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A simple plant-location model for quantity-setting firms subject to price uncertainty
~:
:
: =
39
~, Stanford, California
A~~ ~ e uncapacitated plant location problem under uncertainty is formulated in a mean-variance frame withprices in Various markets correlated via their response to a common random factor. This formulation results in a mixed-i~tteger quadratic programming problem. However, for a given integer solution; the resulting quadratic p r o g r a ~ n g problem is amenable to a very simple solution procedure. The simplicity of this algorithm means that reasonably large problems should be solvable using existing branch-and-bound techniques. Keyworfls: Facilities, location, mixed-integer quadratic programming
Introduction
Consider a single-period model with n markets (indexed by j ) and rn potential plant locations (indexed with i). The selling price for the product in market j is denoted by Pj, wit'h X~j being the quantity shipped from plant i to market j. The unit cost of serving market j from plant i (including production and transportation expenses) is denoted by %. We use ~ to indicate an upper bound on the demand level h~ the j-th market, Y~ (a zero-one decision variable) to indicate whether or not to open plant i, and F ~.he fized cost of opening the i-th plant. With these definitions, the plant location problem can be represemed as
i j subjectto
i
0~
for all j,
Y/=0, 1
for all i.
(1)
This is a standard, uncapacitated, plant location problem with a profit maximization objective. There is a vast ~ o u n t of literature dealing with this and related problems as well as procedures for solving these Fr~cis and Goldstein (1974), Francis, McGinnis and White (1983), Krarup 1983)). The bulk of this literature assumes that quantities demanded, prices, and costs are ~ o w n in actual situations, there may be a considerable lag between when a the facility is in place and operating. There i~ frequently considerable difficulty predicting the length of such a lag or the demand, costs, and prcies wMch wil! obtain when it elapses' ~ o s e who, :have addressed such uncertainty in locadon models have dealt mostly with demand and ReceivedFebruary1984; revisedJuly 1984 European Journal of Operational Research21 (1985)39-46 0377-2217/85/$3;30 ©I985, ElsevierSciencePublishersB.V. (North'Holland)
There are a number of different ways that uncertainty cm In order for the product market to clear, some combination ~ to changes in demand. Firms in both manufacturing and service industries use a variety of mechanisms in an attempt to maintain stable sales and stable production. Consider, for example, the use of discounting, r~bates, and special advertising in the automobile and airline industries. For planning purposes, such firms assume that production will be stable. Consequently, modeling such firms as using a quantity-setting strategy for planning purposes is reasonable. In addition to quantity-setting firms, Jucker and Carlson (1976) considered three other types of firms. For these other firms, uncertainty was realized in the quantity sold at a known price. This assumption of quantity, rather than price adjustment may be appropriate for planning purposes in cases such as regulated industries; however, the quantity-setting planning assumption is probably a much better one for many firms whose prices are not regulated. Another reason for focusing on quantity-setting firms is becaus, ~.it is possible to incorporate interrelated demand uncertaintly across markets via correlated random prkgs in a relatiwfly simple way. The alternative of having uncertainty reflected in correlated random quantities is not amenable to the simple solution procedure described herein. As risk increases, most individuals and even firms change their actions. The possibility of a substantial loss, like the hangman's noose, concentrates one's mind. To incorporate this aspect into the model, we will use a raean-variance objective function to account for the possibility of risk aversion (see Howard (1971), and J~cker and Carlson (1976) for discussions of the mean-variance criterion and its use for complex decis~.on problems and plant-location problems, respectively).
1. The model Changing the deterministic formulation of problem (1) into a formulation consistent with the preceding discussion is accomplished by allowing the Pi to be random and introducing an objective function of the form V = S ( ¢ r ) - hvar(cr). The parameter ~ is a risk aversion measure which describes the decision m~ker's willingness to trade expected return in order to reduce risk as measured by var(~'). We should add that in this new n~odet, the upper bounds on the demand levels in the various markets (the Dj.'s)need n o t be restricted to finite numbers. The algorithm will work whether or not there is a finite upper bound on auantity demanded in any market. Assuming a standard mean-variance objective function with Pj random, this problem can be stated as
i
subject to
j
i
0 ~
for all i and j,
EiXij~Dj
for all j,
Y~= 0,1
for all i.
i
j
-
(2)
i(
:!:~f:r
/:<
:
41
: J.E. Hodder, J. V. Jucker / A siraple plant-tocation model
the Py are i~ndependent. We allow for
(3)
•j).
Inthis for ml llation, ~) is a random factor common to all markets which is statistically distributed with mean 4 . Tho i:oefficient bj > 0 represents an adjustment parameter for the j-th market, with ej ~ independently distrib|ated random ,error term having mean zero and variance of. This use of a common factoi" allows for correlated prices in a manner which will be analytically tractable while representing a substantial improvement over the asumption of complete independence. U t i l ~ n g (3)we can rewrite our objective function as
v=EE(bj i
-c,j)xu-EF, E - x
j
i
EX, jb; o
-xE Ex, jb, o/
" j
j
~,4)
i
2, The nodal subproblem Our problem can be simplified by showing that for a given set of open plants, each market will be served from at most one plant--its 'dominant' plant. This sort of dominance plays an importaat role in the branch-and-bound procedures for the deterministic plant location model (e.g., Efroymson and Ray (1966)) as well as for some location problems with uncertainty (Jucker and Carlson (1976)). Our solution procedure also requires that the identity of the dominant plant (among the open set) does not depend on production levels at that or any other plant. This 'consistent' dominance is perhaps not immediately obvious with the objective function in (4); however, such dominance can easily be demonstrated. Let M denote the set of open plants and d ( j l M ) the (consistently) dominant plant for market j conditional on M. For some market k and an open plant h,
2hbk
aXak = b k P - - C h k - -
EX, jbj o2 - 2hb, EX#b, o,. y
t
Then h = d ( k l M ) i f (OV/OXhk)>I(OV/OX/k) for all other open plants, " f ~ M . But (OV/gXht,)>~ ( O V / O X / , ) if Chk ~ C/,. Therefore, the low cost plant dominates in this problem despite the nonlinear risk adjustments in (4). For cases where multiple plants have identical costs, one is arbitrarily designated as the dominant plant. In what follows, we will use a superscript d to indicate a dominant plant. This change is for notational simplicity. The market for wlfich a particular plant is dominant will be clear from the context. For example, Xf" denotes shipments to market j from its dominant plant. Then for a given set of open plan~s, M, (2) can be rewritten as maximize j subject tO
i~M
j
-
j
(e)
o .< xf.<. Dj for all j.
Since (6) is conditional on the set of open plants, it represents the nodal subprob!em to be solved if a branch-and-bound procedure is used to solve (2). The Kuhn-Tucker conditions for this nod~l subproblem are: bkP_ck-
d - 2 ~ b k o 2 E X j d b j _ 2;~"d'2 2 , + I z k, = O , Ai
k=l,
.. . , ,
J.E.
44
an~
partiall 3 renurnl~
Aj>~Aj+, f o r j = l , : . . , n - 1 . th bc & Aj-H +
E
j
2A%2
j~R(H)
(14)
-
yes(H)
and E
MR=H=
H) 2 A _ 2 +
jER(H)
E
j~S(H)
~ - W j
.
(15)
Notice that tths composite MR curve is piecewise linear and only changes its slope at those points where active variables reach their upper bounds or where inactive variables become active as H is decreased. The slope of this function a~ong a linear segment where the set R (H) is unchanging is
= -
aMa
2xo} -'.
E
(16)
Thus the MR function is non-increasing; and since MC is linearly increasing, the intersection of MR and MC is unique and can be found quite easily. For our three-market example, the results of this process are depicted in Figure 2. For values of H above H 0 = A 1, no market is served. Between H o and H 1, only market 1 is served. At
I~VR MC
HO= A 1
HI = A 2
---
I
H2 = L2
. . . . . . . . . . . . . . . . .
H3=A 3
-_
1. . . . . . . . . . . .
[
o
1
2
3
4
5
s
7
8
9
10
~
12
W(H) = :£jWj{H}
Figure 2. The aggregate MR. curve and {he MC curve corresponding to the example shown in Figure 1.
45 = L 2, where shipments to marke~ 2
, t o market 1, Finally at H 3 = A,. and Ha, determining MC*. Given ~ation (IVi*) as depicted in Figure aent of the M R curve where the intersection occurs. This // Aj Or H = Ly) starting with H 0 = A~ until MC >i MR. ; R(H)and S(H ). Then MC* is easily calculated from with M R = MC. =
6,, Concluding comments in the previous section is quite simple and except for the sorting procedure, the at most linearly with the size of the nodal subproblem. The algorithm has been coded for a popular micro,computer in Pascal; and even in this relatively slow environment, solutions are developed faster than they can be displayed on a CRT. Since the nodal subproblems in (6) can be solved so easily, solutions to the location problem described by ( 2 ) s h o u l d be computationally feasible for reasonably large numbers of potential plant sites axed markets. Indeed' the constraints . . . in . (2) are very sindlar to those of the classic simple plant location probleN described a n d ' s o l v e d ' by Efroymson and Ray (1966), Khumawala (1972), and others. Thug, similar branch-and'bound techniques coupled with our nodal algorithm may be quite efficient. As with many plant location problems, dominance is crucial for a simple solution procedure. Fortunately dominance still holds for our problem, allowing us to relax somewhat the independence, assumption of Jucker and Carlson. Thi~ can be important since the existence of a random factor which i~ common to all markets reduces the firm's ability to 'diversify' its risks. Generally speaking this results in z lower aggregate production than would occur with independence, as well as different production and allocation patterns. We should mention that it is possible to extend the model presented here to incorporate a ~inear downward-Moping demand curve similar to that in Jucker and Carlson (1976), if it is desirable to model a firm with monopoly power. This requires only a algebraic substitution and was not done here for reasons of simplicity and ease o f exposition.
References Carbone, R. (1974), "Public facilities location under stochastic demand", INFOR 12, 261-270. Efroymson, M.A., and Ray, T.L. (1966), "A branch-bound algorithm for plant location", Operation~ Research 14, 361-368. Elton, E.J., Gruber, M.J., and Padberg, M.W. (1976), "Simple criteria for optimal portfolio selection", The Journal of Finance 31, 1341-57. Elton, E.J., Gruber, M.J., and Padberg, M.W. (1977), "Simple criteria for optimal portfolio section with upper bounds", Operao,wns Research 25, 952-967. Erlenkotter, D. (1978), "A dual-based procedure for uncapacitated facifity location '~, Operations Research 26, 992-1009. Francis, R.L., and Goldstein, LM. (1974), "Location theory: a selective bibliography", Operations Research 22, 400-410. Francis, R.L., McGinnis, LF., and White, J.A. (1983), "Locational analysis", European Journal of Operational Research 12, 220-252. Frank, H. (1966), "Optimum locations on a graph with probabilistic demands", Operations Research 14, 402-421. Handler, G.Y., and Mirchandani, P. (1979), Location on Networks: Theory and Aigoritb.ms, M.I.T. Press, Cambridge, MA. Howard, R.A. (1971), "Proximal decision analysis", Management Science 17, 507-541. Jucker, J.V., and Carlson, R.C (1976), "The simple plant-location problem under uncertainty", Operations Researcl~ 24, 1045-55. Jucker, J.V., Carlson, R.C., and Kropp, D.H. (1982), "The simultaneous determination of plant and leased warehouse.' capacities for a firm facing uncertain demand in several regions", lEE Transactions 14, 99-108. Jucker, LV., ~,nd Faro, C. de (1975), "A simplified algorithm for Stone's version of the portfolio selection problem", Journal of Financial and Quantitative Analysis 10, 859-870, Khumawala, B.M. ,1972), An efficient branch-and-bound algorithm for the warehouse location problem", Management Scwnce 18, B.718,B.731. [
'~
•
.
:
: =
39
~, Stanford, California
A~~ ~ e uncapacitated plant location problem under uncertainty is formulated in a mean-variance frame withprices in Various markets correlated via their response to a common random factor. This formulation results in a mixed-i~tteger quadratic programming problem. However, for a given integer solution; the resulting quadratic p r o g r a ~ n g problem is amenable to a very simple solution procedure. The simplicity of this algorithm means that reasonably large problems should be solvable using existing branch-and-bound techniques. Keyworfls: Facilities, location, mixed-integer quadratic programming
Introduction
Consider a single-period model with n markets (indexed by j ) and rn potential plant locations (indexed with i). The selling price for the product in market j is denoted by Pj, wit'h X~j being the quantity shipped from plant i to market j. The unit cost of serving market j from plant i (including production and transportation expenses) is denoted by %. We use ~ to indicate an upper bound on the demand level h~ the j-th market, Y~ (a zero-one decision variable) to indicate whether or not to open plant i, and F ~.he fized cost of opening the i-th plant. With these definitions, the plant location problem can be represemed as
i j subjectto
i
0~
for all j,
Y/=0, 1
for all i.
(1)
This is a standard, uncapacitated, plant location problem with a profit maximization objective. There is a vast ~ o u n t of literature dealing with this and related problems as well as procedures for solving these Fr~cis and Goldstein (1974), Francis, McGinnis and White (1983), Krarup 1983)). The bulk of this literature assumes that quantities demanded, prices, and costs are ~ o w n in actual situations, there may be a considerable lag between when a the facility is in place and operating. There i~ frequently considerable difficulty predicting the length of such a lag or the demand, costs, and prcies wMch wil! obtain when it elapses' ~ o s e who, :have addressed such uncertainty in locadon models have dealt mostly with demand and ReceivedFebruary1984; revisedJuly 1984 European Journal of Operational Research21 (1985)39-46 0377-2217/85/$3;30 ©I985, ElsevierSciencePublishersB.V. (North'Holland)
There are a number of different ways that uncertainty cm In order for the product market to clear, some combination ~ to changes in demand. Firms in both manufacturing and service industries use a variety of mechanisms in an attempt to maintain stable sales and stable production. Consider, for example, the use of discounting, r~bates, and special advertising in the automobile and airline industries. For planning purposes, such firms assume that production will be stable. Consequently, modeling such firms as using a quantity-setting strategy for planning purposes is reasonable. In addition to quantity-setting firms, Jucker and Carlson (1976) considered three other types of firms. For these other firms, uncertainty was realized in the quantity sold at a known price. This assumption of quantity, rather than price adjustment may be appropriate for planning purposes in cases such as regulated industries; however, the quantity-setting planning assumption is probably a much better one for many firms whose prices are not regulated. Another reason for focusing on quantity-setting firms is becaus, ~.it is possible to incorporate interrelated demand uncertaintly across markets via correlated random prkgs in a relatiwfly simple way. The alternative of having uncertainty reflected in correlated random quantities is not amenable to the simple solution procedure described herein. As risk increases, most individuals and even firms change their actions. The possibility of a substantial loss, like the hangman's noose, concentrates one's mind. To incorporate this aspect into the model, we will use a raean-variance objective function to account for the possibility of risk aversion (see Howard (1971), and J~cker and Carlson (1976) for discussions of the mean-variance criterion and its use for complex decis~.on problems and plant-location problems, respectively).
1. The model Changing the deterministic formulation of problem (1) into a formulation consistent with the preceding discussion is accomplished by allowing the Pi to be random and introducing an objective function of the form V = S ( ¢ r ) - hvar(cr). The parameter ~ is a risk aversion measure which describes the decision m~ker's willingness to trade expected return in order to reduce risk as measured by var(~'). We should add that in this new n~odet, the upper bounds on the demand levels in the various markets (the Dj.'s)need n o t be restricted to finite numbers. The algorithm will work whether or not there is a finite upper bound on auantity demanded in any market. Assuming a standard mean-variance objective function with Pj random, this problem can be stated as
i
subject to
j
i
0 ~
for all i and j,
EiXij~Dj
for all j,
Y~= 0,1
for all i.
i
j
-
(2)
i(
:!:~f:r
/:<
:
41
: J.E. Hodder, J. V. Jucker / A siraple plant-tocation model
the Py are i~ndependent. We allow for
(3)
•j).
Inthis for ml llation, ~) is a random factor common to all markets which is statistically distributed with mean 4 . Tho i:oefficient bj > 0 represents an adjustment parameter for the j-th market, with ej ~ independently distrib|ated random ,error term having mean zero and variance of. This use of a common factoi" allows for correlated prices in a manner which will be analytically tractable while representing a substantial improvement over the asumption of complete independence. U t i l ~ n g (3)we can rewrite our objective function as
v=EE(bj i
-c,j)xu-EF, E - x
j
i
EX, jb; o
-xE Ex, jb, o/
" j
j
~,4)
i
2, The nodal subproblem Our problem can be simplified by showing that for a given set of open plants, each market will be served from at most one plant--its 'dominant' plant. This sort of dominance plays an importaat role in the branch-and-bound procedures for the deterministic plant location model (e.g., Efroymson and Ray (1966)) as well as for some location problems with uncertainty (Jucker and Carlson (1976)). Our solution procedure also requires that the identity of the dominant plant (among the open set) does not depend on production levels at that or any other plant. This 'consistent' dominance is perhaps not immediately obvious with the objective function in (4); however, such dominance can easily be demonstrated. Let M denote the set of open plants and d ( j l M ) the (consistently) dominant plant for market j conditional on M. For some market k and an open plant h,
2hbk
aXak = b k P - - C h k - -
EX, jbj o2 - 2hb, EX#b, o,. y
t
Then h = d ( k l M ) i f (OV/OXhk)>I(OV/OX/k) for all other open plants, " f ~ M . But (OV/gXht,)>~ ( O V / O X / , ) if Chk ~ C/,. Therefore, the low cost plant dominates in this problem despite the nonlinear risk adjustments in (4). For cases where multiple plants have identical costs, one is arbitrarily designated as the dominant plant. In what follows, we will use a superscript d to indicate a dominant plant. This change is for notational simplicity. The market for wlfich a particular plant is dominant will be clear from the context. For example, Xf" denotes shipments to market j from its dominant plant. Then for a given set of open plan~s, M, (2) can be rewritten as maximize j subject tO
i~M
j
-
j
(e)
o .< xf.<. Dj for all j.
Since (6) is conditional on the set of open plants, it represents the nodal subprob!em to be solved if a branch-and-bound procedure is used to solve (2). The Kuhn-Tucker conditions for this nod~l subproblem are: bkP_ck-
d - 2 ~ b k o 2 E X j d b j _ 2;~"d'2 2 , + I z k, = O , Ai
k=l,
.. . , ,
J.E.
44
an~
partiall 3 renurnl~
Aj>~Aj+, f o r j = l , : . . , n - 1 . th bc & Aj-H +
E
j
2A%2
j~R(H)
(14)
-
yes(H)
and E
MR=H=
H) 2 A _ 2 +
jER(H)
E
j~S(H)
~ - W j
.
(15)
Notice that tths composite MR curve is piecewise linear and only changes its slope at those points where active variables reach their upper bounds or where inactive variables become active as H is decreased. The slope of this function a~ong a linear segment where the set R (H) is unchanging is
= -
aMa
2xo} -'.
E
(16)
Thus the MR function is non-increasing; and since MC is linearly increasing, the intersection of MR and MC is unique and can be found quite easily. For our three-market example, the results of this process are depicted in Figure 2. For values of H above H 0 = A 1, no market is served. Between H o and H 1, only market 1 is served. At
I~VR MC
HO= A 1
HI = A 2
---
I
H2 = L2
. . . . . . . . . . . . . . . . .
H3=A 3
-_
1. . . . . . . . . . . .
[
o
1
2
3
4
5
s
7
8
9
10
~
12
W(H) = :£jWj{H}
Figure 2. The aggregate MR. curve and {he MC curve corresponding to the example shown in Figure 1.
45 = L 2, where shipments to marke~ 2
, t o market 1, Finally at H 3 = A,. and Ha, determining MC*. Given ~ation (IVi*) as depicted in Figure aent of the M R curve where the intersection occurs. This // Aj Or H = Ly) starting with H 0 = A~ until MC >i MR. ; R(H)and S(H ). Then MC* is easily calculated from with M R = MC. =
6,, Concluding comments in the previous section is quite simple and except for the sorting procedure, the at most linearly with the size of the nodal subproblem. The algorithm has been coded for a popular micro,computer in Pascal; and even in this relatively slow environment, solutions are developed faster than they can be displayed on a CRT. Since the nodal subproblems in (6) can be solved so easily, solutions to the location problem described by ( 2 ) s h o u l d be computationally feasible for reasonably large numbers of potential plant sites axed markets. Indeed' the constraints . . . in . (2) are very sindlar to those of the classic simple plant location probleN described a n d ' s o l v e d ' by Efroymson and Ray (1966), Khumawala (1972), and others. Thug, similar branch-and'bound techniques coupled with our nodal algorithm may be quite efficient. As with many plant location problems, dominance is crucial for a simple solution procedure. Fortunately dominance still holds for our problem, allowing us to relax somewhat the independence, assumption of Jucker and Carlson. Thi~ can be important since the existence of a random factor which i~ common to all markets reduces the firm's ability to 'diversify' its risks. Generally speaking this results in z lower aggregate production than would occur with independence, as well as different production and allocation patterns. We should mention that it is possible to extend the model presented here to incorporate a ~inear downward-Moping demand curve similar to that in Jucker and Carlson (1976), if it is desirable to model a firm with monopoly power. This requires only a algebraic substitution and was not done here for reasons of simplicity and ease o f exposition.
References Carbone, R. (1974), "Public facilities location under stochastic demand", INFOR 12, 261-270. Efroymson, M.A., and Ray, T.L. (1966), "A branch-bound algorithm for plant location", Operation~ Research 14, 361-368. Elton, E.J., Gruber, M.J., and Padberg, M.W. (1976), "Simple criteria for optimal portfolio selection", The Journal of Finance 31, 1341-57. Elton, E.J., Gruber, M.J., and Padberg, M.W. (1977), "Simple criteria for optimal portfolio section with upper bounds", Operao,wns Research 25, 952-967. Erlenkotter, D. (1978), "A dual-based procedure for uncapacitated facifity location '~, Operations Research 26, 992-1009. Francis, R.L., and Goldstein, LM. (1974), "Location theory: a selective bibliography", Operations Research 22, 400-410. Francis, R.L., McGinnis, LF., and White, J.A. (1983), "Locational analysis", European Journal of Operational Research 12, 220-252. Frank, H. (1966), "Optimum locations on a graph with probabilistic demands", Operations Research 14, 402-421. Handler, G.Y., and Mirchandani, P. (1979), Location on Networks: Theory and Aigoritb.ms, M.I.T. Press, Cambridge, MA. Howard, R.A. (1971), "Proximal decision analysis", Management Science 17, 507-541. Jucker, J.V., and Carlson, R.C (1976), "The simple plant-location problem under uncertainty", Operations Researcl~ 24, 1045-55. Jucker, J.V., Carlson, R.C., and Kropp, D.H. (1982), "The simultaneous determination of plant and leased warehouse.' capacities for a firm facing uncertain demand in several regions", lEE Transactions 14, 99-108. Jucker, LV., ~,nd Faro, C. de (1975), "A simplified algorithm for Stone's version of the portfolio selection problem", Journal of Financial and Quantitative Analysis 10, 859-870, Khumawala, B.M. ,1972), An efficient branch-and-bound algorithm for the warehouse location problem", Management Scwnce 18, B.718,B.731. [
'~
•
.