Nagurney Humanitarian Logistics Lecture 7

7/29/2019 Nagurney Humanitarian Logistics Lecture 7

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Lecture 7: Perishable Product Supply

Chains in Healthcare

Professor Anna Nagurney

John F. Smith Memorial Professor

and

Director Virtual Center for Supernetworks

Isenberg School of Management

University of Massachusetts

Amherst, Massachusetts 01003

SCH-MGMT 597LGHumanitarian Logistics and Healthcare

Spring 2012cAnna Nagurney 2012

Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
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Healthcare Supply Chains

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Supply chains may be characterized by decentralizeddecision-making associated with the different economic agents orby centralized decision-making.

Supply chains are, in fact, Supernetworks.

Hence, any formalism that seeks to model supply chains andto provide quantifiable insights and measures must be a

system-wide one and network-based.

Indeed, such crucial issues as the stability and resiliency of supply

chains, as well as their adaptability and responsiveness to events ina global environment of increasing risk and uncertainty canonly be rigorously examined from the view of supply chains asnetwork systems.



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Our Approach to Supply Chain Network Analysis andDesign



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The Pharmaceutical Industry and Issues



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The Pharmaceutical Industry

Pharmaceutical, that is, medicinal drug, manufacturing is animmense global industry.

In 2003, worldwide pharmaceutical industry sales were at $491.8

billion, an increase in sales volume of 9% over the preceding yearwith US being the largest national market, accounting for44% of global industry sales.

In 2011, the global pharmaceutical industry is expected to

experience growth of 5-7% on sales of approximately $880billion (Zacks Equity Research (2011)).



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The Pharmaceutical Industry

Although pharmaceutical supply chains have begun to be coupledwith sophisticated technologies in order to improve both thequantity and the quality of their associated products, despite allthe advances in manufacturing, storage, and distribution methods,

pharmaceutical drug companies are far from effectivelysatisfying market demands on a consistent basis.

In fact, it has been argued that pharmaceutical drug supply chainsare in urgent need of efficient optimization techniques in

order to reduce costs and to increase productivity andresponsiveness (Shah (2004) and Papageorgiou (2009)).



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Pharmaceutical Product Perishability

Product perishability is another critical issue inpharmaceutical / drug supply chains.



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In a 2003 survey, the estimated incurred due to the expirationof branded products in supermarkets and drug stores wasover 500 million dollars.



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In 2007, in a warehouse belonging to the Health Department ofChicago,over one million dollars in drugs, vaccines, and othermedical supplies were found spoiled, stolen, or unaccountedfor.



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In 2007, in a warehouse belonging to the Health Department ofChicago,over one million dollars in drugs, vaccines, and othermedical supplies were found spoiled, stolen, or unaccountedfor.

In 2009, CVS pharmacies in California, as a result of asettlement of a lawsuit filed against the company, had to offerpromotional coupons to customers who had identified expireddrugs, including expired baby formula and childrens medicines, in

more than 42 percent of the stores surveyed the year before.Professor Anna Nagurney SCH-MGMT 597LG Humanitarian Logistics and Healthcare
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Other instances of medications sold more than a year past their

expiration dates have occurred in other pharmacies across the US.

According to the Harvard Medical School (2003), since a law waspassed in the US in 1979, drug manufacturers are required tostamp an expiration date on their products. This is the date at

which the manufacturer can still guarantee the full, that is, 100%,potency and safety of the drug, assuming, of course, that properstorage procedures have been followed.

For example, certain medications, including insulin, must be

stored under appropriate environmental conditions, andexposure to water, heat, humidity or other factors canadversely affect how certain drugs perform in the humanbody.



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Product Shortages

Ironically, whereas some drugs may be unsold and unused and / orpast their expiration dates, the number of drugs that werereported in short supply in the US in the first half of 2011has risen to 211 close to an all-time record with only 58

in short supply in 2004.

According to the Food and Drug Administration (FDA), hospitalshave reported shortages of drugs used in a wide range ofapplications, ranging from cancer treatment to surgery, anesthesia,

and intravenous feedings.



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Some Consequences of Product Shortages

The consequences of such shortages include the postponementof surgeries and treatments, and may also result in the useof less effective or costlier substitutes.

According to the American Hospital Association, all US hospitalshave experienced drug shortages, and 82% have reported delayedcare for their patients as a consequence (Szabo (2011)).



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H1N1 (Swine) Flu

As of May 2, 2010, worldwide,more than 214 countries andoverseas territories orcommunities reported laboratoryconfirmed cases of pandemic

influenza H1N1 2009, includingover 18,001 deaths(www.who.int).

Parts of the globe experienced

serious flu vaccine shortages,both seasonal and H1N1 (swine)ones, in late 2009.



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Disaster Preparedness for Pandemics and Influenza

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Click on underlined text:Disaster Preparedness for Pandemics and Influenza


A E l f C i i l M di i Sh C bi
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An Example of a Critical Medicine Shortage Cytarabine

In the past year, the US experiencedshortages of critical drug, cytarabine,due to manufacturer productionproblems.

Due to the severity of this medical crisis for leukemia patients,Food and Drug Administration is exploring the possibility of

importing this medical product (Larkin (2011)).

Hospira re-entered the market in March 2011 and has made themanufacture of cytarabine a priority ahead of other products.


M di l N l P d S l Ch i


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Medical Nuclear Product Supply Chains

Technetium, 99mTc, which is a decay product of Molybdenum,99Mo, is the most commonly used medical radioisotope,accounting for over 80% of the radioisotope injections andrepresenting over 30 million procedures worldwide each year.

There have been shortages of this critical radioisotope dueto nuclear production problems.


S P ibl C f Sh


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Some Possible Causes of Shortages

While the causes of many shortages are complex, most cases

appear to be related to manufacturers decisions to ceaseproduction in the presence of financial challenges.

It is interesting to note that, among curative cancer drugs,only the older generic, yet, less expensive, ones, have

experienced shortages.

As noted by Shah (2004), pharmaceutical companies secure

notable returns solely in the early lifetime of a successful drug,before competition takes place. This competition-free time-span,however, has been observed to be shortening, from 5 years to only1-2 years.


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Some Possible Causes of Shortages

Hence, the low profit margins associated with such drugsmay be forcing pharmaceutical companies to make a difficultdecision: whether to lose money by continuing to produce alifesaving product or to switch to a more profitable drug.

Unfortunately, the FDA cannot force companies to continue toproduce low-profit medicines even if millions of lives rely on them.

On the other hand, where competition has been lacking, shortagesof some other lifesaving drugs have resulted in spikes in prices,ranging from a 100% to a 4,500% increase with an average of650% (Schneider (2011)).


E i d Fi i l P


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Economic and Financial Pressures

Pharmaceutical companies are expected to suffer a significantdecrease in their revenues as a result of losing patent protection forten of the best-selling drugs by the end of 2012 (De la Garza(2011)).

These include Lipitor and Plavix, that, presently, generate morethan $142 billion in sales, are expected, over the next five years, tobe faced with generic competition.

In 2011, pharmaceutical products valued at more than $30 billionare losing patent protection, with such products generating morethan $15 billion in sales in 2010.


Safety Issues


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Safety Issues

More than 80% of the ingredients of drugs sold in the US aremade overseas, mostly in remote facilities located in China andIndia that are rarely if not ever visited by governmentinspectors.

Supply chains of generic drugs, which account for 75 percent ofthe prescription medicines sold in the US, are, typically, moresusceptible to falsification with the supply chains of some of theover-the-counter products, such as vitamins or aspirins, alsovulnerable to adulteration.

The amount of counterfeit drugs in the European pharmaceuticalsupply chains has considerably increased.


In the past, product recalls were mainly related to local errors in


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p , p ydesign, manufacturing, or labeling, a single product safety issuemay result in huge global consequences.


Waste and Environmental Impacts


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Waste and Environmental Impacts

Another pressure faced by pharmaceutical firms is theenvironmental impact of their medical waste, which includes the

perished excess medicine, and inappropriate disposal on the retailer/ consumer end.


A Generalized Network Oligopoly Model for


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A Generalized Network Oligopoly Model forPharmaceutical Supply Chains

The supply chain generalized network oligopoly model has thefollowing novel features:




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1. it handles the perishability of the pharmaceutical product

through the introduction of arc multipliers;




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2. it allows each firm to minimize the discarding cost of waste /perished medicine;




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2. it allows each firm to minimize the discarding cost of waste /perished medicine;

3. it captures product differentiation under oligopolisticcompetition through the branding of drugs, which can also includegenerics as distinct brands.




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References can be found in the paper, A supply chain generalized

network oligopoly model for pharmaceuticals under branddifferentiation and perishability, A. H. Masoumi, M. Yu, and A.Nagurney, 2011.




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The supply chain network model can be applied to similar cases ofoligopolistic competition in which a finite number of firms provide

perishable products.

However, proper minor modifications may have to be madein order to address differences in the supply chain networktopologies in related industries.


Some Examples of Oligopolies


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Some Examples of Oligopolies

airlines freight carriers

automobile manufacturers

oil companies

beer / beverage companies

wireless communications

fast fashion brands

certain food companies.




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g p yPharmaceutical Supply Chains

We consider I pharmaceutical firms, with a typical firm denoted byi.

The firms compete non-cooperatively, in an oligopolistic manner,and the consumers can differentiate among the products of thepharmaceutical firms through their individual product brands.

The supply chain network activities include manufacturing,

shipment, storage, and, ultimately, the distribution of the brandname drugs to the demand markets.


Pharmaceutical Firm 1 Pharmaceutical Firm I


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R1 RnRDemand Markets

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M11

M1n1M

MI1

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Figure: The Pharmaceutical Supply Chain Network Topology


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Each pharmaceutical firm i; i = 1, . . . , I, utilizes niM manufacturingplants and niD distribution / storage facilities, and the goal is toserve nR demand markets consisting of pharmacies, retail stores,

hospitals, and other medical centers.

Li denotes the set of directed links corresponding to the sequenceof activities associated with firm i. Also, G = [N, L] denotes thegraph composed of the set of nodes N, and the set of links L,

where L contains all sets of Lis: L i=1,...,ILi.




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In the Figure, the first set of links connecting the top two tiers ofnodes corresponds to the process of production of the drugs ateach of the manufacturing units of firm i; i = 1, . . . , I. Suchfacilities are denoted by Mi1, . . . , M

iniM

, respectively, for firm i.

We emphasize that the manufacturing facilities may be located notonly in different regions of the same country but also in different

countries.


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Pharmaceutical Supply Chains

The next set of nodes represents the distribution centers, and,thus, the links connecting the manufacturing nodes to thedistribution centers are shipment-type links. Such distributionnodes associated with firm i; i = 1, . . . , I are denoted by

Di1,1, . . . , D

iniD,1 and represent the distribution centers that the

produced drugs are shipped to, and stored at, before beingdelivered to the demand markets.

There are alternative shipment links to denote different possible

modes of transportation. In the shipment of pharmaceuticals thatare perishable one may wish, for example, to ship by air, but at ahigher cost.




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The next set of links connecting nodes Di1,1, . . . , DiniD,1

to

Di1,2, . . . , DiniD,2

; i = 1, . . . , I represents the process of storage.

Since drugs may require different storage conditions / technologies

before being ultimately shipped to the demand markets, werepresent these alternatives through multiple links at this tier.

The last set of links connecting the two bottom tiers of the supplychain network corresponds to distribution links over which the

stored products are shipped from the distribution / storagefacilities to the demand markets. Here we also allow for multiplemodes of shipment / transportation.




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T here are direct links connecting manufacturing units with variousdemand markets in order to capture the possibility of direct mailshipments from manufacturers and the costs should be adjusted

(see below) accordingly.

While representing a small percentage of the total filledprescriptions (about 6.1 percent in 2004), mail-order pharmacysales remained the fastest-growing sector of the US prescription

drug retail market in 2004, increasing by 18 percent over thepreceding year (The Health Strategies Consultancy LLC (2005)).


How We Handle Perishability


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Although pharmaceutical products may have different life-times,we can assign a multiplier to each activity / link of the supplychain to represent the fraction of the product that may perish / be

wasted / be lost over the course of that activity.

The fraction of lost product depends on the type of the activitysince various processes of manufacturing, shipment, storage, anddistribution may result in dissimilar amounts of losses.




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In addition, this fraction need not be the same among various linksof the same tier in the supply chain network since different firmsand even different units of the same firm may experiencenon-identical amounts of waste, depending on the brand of drug,

the efficiency of the utilized technology, and the experience of thestaff, etc.

Also, such multipliers can capture pilferage / theft, a significantissue in drug supply chains.




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As in Nagurney, Masoumi, and Yu (2011), we associate with everylink a in the supply chain network, a multiplier a, which lies in the

range of (0,1]. The parameter a may be interpreted as athroughput factor corresponding to link a meaning that a 100%of the initial flow of product on link a reaches the successor nodeof that link.

Let fa denote the (initial) flow of product on link a with f

a denotingthe final flow on link a; i.e., the flow that reaches the successornode of the link after wastage has taken place. Therefore, we have:

fa = afa, a L. (1)

Consequently, the waste / loss on link a, denoted by wa, which isthe difference between the initial and the final flow, can be derivedas:

wa = fa fa = (1 a)fa, a L. (2)


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Associated with the medical waste, wa, is a discarding cost, ya,which is a function of flow on that link, fa:

ya(wa) = ya(fa fa) = ya

(1 a)fa

, a L. (3a)

The parameter a is assumed to be constant and known a priori;therefore, a new total discarding cost function, za, is definedaccordingly, which is a function of the flow, fa, and is assumed tobe convex and continuously differentiable:

za = za(fa), a L. (3b)




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Let xp represent the (initial) flow of product on path p joining anorigin node, i, with a destination node, Rk. The path flows mustbe nonnegative, that is,

xp 0, p Pik; i = 1, . . . , I; k = 1, . . . , nR, (4)

where Pik is the set of all paths joining the origin node i;

i = 1, . . . , I with destination node Rk.

Also, p denotes the multiplier corresponding to the throughput onpath p, defined as the product of all link multipliers on linkscomprising that path, that is,

p ap

a, p Pik; i = 1, . . . , I; k = 1, . . . , nR. (5)




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We define the multiplier, ap, which is the product of themultipliers of the links on path p that precede link a in that path,

as follows:

ap

apa


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Hence, the relationship between the link flow, fa, and the pathflows can be expressed as:

fa =Ii=1

nR

k=1

pPi

k

xp ap, a L. (7)




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Note that the arc multipliers may be obtained from historical and

statistical data.

They may also, in the case of certain perishable products, berelated to an exponential time decay function where the time, inour framework, is associated with each specific link activity (see,

for instance, Blackburn and Scudder (2009) and Bai and Kendall(2009)).

For example, Nagurney and Nagurney (2011) constructed explicitarc multipliers for molybdenum, which is used in nuclear medicine,

which were based on the physics of time decay for thispharmaceutical product used in cancer and cardiac diagnostics,among other procedures.




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Let dik denote the demand for pharmaceutical firm is brand drug;

i = 1, . . . , I, at demand market Rk; k = 1, . . . , nR. The consumersdifferentiate the products by their brands.

The following equation reveals the relationship between the pathflows and the demands in the supply chain network:

pPik

xpp = dik, i = 1, . . . , I; k = 1, . . . , nR, (8)

that is, the demand for a brand drug at the demand market Rk is

equal to the sum of all the final flows subject to perishability on paths joining (i, Rk). We group the demands dik;i = 1, . . . , I; k = 1, . . . , nR into the nR I-dimensional vector d.


The Demand Price Functions


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A demand price function is associated with each firmspharmaceutical at each demand market. We denote the demandprice of firm is product at demand market Rk by ik and assumethat

ik = ik(d), i = 1, . . . , I; k = 1, . . . , nR. (9)


The Total Cost Functions


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The total operational cost on link a may, in general, depend uponthe product flows on all the links, that is,

ca = ca(f), a L, (10)

where f is the vector of all the link flows. The total cost on each

link is assumed to be convex and continuously differentiable.

Xi denotes the vector of path flows associated with firm i;i = 1, . . . , I, where Xi {{xp}|p P

i}} RnPi

+ , andPi k=1,...,n

R

Pik

. In turn, nPi, denotes the number of paths from

firm i to the demand markets. Thus, X is the vector of all thefirm strategies, that is, X {{Xi}|i = 1, . . . , I}.


The Profit Function


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The profit function of firm i, denoted by Ui, is expressed as:

Ui =

nRk=1

ik(d)pPi

k

pxp aLi

ca(f) aLi

za(fa). (11)

In lieu of the conservation of flow expressions (7) and (8), and thefunctional expressions (3b), (9), and (10), we may defineUi(X) = Ui for all firms i; i = 1, . . . , I, with the I-dimensionalvector U being the vector of the profits of all the firms:

U = U(X). (12)


Supply Chain Generalized Network Cournot-NashEquilibrium


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Equilibrium

In the Cournot-Nash oligopolistic market framework, each firmselects its product path flows in a noncooperative manner, seekingto maximize its own profit, until an equilibrium is achieved,according to the definition below.

Definition 1: Supply Chain Generalized Network

Cournot-Nash EquilibriumA path flow pattern X K =

Ii=1 Ki constitutes a supply chain

generalized network Cournot-Nash equilibrium if for each firm i;i = 1, . . . , I :

Ui(Xi , Xi ) Ui(Xi, Xi ), Xi Ki, (13)

where Xi (X1 , . . . , X

i1, X

i+1, . . . , X

I ) and

Ki {Xi|Xi RnPi

+ }.



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An equilibrium is established if no firm can unilaterally improve itsprofit by changing its production path flows, given the productionpath flow decisions of the other firms.

Next, we present the variational inequality formulations of the

Cournot-Nash equilibrium for the pharmaceutical supply chainnetwork under oligopolistic competition satisfying Definition 1, interms of both path flows and link flows (see Cournot (1838), Nash(1950, 1951), Gabay and Moulin (1980), and Nagurney (2006)).


The Variational Inequality Formulation


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Theorem 1

Assume that, for each pharmaceutical firm i; i = 1, . . . , I , theprofit function Ui(X) is concave with respect to the variables inXi, and is continuously differentiable. Then X

K is a supplychain generalized network Cournot-Nash equilibrium according toDefinition 1 if and only if it satisfies the variational inequality:

I

i=1

XiUi(X)T, Xi X

i 0, X K, (14)

where , denotes the inner product in the correspondingEuclidean space and XiUi(X) denotes the gradient of Ui(X) withrespect to Xi.



( )


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Variational inequality (14), in turn, for our model, is equivalent tothe variational inequality: determine x K1 such that:

Ii=1

nRk=1

pPi

k

Cp(x

)

xp+

Zp(x)

xp ik(x

)p

nRl=1

il(x

)dik

ppPi

l

pxp [xp xp] 0, x K1, (15)

where K1 {x|x RnP+ }, and, for notational convenience, wedenote:

Cp(x)

xpbLi

aLi

cb(f)

faap and

Zp(x)

xpaLi

za(fa)

faap.

(16)




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Variational inequality (15) can also be re-expressed in terms oflink flows as: determine the vector of equilibrium link flows and thevector of equilibrium demands (f, d) K2, such that:

I

i=1aLi

bLi

cb(f)

fa+

za(fa )

fa [fa f

a

]

+I

i=1nR

k=1

ik(d)

nR

l=1il(d

)

dikdil

[dikd

ik] 0, (f, d) K

2,

(17)where K2 {(f, d)|x 0, and(7), and(8) hold}.




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Variational inequalities (15) and (17) can be put into standard

form (see Nagurney (1999)): determine X K such that:

F(X)T, X X 0, X K, (18)

where , denotes the inner product in n-dimensional Euclidean

space. Let: X x and

F(X) Cp(x)

xp+

Zp(x)

xpik(x)p

nRl=1

il(x)

dikppPi

l

pxp; p Pik;

and K K1.




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Similarly, for the variational inequality in terms of link flows, if wedefine the column vectors: X (f, d) andF(X) (F1(X), F2(X)):

F1(X) =bLi

cb(f)

fa +

za(fa)

fa ; a Li

; i = 1, . . . , I

,

F2(X) =

ik(d)

nR

l=1il(d)

dikdil; i = 1, . . . , I; k = 1, . . . , nR

,

and let K K2.



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Relationship of the Model to Others in the Literature




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The above model is now related to several models in the literature.

If the arc multipliers are all equal to 1, in which case the product isnot perishable, then the model is related to the sustainable fashionsupply chain network model of Nagurney and Yu (2011). In thatmodel, however, the other criterion, in addition to the profitmaximization one, was emission minimization, rather than waste

cost minimization, as in the model in this paper.



If the demands are fixed and there is a single organization but


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If the demands are fixed, and there is a single organization, butthere are additional processing tiers, as well as capacity

investments as variables, the model is the medical nuclear supplychain design model of Nagurney and Nagurney (2011).


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RadioisotopeProduction


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c

% c

rrrrrjc

%

rrrrrjTransportation

hC11 h C1j hC

1nC

c c c

ProcessingFacilities99Mo Extractionand PurificationhC21 hC2j C2nC h

c

rrrrrj

zc

%

rrrrrjc

%

$$$$$$$$$$W jTransportation

hG11 G1kh hG1nG

c c cGeneratorManufacturinghG21 G2kh hG2nG

c

rrrrrj

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%

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hH11 H1kh hH1nHHospitals orImaging Facilities

GeneratorManufacturing Facilities

c c c

99mTcElucitation

hH12 H2kh hH2nHPatient Demand PointsFigure: The Medical Nuclear Supply Chain Network Topology



If there is only a single organization / firm and the demands are
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If there is only a single organization / firm, and the demands aresubject to uncertainty, with the inclusion of expected costs due to

shortages or excess supplies, the total operational cost functionsare separable, and a criterion of risk is added, then the modelabove is related to the blood supply chain network operationsmanagement model of Nagurney, Masoumi, and Yu (2011).


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sBlood Collection

CS1 CS2 CS3 CSnCS Blood Collection Sitesd d d d


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CS1 CS2 CS3 CSnCS Blood Collection Sitesd d d dd

dd

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dd

dc

%

Shipment ofCollected Blood

BC1 BCnBC Blood Centersd d

c c

Testing & Processing

CL1 CLnCL Component Labsd d

c c

Storage

SF1 SFnSF Storage Facilitiesd d

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s

C

)

tt

tt

Shipment

DC1 DC2 DCnDC Distribution Centersd d d%

e

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e

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ee

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qDistribution d d d dR1 R2 R3 RnR Demand PointsFigure: Supply Chain Network Topology for a Regionalized Blood Bank



If the product is homogeneous and all the arc multipliers are


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If the product is homogeneous, and all the arc multipliers are,again, assumed to be equal to 1, and the total costs are assumedto be separable, then the above model collapses to the supplychain network oligopoly model of Nagurney (2010) in whichsynergies associated with mergers and acquisitions were assessed.


The Original Supply Chain Network Oligopoly Model

m1 mI Firm 1 Firm I


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mR1 RnR m

rrrrrrj

c

qc

A

%

222222222222222

D11,2 m mD1n1D,2 D

I1,2

m mDInID,2

c c c c

D11,1 m mD1n1D,1 D

I1,1

m mDInID,1

c

rrrrrrjc

%

c

rrrrrrjc

%

M11 m mM1n1M MI1

m mMInIM

dd

d

dd

d

m m

Figure: Supply Chain Network Structure of the Oligopoly WithoutPerishability; Nagurney Computational Management Science, 2010, 7,377-401.


Mergers Through Coalition Formation

jFirm 1

jFirm 2


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jR1 RnR jd

dd

c

qc

A

$$$$$$$$$

D11,2 j jD

1

n1D,2D

n1

1,2 j jDn1

n1

D ,2

c c c c

D11,1 j jD1n1D,1D

n11,1

j jDn1nn1D

,1

cd

ddc

q

cd

ddA$$$$$$$$$ c

$$$$$$$$$

M1

1 j jM1

n1MM

n1

1 j jM

n1

nn

1

M

eee

eee

j1 jn1

d

dddd

j1

Firm 1 Firm n1

j jA$$$$$$$$$W

j j

c c

j jcd

ddqz

c

rrrrrjq

jFirm n1 + 1

eee

j j

j j

cd

ddc

j jMI

nIM

DInID,1

DI

nID,2

cd

dd%A

c

A$$$$$$$$$W

jI

Firm I

eee

G

w

j2

Figure: Mergers of the First n1 Firms and the Next n2 Firms

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A Simple Perishable Product Numerical Example



Pharmaceutical Firm 1 Pharmaceutical Firm 2


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lR1

rrr

rrrj

%7 8

D11,2 l lD21,2c c

5 6

D11,1 l D21,1lc c

3 4

M11l M21l

c c1 2

l1 l2Pharmaceutical Firm 1 Pharmaceutical Firm 2

Figure: Supply Chain Network Topology for the Pharmaceutical Duopolyin the Illustrative Example



In this example, two pharmaceutical firms compete in a duopoly


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with a single demand market (See Figure). The two firms producedifferentiated, but substitutable, brand drugs 1 and 2,corresponding to Firm 1 and Firm 2, respectively.The total cost functions on the various links of manufacturing,shipment, storage, and distribution are:

c1(f1) = 5f2

1 + 8f1, c2(f2) = 7f2

2 + 3f2, c3(f3) = 2f2

3 + f3,

c4(f4) = 2f2

4 + 2f4,

c5(f5) = 3f2

5 + 4f5, c6(f6) = 3.5f2

6 + f6, c7(f7) = 2f2

7 + 5f7,

c8(f8) = 1.5f2

8 + 4f8.

The arc multipliers are given by:

1 = .95, 2 = .98, 3 = .99, 4 = 1.00, 5 = .99, 6 = .97,

7 = 1.00, 8 = 1.00.


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The total discarding cost functions on the links are assumedidentical, that is,za(fa) = .5f

2a , a.

The firms compete in the demand market R1, and the consumersreveal their preferences for the two products through the followingnonseparable demand price functions:

11(d) = 3d11 d21 + 200, 21(d) = 4d21 1.5d11 + 300.

In this supply chain network, there exists one path correspondingto each firm, denoted by p1 and p2.




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Thus, variational inequality (15), here takes the form:Cp1 (x

)

xp1+

Zp1 (x)

xp1 11(x

)p1 11(x)

d11p1 p1 x

p1

[xp1 xp1 ]

+

Cp2 (x

)

xp2+

Zp2 (x)

xp2 21(x

)p2 21(x)

d21p2 p2 x

p2

[xp2 x

p2 ] 0, x K1

.



Under the assumption that xp1 > 0 and xp2

> 0, the two


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expressions on the left-hand side of inequality (27) must be equalto zero, that is:

Cp1 (x)

xp1+

Zp1 (x)

xp1 11(x

)p1 11(x)

d11p1 p1 x

p1

[xp1 xp1

] = 0,

andCp2 (x

)

xp2+

Zp2 (x)

xp2 21(x

)p2 21(x)

d21p2 p2 x

p2

[xp2 xp2 ] = 0.

Since each of the paths flows must be nonnegative, we know thatthe term preceding the multiplication sign in both of the abovemust be equal to zero.



Calculating the values of the multipliers from (6), and then,


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substituting those values, as well as, the given functions into (16),we can determine the partial derivatives of the total operationalcost and the total discarding cost functions. Furthermore, thepartial derivatives of the given demand price functions can becalculated and substituted into the above. Applying (5), the pathmultipliers are equal to:

p1 = 1 3 5 7 = .95 .99 .99 1 = .93,

p2 = 2 4 6 8 = .98 1 .97 1 = .95.

Simple arithmetic calculations, with the above substitutions, yieldthe below system of equations:

31.24xp1 + 0.89xp2

= 168.85,

1.33xp1 + 38.33xp2

= 274.46.



Thus, the equilibrium solution corresponding to the path flow of


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brand drugs produced by firms 1 and 2 is:

xp1 = 5.21, xp2 = 6.98.

Using (7), the equilibrium link flows can be calculated as:

f1 = 5.21, f

3 = 4.95, f

5 = 4.90, f

7 = 4.85,

f2 = 6.98, f4 = 6.84, f6 = 6.84, f8 = 6.64.

From (8), the equilibrium values of demand for products of the twopharmaceutical firms are equal to:

d

11 = 4.85, d

21 = 6.64.

Finally, the equilibrium prices of the two branded drugs are:

11 = 178.82, 21 = 266.19.



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Note that, even though the price of Firm 2s product is observed tobe higher, the market has a slightly stronger tendency toward thisproduct as opposed to the product of Firm 1.

This is due to the willingness of the consumers to spend more on

one product which can be a consequence of the reputation, or theperceived quality, of Firm 2s brand drug.



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The Algorithmwith

Explicit Formulae


The Algorithm


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We now recall the Euler method, which is induced by the general

iterative scheme of Dupuis and Nagurney (1993). Its realization forthe solution of the supply chain generalized network oligopolymodel with brand differentiation governed by variational inequality(15) induces subproblems that can be solved explicitly and inclosed form.

Specifically, iteration of the Euler method (see also Nagurneyand Zhang (1996)) is given by:

X+1 = PK(X aF(X

)),

where PK is the projection on the feasible set K and F is thefunction that enters the variational inequality problem (18).


The Algorithm


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As shown in Dupuis and Nagurney (1993) and Nagurney andZhang (1996), for convergence of the general iterative scheme,which induces the Euler method, the sequence {a} must satisfy:

=0 a = , a > 0, a 0, as .

Conditions for convergence of this scheme as well as variousapplications to the solutions of network oligopolies can be found inNagurney and Zhang (1996), Nagurney, Dupuis, and Zhang(1994), Nagurney (2010a), and Nagurney and Yu (2011).


The Algorithm

E li i F l f h E l M h d A li d h


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Explicit Formulae for the Euler Method Applied to the

Supply Chain Generalized Network Oligopoly VariationalInequality (15)

The elegance of this procedure for the computation of solutions toour supply chain generalized network oligopoly model with product

differentiation can be seen in the following explicit formulae. Inparticular, we have the following closed form expressions for all thepath flows p Pik, i, k:

x+1p = max

{0, xp+a(ik(x)p+

nRl=1

il(x)

dikppPi

l

pxp

Cp(x)

xp

Zp(x)

xp)}.



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A Case Study


A Case Study Case I

Firm 1 represents a multinational pharmaceutical giant,


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hypothetically, Pfizer, Inc., which currently possesses the patent for

Lipitor, the most popular brand of cholesterol-lowering drug. Withmore than $5 billion of sales in the US alone in 2011, this drug wasonce the top-selling pharmaceutical brand in the world (Rossi(2011)).

Firm 2, on the other hand, which might represent, for example,Merck & Co., Inc., also is one of the largest global pharmaceuticalcompanies, and has been producing Zocor, another cholesterolregulating brand, whose patent expired in 2006.

We consider two competing brands in three demand marketslocated across the US. Each of the two firms is assumed to havetwo manufacturing units and three storage / distribution centers,as illustrated in the Figure.


The Pharmaceutical Supply Chain Network Topology for Case I

1 d 2 3 d

m1 m2Pharmaceutical Firm 1 Pharmaceutical Firm 2


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mR1 mR2 mR3

s

hhhhhhhhhhhhhhhhhh

2324 25

c

rrrr

rrj

z

262728

ee

e

2930 31$$$$$$$$$

$$$

C

ee

e

32333422222222222

2222

A

35 3637@@@@@@@@@@@@@@@

@@@@

222222222222222

33333333

38 3940

D11,2m mD12,2 mD13,2 D21,2 m mD22,2 mD23,2c

17

c18

c19

c20

c21

c22

D11,1m mD12,1 mD13,1 D21,1 m mD22,1 mD23,1

dd

d

33333333

ee

e

5 6 7 8 9 10

dd

d

33333333

ee

e

11 12 13 14 15 16

M11m mM12 M21m mM22

1 dd

d

2

3 dd

d

4

Figure: Case I Supply Chain Network


A Case Study Case I

The demand price functions corresponding to the three demand


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markets for each of the two brands 1 and 2 were as follows:

11(d) = 1.1d11 0.9d21 +275; 21(d) = 1.2d21 0.7d11 +210;

12(d) = 0.9d12 0.8d22 +255; 22(d) = 1.0d22 0.5d12 +200;

13(d) = 1.4d13 1.0d23 +265; 23(d) = 1.5d23 0.4d13 +186.

The Euler method for the solution of variational inequality (15)was implemented in Matlab on a Microsoft Windows 7 Systemwith a Dell PC at the University of Massachusetts Amherst. We

set the sequence a = .1(1,12 ,

12 , ), and the convergence

tolerance was 106. In other words, the absolute value of thedifference between each path flow in two consecutive iterations wasless than or equal to this tolerance.


A Case Study Case I


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The arc multipliers, the total operational cost functions, and thetotal discarding cost functions were as reported in the next Table.These cost functions have been selected based on the averagevalues of the data corresponding to the prices, the shipping costs,etc., available on the web. The values of arc multipliers, in turn,

although hypothetical, are constructed in order to reflect thepercentage of perishability / waste / loss associated with thevarious supply chain network activities in medical drug supplychains.

The equilibrium pattern is also reported in the Table.


Link Multipliers, Total Cost Functions and Link Flow Solution for Case I

Link a a ca(fa) za(fa) fa1 .95 5f21 + 8f1 .5f

21 13.73

2 .97 7f 22 + 3f2 .4f2

2 10.77


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2 .97 7f2 + 3f2 .4f2 10.773 .96 6.5f23 + 4f3 .3f

23 8.42

4 .98 5f2

4 + 7f4 .35f2

4 10.555 1.00 .7f25 + f5 .5f

25 5.21

6 .99 .9f26 + 2f6 .5f2

6 3.367 1.00 .5f27 + f7 .5f

27 4.47

8 .99 f28 + 2f8 .6f2

8 3.029 1.00 .7f29 + 3f9 .6f

29 3.92

10 1.00 .6f2

10

+ 1.5f10 .6f2

10

3.5011 .99 .8f211 + 2f11 .4f

211 3.10

12 .99 .8f212 + 5f12 .4f2

12 2.3613 .98 .9f213 + 4f13 .4f

213 2.63

14 1.00 .8f214 + 2f14 .5f2

14 3.7915 .99 .9f215 + 3f15 .5f

215 3.12

16 1.00 1.1f216 + 3f16 .6f2

16 3.43

17 .98 2f217 + 3f17 .45f217 8.2018 .99 2.5f218 + f18 .55f

218 7.25

19 .98 2.4f219 + 1.5f19 .5f2

19 7.9720 .98 1.8f220 + 3f20 .3f

220 6.85


Link Multipliers, Total Cost Functions and Solution for Case I (contd)

Link a a ca(fa) za(fa) fa21 .98 2.1f221 + 3f21 .35f

221 5.42

22 .99 1.9f 222 + 2.5f22 .5f2

22 6.00


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22 .99 1.9f22 + 2.5f22 .5f22 6.0023 1.00 .5f223 + 2f23 .6f

223 3.56

24 1.00 .7f2

24 + f24 .6f2

24 1.6625 .99 .5f225 + .8f25 .6f

225 2.82

26 .99 .6f226 + f26 .45f2

26 3.3427 .99 .7f227 + .8f27 .4f

227 1.24

28 .98 .4f228 + .8f28 .45f2

28 2.5929 1.00 .3f229 + 3f29 .55f

229 3.45

30 1.00 .75f2

30

+ f30 .55f2

30

1.2831 1.00 .65f231 + f31 .55f

231 3.09

32 .99 .5f232 + 2f32 .3f2

32 2.5433 .99 .4f233 + 3f33 .3f

233 3.43

34 1.00 .5f234 + 3.5f34 .4f2

34 0.7535 .98 .4f235 + 2f35 .55f

235 1.72

36 .98 .3f236 + 2.5f36 .55f2

36 2.64

37 .99 .55f237 + 2f37 .55f237 0.9538 1.00 .35f238 + 2f38 .4f

238 3.47

39 1.00 .4f239 + 5f39 .4f2

39 2.4740 .98 .55f240 + 2f40 .6f

240 0.00


A Case Study Case I

The values of the equilibrium link flows in the Table demonstrateh i f i h bili f h d h h h l


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the impact of perishability of the product throughout the supply

chain network links of each pharmaceutical firm. Under the abovedemand price functions, the computed equilibrium demands foreach of the two brands were:

d11 = 10.32, d21 = 7.66, d

12 = 4.17, d

22 = 8.46, d

13 = 8.41,

d23 = 1.69.

Furthermore, the incurred equilibrium prices associated with thebranded drugs at each demand market were as follows:

11 = 256.75, 21 = 193.58, 12 = 244.48, 22 = 189.46, 13 = 251.52,

23 = 180.09.


A Case Study Case I

Firm 1, which produces the top-selling product, captures themajority of the market share at demand markets 1 and 3, despite


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j y , pthe higher price. While this firm has a slight advantage over itscompetitor in demand market 1, it has almost entirely seizeddemand market 3. Consequently, several links connecting Firm 2 todemand market 3 have insignificant flows including link 40 with aflow equal to zero.

In contrast, Firm 2 dominates demand market 2, due to theconsumers willingness to lean towards this product there, perhapsas a consequence of the lower price, or the perception of quality,etc., as compared to the product of Firm 1.

The profits of the two firms are:

U1 = 2, 936.52 and U2 = 1, 675.89.



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Recall that Firm 1 still holds the patent rights of its branded drug,

and, thus, makes a higher profit from selling cholesterol regulators.

In contrast, Firm 2 has completed the competition-free timespanfor its brand of cholesterol medicine a few years ago as aconsequence of losing the patent rights to the manufacturers of

generic drugs. Hence, fewer numbers of consumers choose thisproduct as compared to the product of Firm 1 leading to a higherprofit for the producer of the newer brand.


A Case Study Case II


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In this case, we consider the scenario in which Firm 1 has just lostthe exclusive patent right of its highly popular cholesterolregulator. A manufacturer of generic drugs, say, Sanofi, here

denoted by Firm 3, has recently introduced a generic substitute forLipitor by reproducing its active ingredient Atorvastatin (Smith(2011)). Firm 3 is assumed to have two manufacturing plants, twodistribution centers as well as two storage facilities in order tosupply the same three demand markets as in Case I (See Figure).

Professor Anna Nagurney SCH MGMT 597LG Humanitarian Logistics and Healthcare

The Pharmaceutical Supply Chain Network Topology for Cases II and III

Pharmaceutical Firm 1 Pharmaceutical Firm 2Pharmaceutical Firm 3


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mR1 mR2 mR3

23

24hhhhhhhhhhhhhhhhhhhhh

25

s

26

27hhhhhhhhhhhhhhhhhh

28

c

29rrrrrrj

30z

31

c

32%

33$$$$$$$$$$$$W

34

C

35 $$$$$$$$$$$$

36@@@@@@@@@@@@@@@@@@

37 @@@@@@@@@@@@@@@@@@@@@@

38 2222222222222222

39

40

49dd

d

50q

51 A

52

53dd

d

54

D11,2 m m

D12,2 m

D13,2

D21,2m m

D22,2 m

D23,2

D31,1m

D32,1 m

c17

c18

c19

c20

c21

c22

c47

c48

D11,1m mD12,1 mD13,1 D21,1m mD22,1 mD23,1D31,1m D32,1 m

d

dd

33333333

e

ee

5 6 7 8 9 10

d

dd

33333333

e

ee

11 12 13 14 15 16

c

43rrr

rrrj

44

c

46

%

45

M11m mM12 M21m mM22M31m mM32

1 dd

d

2

3 dd

d

4

41 dd

d

42m1

m2

m3



Since, in Case II, the new generic drug has just been released, weassume that the demand price functions for the products of Firm 1


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assume that the demand price functions for the products of Firm 1

and 2 will stay the same as in Case I. On the other hand, thedemand price functions corresponding to the product of Firm 3 fordemand markets 1, 2, and 3 are as follows:

31(d) = 0.9d31 0.6d11 0.8d21 + 150;

32(d) = 0.8d32 0.5d12 0.6d22 + 130;

33(d) = 0.9d33 0.7d13 0.5d23 + 133.

The next Table displays the arc multipliers, the total operationaland the total discarding cost functions with regards to the existinglinks as well as the new links. The computed values of theequilibrium link flows are also reported in the next Table.


Link Multipliers, Total Cost Functions and Link Flow Solution for Case II

Link a a ca(fa) za(fa) f

a

1 .95 5f21 + 8f1 .5f2

1 13.73

2 .97 7f22 + 3f2 .4f2

2 10.77

3 .96 6.5f23 + 4f3 .3f2

3 8.42

4 .98 5f24 + 7f4 .35f2

4 10.552 2


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5 1.00 .7f25 + f5 .5f2

5 5.21

6 .99 .9f2

6+ 2f

6.5f2

63.36

7 1.00 .5f27 + f7 .5f2

7 4.47

8 .99 f28 + 2f8 .6f2

8 3.02

9 1.00 .7f29 + 3f9 .6f2

9 3.92

10 1.00 .6f210 + 1.5f10 .6f2

10 3.50

11 .99 .8f211 + 2f11 .4f2

11 3.10

12 .99 .8f212 + 5f12 .4f2

12 2.36

13 .98 .9f213 + 4f13 .4f2

13 2.63

14 1.00 .8f2

14 + 2f14 .5f2

14 3.7915 .99 .9f215 + 3f15 .5f

215 3.12

16 1.00 1.1f216 + 3f16 .6f2

16 3.43

17 .98 2f217 + 3f17 .45f2

17 8.20

18 .99 2.5f218 + f18 .55f2

18 7.25

19 .98 2.4f219 + 1.5f19 .5f2

19 7.97

20 .98 1.8f220 + 3f20 .3f2

20 6.85

21 .98 2.1f221 + 3f21 .35f2

21 5.42

22 .99 1.9f222 + 2.5f22 .5f222 6.0023 1.00 .5f223 + 2f23 .6f

223 3.56

24 1.00 .7f224 + f24 .6f2

24 1.66

25 .99 .5f225 + .8f25 .6f2

25 2.82

26 .99 .6f226 + f26 .45f2

26 3.34

27 .99 .7f227 + .8f27 .4f2

27 1.24


Link Multipliers, Total Cost Functions and Solution for Case II (contd)

Link a a ca(fa) za(fa) f

a

28 .98 .4f228 + .8f28 .45f2

28 2.59

29 1.00 .3f229 + 3f29 .55f2

29 3.45

30 1.00 .75f230 + f30 .55f2

30 1.28

31 1.00 .65f231 + f31 .55f2

31 3.092 2


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32 .99 .5f232 + 2f32 .3f2

32 2.54

33 .99 .4f2

33+ 3f

33.3f2

333.43

34 1.00 .5f234 + 3.5f34 .4f2

34 0.75

35 .98 .4f235 + 2f35 .55f2

35 1.72

36 .98 .3f236 + 2.5f36 .55f2

36 2.64

37 .99 .55f237 + 2f37 .55f2

37 0.95

38 1.00 .35f238 + 2f38 .4f2

38 3.47

39 1.00 .4f239 + 5f39 .4f2

39 2.47

40 .98 .55f240 + 2f40 .6f2

40 0.00

41 .97 3f2

41 + 12f41 .3f2

41 6.1742 .96 2.7f242 + 10f42 .4f

242 6.23

43 .98 1.1f243 + 6f43 .45f2

43 3.23

44 .98 .9f244 + 5f44 .45f2

44 2.75

45 .97 1.3f245 + 6f45 .5f2

45 3.60

46 .99 1.5f246 + 7f46 .55f2

46 2.38

47 .98 1.5f247 + 4f47 .4f2

47 6.66

48 .98 2.1f248 + 6f48 .45f2

48 5.05

49 .99 .6f249 + 3f49 .55f249 3.7950 1.00 .7f250 + 2f50 .7f

250 1.94

51 .98 .6f251 + 7f51 .45f2

51 0.79

52 .99 .9f252 + 9f52 .5f2

52 1.43

53 1.00 .55f253 + 6f53 .55f2

53 1.23

54 .98 .8f254 + 4f54 .5f2

54 2.28



The equilibrium product flows of Firms 1 and 2 on links 1 through40 id ti l t th di l i C I


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40 are identical to the corresponding values in Case I.

When the new product produced by Firm 3 is just introduced, themanufacturers of the two existing products will not experience animmediate impact on their respective demands of branded drugs.

Consequently, the equilibrium computed demands for the productsof Firms 1 and 2 at the demand markets will remain as in Case I.

However, the equilibrium amounts of demand for the new productof Firm 3 at each demand market is equal to:

d31 = 5.17, d32 = 3.18, and d

33 = 3.01.


A Case Study Case IIIUnder the above assumptions, the equilibrium prices associatedwith the branded drugs 1 and 2 at the demand markets will not

h h th i d ilib i i f i d 3


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change, whereas the incurred equilibrium prices of generic drug 3

are as follows:

31 = 133.02, 32 = 120.30, and 33 = 123.55,

which is significantly lower than the respective prices of itscompetitors in all the demand markets.

Thus, the profit that Firm 3 derived from manufacturing anddelivering the new generic substitute to these 3 markets is:

U3 = 637.38,

while the profits of Firms 1 and 2 remain unchanged. In the nextcase, we will investigate the situation in which the consumers arenow more aware of the new generic substitute of cholesterolregulators.


A Case Study Case IIIWe assumed that the generic product of Firm 3 has now beenwell-established, and has affected the behavior of the consumersth h th d d i f ti f th l ti l


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through the demand price functions of the relatively more

recognized products of Firms 1 and 2. The demand price functionsassociated are now given by:

Firm 1: 11(d) = 1.1d11 0.9d21 1.0d31 + 192;

21(d) = 1.2d21 0.7d11 0.8d31 + 176;

31 = 0.9d31 0.6d11 0.8d21 + 170;

Firm 2: 12(d) = 0.9d12 0.8d22 0.7d32 + 166;

22(d) = 1.0d22 0.5d12 0.8d32 + 146;

32

(d) = 0.8d32

0.5d12

0.6d22

+ 153;

Firm 3: 13(d) = 1.4d13 1.0d23 0.5d33 + 173;

23(d) = 1.5d23 0.4d13 0.7d33 + 164;

33(d) = 0.9d33 0.7d13 0.5d23 + 157.

Professor Anna Nag rne SCH MGMT 597LG H manitarian Logistics and Healthcare

A Case Study Case III
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Nagurney Humanitarian Logistics Lecture 7

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Transcript of Nagurney Humanitarian Logistics Lecture 7