record #2 from Eagle River with a 100% occurrence rate, so if the business


Fig. 6. Optimized supply chain under uncertainty.

intent is to reduce the complexity and other unmodeled costs of the network, Eagle River would be used to source both records 2and 6.

The recommendation for Record #4 is to direct source from Calumet plant rather than Dollar Bay plant with a 99% occurrencerate, with only 1% of the scenarios actually retaining the original configuration.

Finally, the results for Record #1 are clear for the deterministic case: direct source from the Dollar Bay plant. However, thestochastic runs suggest a wide array of routes due to the uncertainty in our freight rates. The most frequent route is Dollar Bay toAllouez (DA) at 31% which disagrees with the deterministic result which was favored only 21% of the time. The remaining routesare of lower frequency, and also much lower savings. Whether we choose to direct source or use Allouez, the savings are around$71–77, while any other proposed route is about half that, ranging from $25 to $33.

7.6.3. Summary of proposed changes to the Peninsular Supply ChainThe output from an optimization model can suggest numerous changes to the supply chain with a wide range of economic impact.

Typically only those changes that represent the largest opportunities are actually implemented to avoid major disruptions to thesystem. The record reports presented in Tables 5 and 6 rank the opportunities for changing a customer’s sourcing and can be helpfulin identifying the key changes that will make the largest impact.

Based on the previous discussion, a proposed optimal network that will provide economic savings even with demand and freightuncertainty is given in Fig. 6. In this network, most of highest frequency routes in Table 6 have been applied, with the exceptionof Record #1, where it was chosen to simply direct source this from the Dollar Bay plant and avoid using a terminal. Although theoptimization suggests using the terminal might result in slightly lower costs, the unmodeled benefit of a simpler network is usuallyfavored—and the sensitivity regarding the sourcing record as demonstrated by the six alternative routes posed in Table 6 supportsthis conclusion.

Although the record report provides a ranked summary of opportunities, one must keep in mind that changes in the networkcannot always be made individually and expect to receive the reported savings. The network as a whole has been optimized, andchanges may have occurred in other records to achieve that savings. For example, one record may have been sourced to a terminal,but so may have 50 other customers to the same terminal, perhaps resulting in a higher throughput to receive a volume discount.Changing only one record in isolation may not provide savings.

However, the network flows presented in Tables 5 and 6 provide a good overall picture to help understand the trends, andhow sensitive a route is under uncertainty. The top opportunities in the record report are often very good opportunities. In

Table 7Supply chain network dimensions

Minimum Average Maximum

Customer records 50 500 7000+Distribution centers 0 10 80Production plants 1 4 80Products 1 25 500


any event, it is good practice to decide which changes the business would like to pursue, and then propose the new net-work to the model in the same way the historical base case was run to ensure feasibility and to calculate the actual costsavings.

8. Business results

During the past several years the approach described in this paper has been used to improve the supply chain networks of nearlyall the business units in The Dow Chemical Company. The value captured so far is around 40 MM$ using 10-year NPV basis. Theemphasis has been the cost reduction of existing networks.

Supply chain networks with a wide range of dimension have been analyzed (Table 7). Smaller networks involve a single businessunit in one particular geographic region, such as a specialty chemical business in Europe. Extremely large networks covered multiplebusinesses with similar products, such as all bulk liquid products in North America. Generally each network is analyzed on an annualbasis but more frequent analysis is used for sudden large changes in market conditions or logistic costs.

The tool used by the supply chain analysts is based on the model implemented in GAMS/CPLEX MILP solver. A customized MSExcel spreadsheet is used to extract the relevant data from the enterprise resource planning system, to analyze and consolidate thedata, and to populate the model parameters. Supply chain analysts generally run the model on their own workstations to optimize thenetwork of a specific business but sometimes use a multi-processor server with a multi-thread version of CPLEX for enterprise-widenetworks. Solve times for a single scenario range from several minutes to several hours. For the 2000-scenario supply chain describedin Section 7, the solve time was 4.5 min using GAMS/CPLEX on a dual core 2004 MHz AMD OpteronTM 64-bit processor with16 GB RAM.

9. Concluding remarks

In this paper we have presented a single period network design MILP model for multi-product supply chains. The networkis comprised of production plants, an arbitrary number of echelons of distribution centers and customer locations. The model,containing logistics costs, is more detailed than is typically reported in the literature. The main use of the model is for identifyingcost-cutting opportunities in existing supply chain networks. To that end, the model does not consolidate customer demand intozones as is typically done. Instead, the model works with individual customer records to support (1) the quantification of an accuratebase case scenario using historical records, (2) the modeling of individual logistic capabilities (and cost) of each customer, and (3)the partitioning and prioritization of optimization results for implementation.

To support (3) above, the model contains novel cost calculations that assign network cost to individual customer records. Thesecalculations, used in combination with Monte Carlo base case simulation/optimization, provide decision makers with improvedinsight regarding all of the proposed changes resulting from a network optimization. This approach emphasizes making only thosechanges that provide sure benefit relative to the cost of implementation or the risk of uncertainty. We provided a simple contrivedexample to illustrate these concepts.

The utility of the model has been demonstrated by many applications to real world problems of varying size and complexity.Using the opportunities identified from supply chain network optimizations, The Dow Chemical Company has produced significantcost reductions in its operations.

As we stated in the beginning there are numerous value creating changes that can be pursued in a supply chain. We have extendedthe model presented here to support multi-period problems, inventory level setting, and tactical planning. Our future research willinvestigate more formal approaches of dealing with the stochastic nature of supply chains.

Acknowledgements

The authors gratefully acknowledge the collaboration with Dow’s Supply Chain Expertise Center. We thank Bill Meinhart, JimMann, Kurt Zetah and Cary Slade for their support, their technical input, and the development of the Excel front to the model whichmakes it a well used tool.

References

Arntzen, B. C., Brown, G. G., Harrison, T. P., & Trafton, L. L. (1995). Global supply chain management at Digital Equipment Corporation. Interfaces, 25, 69–93.Cohen, M. A., & Moon, S. (1991). An integrated plant loading model with economics of scale and scope. European Journal of Operations Research, 50, 266–279.Geoffrion, A. M., & Graves, G. W. (1974). Multicommodity distribution system by Benders Decomposition. Management Science, 20(5), 822–844.Meixell, M. J., & Gargeta, V. B. (2005). Global supply chain design: A literature review and critique. Transportation Research Part E, 41, 531–550.Tsiakis, P., Shah, N., & Pantelides, C. C. (2001). Design of multi-echelon supply chains networks under demand uncertainty. Industrial Engineering and Chemistry

Research, 40, 3585–3604.Vidal, C. J., & Goetschalckx, M. (1997). Strategic production-distribution models: A critical review with emphasis on global supply chain models. European Journal

of Operations Research, 98, 1–18.

Chemcal Supply

Documents

wassick computers

piecewise

bulk liquid

linear cost

exceeds breakpoint

supply chain

dow chemical

assign cost