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Supply chain coordination in spatial games
Int. J. Production Economics 71 (2001) 277}285
Supply chain coordination in spatial games Marija Bogataj*, Ludvik Bogataj Dipartimento marittimo e dei transporti Pirano, FPP, EF, University of Ljubljana, pot Pomorscakov 4, 66320 Portoroz, Slovenia
Abstract The notion that economies consist of the systems of cities originates from a theory by Christaller and Beckmann, based on a LoK chean framework of retailers having an endogenously determined market area. In 1968 it was completed by Tinbergen, who has suggested production approach. This original theory explains the location pattern of the market and manufacturing centers selling goods to the local populations within given market areas. The theory was based on the idea of the urban area being a trading center. Our paper, in#uenced by the ideas, presents the quantitative method of building up the model of spatial hierarchy as the result of spatial games. The paper by Girlich: `On the Metric Transportation Problems and Their Solutiona (Inventory Modelling, Lecture Notes of the International Postgraduate Summer School, Vol. 2, ISIR, Budapest-Portoroz\ , 1995, pp. 13}24), gives us proper foundations for our present research. In this paper customer traveling problem is described, which helps us to de"ne a market area. A mathematical description of market areas, de"nes the border between two areas under the assumption of shortages of goods in at least one market area. The study is in#uenced by GrubbstroK m's work on MRP optimization (for details see R.W. GubbstroK m, Management Science 13(7) (1967) 558}567; The Mathematical Scientist 16 (1991) 118}129; International Journal of Production Economics, in press; International Encyclopaedia of Business and Management, Routledge, London, 1996; R.W. GrubbstroK m, A. Bogataj (Eds.), Storlien, 1997, FPP, Portoroz\ , 1998; R.W. GrubbstroK m, A. Molinder, International Journal of Production Economics 35 (1994) 289}311.). In our paper the problem is formulated as a non-constant sum game, which is the case of the total market area being constant and the conditions for lower central places, described for di!erent positions of this central place in an urban hierarchy, in the case of two-level production. Di!erential games are suggested when the total market area changes over the time. 2001 Elsevier Science B.V. All rights reserved. Keywords: Non-constant sum games; Location; Inventory; Logistics; MRP; Input}output analysis; Laplace transforms; Annuity stream; Shortage of goods; Duopoly; Customer traveling problem
1. Introduction In any national economy there is an urban hierarchy with one or two large cities, more medium-sized cities and towns and a larger number of smaller towns and villages. A number of factors
* Corresponding author. Tel.: # 386-66-477-100; fax: 386-66477-130. E-mail address: [email protected] (M. Bogataj).
in#uence the nature of a hierarchical urban system. The central place theory, developed originally by Christaller [1] and LoK sch [2] is explaining the location pattern of market and manufacturing centers selling goods to the local populations within given market areas. This theory is based on the idea of the urban area as a trading center. According to the theory, the smallest urban places, such as villages, sell commonplace goods like food, to the population inside it and to the population of its own small surroundings. Above the villages there is
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 1 2 5 - 0
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a smaller number of larger central places, these are small towns, which sell less commonly purchased goods, e.g. clothing, to the populations of its market areas. Larger towns sell even less frequently purchased goods, e.g. furniture, to still larger market areas. In the context of the central place theory, the whole land area is divided into a series of nested sets of hexagonal market areas. Market areas of all urban places are supposed to be hexagons since any area could be divided into a regular hexagon with no gaps or overlaps, while the partition to circles, though theoretically more understandable would lead to overlaps and gaps. The fact is, the hexagon is the nearest to a circle and it provides a pattern without gaps. While in non-industrial areas the predictions of the theory describing spatial system are in agreement with the empirical results, the explanation for industrialized economies is not good enough as industrial production in towns and cities is not only for distribution in the surrounding market areas but may also be for sale throughout the nation and for export all over the world. How to construct a theory, which would explain the urban system in an industrial and postindustrial economy? In the model suggested by Tinbergen [3] production approach was suggested as follows: The smallest towns and villages are producing some goods, e.g. the goods of type TY . The next largest towns are producing goods of type TY and TY . They meet the de mands of their own population and they are exporting goods of TY to the smaller central places and villages. Larger towns manufacture three kinds of goods, TY ,TY and TY ! for their own population, exporting TY and ! so on. The market area of goods is the result of the competition of spatial oligopoly. In case that some central places of a certain level are stronger than others, their market area starts growing and attracting customers from other central places which are on the same level. This process will be described by the models of spatial games. In real-world problems production and competition for customers is not so simple. The #ow of goods between central places can be described with
input}output matrices. The availability of certain goods in a certain central place depends on production } transportation and supply chains. The cumulative demand of a certain product or raw material can exceed the cumulative supply. In this case the shortage of goods can appear. If such a situation arises frequently, result in some market areas of the same level growing bigger and some of them becoming smaller. Various approaches describing production, inventory, transportation and consumption as the transformation of one set of resources to another set of resources and its distribution can be found in literature (for details see [4]). The study of spatial interactions and their in#uence on optimal inventory policy was presented in the paper `Inventories in Spatial Modelsa [5] and it was analyzed in detail in [6]. The framework of this problem as a market game was pointed out by Horvat and Bogataj [7}9] derived from [7,10], but there the net present value approach was not explicitly connected with spatial games though it appears in their approach. This has to be the main contribution of our paper. Many production}supply units, which form a spatial oligopoly, need shops, retails and warehouses to secure the "nal production stage and supply of "nal products. In our paper we are describing a two-level competition, used in the framework of Tinbergen consideration mentioned above. In central place B on a higher level, with more specialized and costly human resources, the manufacture of di!erent kinds of components takes place. TY denotes all these components. Final production and supply can take place in A with lower wages and land rent (warehousing), or in B. The games answer these questions: `What should the parameters in MRP for A and B that agree with "nal production in A be? What should the parameters in MRP for A and B that agree only marketing of "nal product in A be?a These conditions describe the hierarchical position of A in spatial hierarchy relative to B. The main answers we are looking for are: 1. When is A supposed to lose the position of a central place for a certain group of products?
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2. When is it supposed to be only a local distributor of a "nal product? 3. What are the conditions to maintain the role of the producer of "nal products? In case the "nal products are assembled at the location of supply, the components for the production of these items have to be available in time (JIT) with an acceptable shortage level. This production and supply policies consist of MRP decisions, determining the price and acceptable shortage of goods. Annuity stream approach is used here as it is often used in the papers of GrubbstroK m [4,11}15].
2. Customer:s traveling problem (CTP) In von Girlich's study [16] the in#uence of shortage of goods on the market area was presented as follows. We assume that customers locations are continuously distributed with the density of demand r(z)"r'0 on the area S , S "z3R: r(z)'0. Retailer or production unit of ith "rm, located at the point z of this area, is selling the "nal product G which is available at ith "rm at an arbitrary chosen moment with probability . This probability G depends on MRP policy. The set of all customers' locations supplied primarily from z is denoted by G S , de"ned as G S "z3S: EC (z))EC (z), jOi, (1) G G H where C (z) denotes the random cost for a unit of G the commodity for a customer with residence z patronizing warehouse i. EC (z) is its mathematical G expectation (see von Girlich [16]). We assume A"A , A A ", iOj, i"1, 2,2, n. (1a) G G H In this paper, the distance between the retailer i location (z "(x , y )) and consumer's location G G G z"(x, y) is assumed to be Euclidean. We are looking for a rational partition of total demand corresponding to market areas described
279
by (1). Every customer has to patronize one retail. His decision depends on E the mill price p for the commodity at i, G E the transportation cost, E the uncertainty at each i. G When a customer arrives at z , there are two G possible events: 1. supply unit i is empty .........................= "0 G 2. the commodity is available at i...........= "1. G The uncertainty of this oligopoly system can be described by random variables = , = , = ,2, = , with the state space 0, 1 of each L random variable. P(= "0)" 3[0, 1] and G G P(= "1)"1! 3[0, 1], i"1, 2,2, n. (2) G G Here is the probability of shortage of goods at ith G supply unit. In his work, the author simpli"es the situation so that there are only two production}supply enterprises with their supply in z "(1, 0) for the "rst unit and z "(0, 0) and he describes their market area if P(= "1= "0)"1 G H for iOj, i"1, 2, j"1, 2.
(3)
This assumption ensures that the commodity is available in the other supply unit in case there is a shortage of production in jth unit, j"1, 2. We call it a D model. In the games approach, which will describe a two-level model, we will assume that there are two centers, A on a lower level and B on a higher level and that B is a reliable supplier (has the goods with certainty). (i"A, j"B). Take C (z) as the cost of a unit of commodity for G a customer with residence z patronizing supply unit i. Because of uncertain supply in i, where he is going to buy this product at his "rst travel, C is a random G variable. If l is the Euclidean distance to the ith G supply unit and t the corresponding transportaG tion cost factor, we obtain for iOj:
p #2t l (z) for = "1, GG G C (z)" G G t l (z)#t l #p #t l (z) for = "0. GG GH GH H HH G (4)
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Here t is the transportation cost factor for travelGH ing from i to j which arises in case that the good are not available in i and l is the distance from ith to GH jth unit. Every customer from area S "rst travels to A and in case there is a shortage of this item there, he travels to B , where (according to our assumpH tion) the product is certainly available. From (4) follows the boundary between S and S : EC (z)"EC (z), (2! ! )(t l (z)!t l (z)) G H GG HH "t l ( ! )#(p !p )(1! ! ), GH GH H G H G G H i"A, j"B.
(5)
c
v
v
c
(5a)
In the paper [17] we have assumed "xed parameters for production and supply unit B. What is the result when B acts as a player competing for the same market area as A and is changing parameters? Which conditions have to be ful"lled for the market area for A, S '0, if B is able to change its para meters and if "; "0? As the transportation costs are known, boundaries S and S depend only on the shortage , and G "nal price p , i"A, B of duopolist. The boundary G divides S and S and therefore, the boundary determines the demand d , i"A, B at z . G G 3. MRP approach We use the following notation: A
c
production and supply unit (town) on lower level B production and supply unit (town) on higher level NPV annuity stream of production and supply unit A ("rst-order approximation) NPV annuity stream of production and supply unit B ("rst-order approximation) d external demand rate for "nal products in A d external demand rate for "nal products in B c the echelon stock value of a "nal product in A (value added in the process of manufacturing the item)
the echelon stock value of a "nal product in B (value added in the process of manufacturing the item), the echelon stock value of a component in A, in our case the following is valid: c "0, because components are only produced in B (value added in the process of manufacturing the item), the price of all components for one assembly into the "nal product (the purchase price of components, bought by A) the price of the "nal product (the purchase price of A, which can be reduced) the echelon stock value of a component in B (a value added in the process of manufacturing the item)
We can expect that the following two conditions are valid: v
'c , (6) c (c . (7) But, are they strong enough? In this study we show that those conditions are not strong enough. Some further notations:
K K K K ¹ ¹ ¹ ¹ K 20
continuous interest rate the set-up cost of "nal products in A the set-up cost of "nal products in B The set-up cost of components in A the set-up cost of components in B ordering interval for "nal products in A ordering interval for "nal products in B ordering interval for components in A ordering interval for components in B the cost of transportation of components from B to A (which is a part of setup costs for the production of a "nal product).
We have two cases. First, A is only ordering "nal products and secondly, A is only ordering components and assembles them into "nal products, therefore setup costs are K "0. m "¹ /¹ , (8) p selling price for "nal product in A, p selling price for "nal product in B, K "K #K !K . 20
(9)
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281
Here K is the di!erence of set-up costs between locations because of lower wages for workers performing setups (di!erent labor costs), and lower prices of land use at required warehouse facilities for A. From the theory of urban economics it follows that c 'c because of specialized human re sources on a higher level that is not available in smaller towns. Manufacturing the "nal product does not need specialized workers, so it can also be produced in A where wages are supposed to be lower. We assume that
More speci"cally: Duopolist i, i"A, B, determines the value of and p so that demand d , G G G
c "c #c , c '0. (10) Duopolist i (i"A, B) determines the value of shortage of goods so that demand d (i"A, B), G G d "r((1! )S # S ) (11) G G G H H will give the optimal annuity stream NPV . G Instead of minimizing average costs involved in production, inventory and distribution processes, a more correct procedure is to maximize the net present value (NPV) of all payments associated with the processes concerned (see [4,8,9,11}15]). Multiplying the NPV by an annuity factor , which in the in"nite horizon case is equal to the interest rate, we obtain the payment stream yielding the same NPV. This stream is called the annuity stream. When the payments are described in terms of their Laplace transforms, the NPV is directly obtained by exchanging the complex frequency s for the continuous interest rate in all relevant transform expressions [4].
will give the optimal annuity stream NPVH, G i"A, B. The annuity stream formulas (12) and (13) are derived from the general expression for the annuity stream using MRP theory, input output analysis and Laplace transform given in [7] which is a special two-level case of general expression of NPV, presented in (13). The annuity stream expressions here are "rst-order approximations in . Possible decisions of A are the following: (1) Decision A1: A is only buying the "nal products from B and can expect a shortage probability of this product according to uncertain demand and time intervals between orders. In this case his annuity stream is
NPV ( , , p , p ) "d ( , , p , p )(p !c !v ) 1 ¹ c , K !d ! K ! 2 2 ¹ NPV ( , , p , p , ¹ , m ) "d ( , , p , p )(p !c !c ) 1 K ! (K #K )! K # 2 ¹ m ¹ (c #m c ). !d 2
(12)
d "r(1!)S ,
(14)
d "r(S #S )"r(S #S!S )
(15)
and "nally d "r(S!(1!)S )
(16)
NPV (1, .)"d ( , , p , p )(p !v )
!
1 # K . 2 ¹
(17)
(2) Decision A2: A is only buying components and is involved in "nal production, selling the goods to its market area with a shortage probability of this product dependent on the uncertainty of demand. In this case, its annuity stream is NPV (2, .)"d ( , , p , p )(p !c !v )
(13)
1 ¹ c . ! K ! K !d 2 2 ¹ (18)
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Possible decisions of B are the following: (1) Decision B1: B is selling to A a quantity of "nal products at reduced price v : NPV (., 1)"d ( , , p , p )(p !c !c ) # d ( , , p , p )v ! (K #K ) 2
1 K ! K # ¹ m ¹ (c #m c ). !d 2
(19)
(2) Decision B2: B is selling to A a quantity of components at price v . NPV (., 2)"d ( , , p , p )(p !c !c ) ! (K #K ) 2 1 ! ¹
K K # m
¹ ! (d !d ) (c #m c ) 2 #d ( , , p , p )v . (20) For the sake of clarity of expressions, we choose the special case of equal mill prices and no shortage in B. The annuity streams are the following, concerning the optimal ordering policy (¹H, mH) for each reward matrix position separately: NPV (1, 1)"r(1!)S ((p !v )! K 2 1 r(1!)S c' ¹ , ! K ! ¹ 2 (21) where c' denotes storage costs of a unit of "nal product in A. NPV (1, 2)"r(1!)S (p !p )! K 2 1 ! K (0, ¹
(22)
NPV (2, 1)"r(1!)S (p !c !v ) 1 ! K ! K !r 2 ¹ ¹ c , (1!)S 2 NPV (2, 2)"r(1!)S (p !c !v ) 1 ! K ! K 2 ¹ ¹ c "0. !r(1!)S 2
(23)
(24)
In (23) and (24) the inventory costs are included in c . NPV (1, 1)"r(S!(1!)S )p !rS(c !c )#r(1!)S v ! (K #K ) 2
1 K ! K # ¹ m ¹ !rS (c #m c ), 2
(25)
NPV (2, 1)"r(S!(1!)S )(p !c ) !rSc #r(1!)S v ! (K #K ) 2 1 ! ¹
K K # m
¹ !r(S!(1!)S ) c 2 ¹ !rS m c . 2
(26)
Here v denotes the price by which B delivers components to A. When B covers the total market area S because B is not selling any goods to A, B acts as monopolist.
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annuity stream under constraints given by the second player. It is easy to see that the following conditions:
Because of equal mill prices we can write NPV (1, 2)"rS(p !c !c ) ! (K #K ) 2
1 ! ¹
NPVH (2, 1)' NPVH (1, 1), NPVH(2, 1)' NPVH(2, 2)
1 K ! K # ¹ m ¹ !rS (c #m c ), 2 NPV (2, 2)"rS(p !c !c ) ! (K #K ) 2 K K # m
(27)
(28)
4. The results of spatial games The natural question arises what kind of conditions should be ful"lled for A to patronize an area according to Christaler model on a lower level than B, so that it will manufacture the "nal product and sell it to his market area S . We have the following reward matrix of the game:
(30) (31)
should be ful"lled to achieve that neither player can gain advantage by unilateral change in strategy. Thus ( NPVH (2, 1), NPVH(2, 1)) is an equilibrium point. We get the optimal annuity stream NPVH (1, 1) considering the necessary condition of optimality NPVH (1, 1) "0 (32) ¹ from which follows that the optimal time between orders is
¹ !rS (c #m c ). 2
283
( NPVH (1, 1), NPVH(1, 1)) ( NPVH (1, 2), NPVH(1, 2)) ( NPVH (2, 1), NPVH(2, 1)) ( NPVH (2, 2), NPVH(2, 2))
.
(29) When (1, 1) is the solution of the game, A is selling only "nal products. When (2, 1) is the solution of the game, A is producing "nal products. When (1, 2) or (2, 2) is the solution of the game, A can only sell the goods in a small local area, but is not producing the "nal products. We have to answer the question, what conditions should be ful"lled so that the optimal strategy would be ( NPVH (2, 1), NPVH(2, 1)), which will make A the producer of a "nal product. Here the superscript H denotes the optimal value of the
2K ¹H " (33) r(1!)S c' and so the optimal value of the annuity stream NPVH (1, 1) can be expressed as NPVH (1, 1)"r(1!)S (p !v ) ! K !(2r(1!)S c' K . 2 (34) In a similar way, we also get the optimal annuity stream NPVH (2, 1) from the necessary condition NPVH (2, 1) "0, (35) ¹ from which we get
2K ¹H " r(1!)S c and
(36)
NPVH (2, 1)"r(1!)S (p !v ) ! K !(2r(1!)S c K . 2 (37) Considering (34) and (37) in (30) we "nd out that after some mathematical manipulations the follow-
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ing condition should be ful"lled:
We also get
r(1!)S (v !v !c )
1 mH " 0 ¹H 0 and
#(2r(1!)S K ((c !(c' )'0. (38) From (38) it follows: v
#c (v
2K Src
(48)
(S!(1!)S )c K . (49) K Sc The connection between optimal values for situations (2, 1) and (2, 2) is the following: mH " 0
(2r(1!)S K ((c !(c' ) . (39) # r(1!)S For player B, condition (31) should be ful"lled for the game. Let us denote situation (2, 1) by index ¸ and situation (2, 2) by index R. After stating the necessary conditions of optimality for situation (2, 1),
Let q denote the percentage of market area that belongs to B:
NPVH(2, 1) "0, m
(40)
S!(1!)S . q " S
(41)
It follows that the market share also strongly in#uences optimal parameters mH and mH , which * 0 can be seen from the relation
NPVH(2, 1) "0, ¹ from which follows 1 mH " * ¹H * and
2K Sr
(K #K /mH )2 * c SmH c ), ¹H " * * r((S!(1!)S we also get
(K #K /mH )2 0 . rS(c #mH c ) 0
(51)
(43)
(K #K /mH 1/q )2 0 . (53) ¹H " * r(S!(1!)S )c #mH c Sq ) 0 From (47) and (53) it is easy to see that the relationship among the times between two orders is the following:
(S!(1!)S )c K . (44) K SC To "nd the optimal value of NPV (2, 2), see Eq. (28), the necessary optimality conditions are
¹H " 0
(50)
mH "(q mH . * 0 In this case, we can also write
NPVH(2, 2) "0, ¹ from which follows
S!(1!)S mH . 0 S
(42)
mH " *
NPVH(2, 2) "0, m
mH " *
(45) (46)
¹H '¹H . (54) * 0 The condition that follows from (31) is the following: r(1!)S (v #c !p ) c K #(¹H !¹H ) Sr # '0, (55) 0 * 2 ¹H ¹H 0 * or, in a more explicit form,
v
(47)
(52)
#c 'p (¹H !¹H )(Src /2#K /¹H ¹H ) * 0 * . # 0 r(1!)S (56)
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Putting together both conditions (39) and (56) that should be ful"lled so that the optimal strategy for the game would be (2, 1), we get the main result: v
#c (v (2r(1!)S K ((c !(c' ) , (57) # r(1!)S v #c 'p (¹H !¹H )(Src /2#K /¹H ¹H ) * 0 * . # 0 r(1!)S
6. Conclusions and directions to di4erential games In this paper we have shown how to use MRP and input}output analysis (the NPV model) to investigate the results of customers' behavior when the supply units compete for customers and are exposed to an uncertain demand which results in a possible shortage of goods, assuming the total market area of both players to be constant. Player A will chose the optimal shortage of goods on that acceptable range. From the second term of right sides of conditions (52) we can see how important are the set-up costs which also include transportation expenses. This theory is only valid, if the total market area belonging to both players is constant. In case of the expansion of one or both players on the market area of a third central place, the theory of di!erential games has to be used. On the basis of these results, the optimal ordering policy which also includes the optimal shortage of goods can be worked out. These results can also be extended to the other customers' traveling behavior described by [16] and even to more general cases.
References [1] W. Chrtistaller, The Central Places of Southern Germany, Prentice-Hall, Englewood Cli!s, NJ, 1966. [2] A. LoK sch, The Economics of Location, New Haven, 1954.
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[3] J. Tinbergen, The hierarchy model of the size distribution of centres, Papers and Proceedings of the Regional Science Association 20 (1968). [4] R.W. GrubbstroK m, A. Molinder, Further theoretical consideration on the relationship between MRP, input} output analysis and multiechelon inventory systems, International Journal on Production Economics 35 (1994) 299}311. [5] M. Bogataj, Inventories in spatial models, International Journal of Production Economics 45 (1996) 337}342. [6] M. Bogataj, Location-inventory problems in MRP, International Journal of Production Economics, to appear. [7] L. Bogataj, L. Horvat, Stochastic considerations of the GrubbstroK m}Molinder model of MRP, input}output analysis and multi-echelon inventory systems, International Journal of Production Economics 45 (1996) 329}336. [8] L. Horvat, L. Bogataj, MRP, input}output analysis and multi-echelon inventory systems with exponentially distributed external demand, Proceedings of GLOCOSM Conference, Bangalore, India, 1996, pp. 149}154. [9] L. Horvat, L. Bogataj, A market game with characteristic function according to the MRP and input}output analysis model, International Journal on Production Economics 59 (1/3) (1999) 281}288. [10] R.W. GrubbstroK m, A. Molinder, Further theoretical consideration on the relationship between MRP, input}output analysis and multiechelon inventory systems, International Journal of Production Economics 35 (1994) 299}311. [11] R.W. GrubbstroK m, On the application of the Laplacetransforms to certain economic problems, Management Science 13 (7) (1967) 558}567. [12] R.W. GrubbstroK m, The z-Transform of tI, The Mathematical Scientist 16 (1991) 118}129. [13] R.W. GrubbstroK m, Stochastic Properties of a production}inventory process with planned production using transform methodology, International Journal on Production Economics, in preparation. [14] R.W. GrubbstroK m, Material requirements planning and manufacturing resources planning, in: M. Warner (Ed.), International Encyclopaedia of Business and Management, Routladge, London, 1996. [15] R.W. GrubbstroK m, L. Bogataj (Eds.), Input}output analysis and Laplace transforms in material requirements planning, Storlien, 1997, FPP, Portoroz\ , 1998. [16] H.J. von Girlich, On the metric transportation problems and their solution, in: L. Bogataj (Ed.), Inventory Modelling, Lecture Notes of the International Postgraduate Summer School, Vol. 2, ISIR, Budapest- Portoroz\ , 1995, pp. 13}24. [17] M. Bogataj, Uncertainty of delivery determining the market area, in: R.W. GrubbstoK m, L. Bogataj (Eds.), Input}Output Analysis and Laplace Transforms in Material Requirements Planning, Storlien 1997, FPP Portoroz\ , 1998, 115}122.
Supply chain coordination in spatial games Marija Bogataj*, Ludvik Bogataj Dipartimento marittimo e dei transporti Pirano, FPP, EF, University of Ljubljana, pot Pomorscakov 4, 66320 Portoroz, Slovenia
Abstract The notion that economies consist of the systems of cities originates from a theory by Christaller and Beckmann, based on a LoK chean framework of retailers having an endogenously determined market area. In 1968 it was completed by Tinbergen, who has suggested production approach. This original theory explains the location pattern of the market and manufacturing centers selling goods to the local populations within given market areas. The theory was based on the idea of the urban area being a trading center. Our paper, in#uenced by the ideas, presents the quantitative method of building up the model of spatial hierarchy as the result of spatial games. The paper by Girlich: `On the Metric Transportation Problems and Their Solutiona (Inventory Modelling, Lecture Notes of the International Postgraduate Summer School, Vol. 2, ISIR, Budapest-Portoroz\ , 1995, pp. 13}24), gives us proper foundations for our present research. In this paper customer traveling problem is described, which helps us to de"ne a market area. A mathematical description of market areas, de"nes the border between two areas under the assumption of shortages of goods in at least one market area. The study is in#uenced by GrubbstroK m's work on MRP optimization (for details see R.W. GubbstroK m, Management Science 13(7) (1967) 558}567; The Mathematical Scientist 16 (1991) 118}129; International Journal of Production Economics, in press; International Encyclopaedia of Business and Management, Routledge, London, 1996; R.W. GrubbstroK m, A. Bogataj (Eds.), Storlien, 1997, FPP, Portoroz\ , 1998; R.W. GrubbstroK m, A. Molinder, International Journal of Production Economics 35 (1994) 289}311.). In our paper the problem is formulated as a non-constant sum game, which is the case of the total market area being constant and the conditions for lower central places, described for di!erent positions of this central place in an urban hierarchy, in the case of two-level production. Di!erential games are suggested when the total market area changes over the time. 2001 Elsevier Science B.V. All rights reserved. Keywords: Non-constant sum games; Location; Inventory; Logistics; MRP; Input}output analysis; Laplace transforms; Annuity stream; Shortage of goods; Duopoly; Customer traveling problem
1. Introduction In any national economy there is an urban hierarchy with one or two large cities, more medium-sized cities and towns and a larger number of smaller towns and villages. A number of factors
* Corresponding author. Tel.: # 386-66-477-100; fax: 386-66477-130. E-mail address: [email protected] (M. Bogataj).
in#uence the nature of a hierarchical urban system. The central place theory, developed originally by Christaller [1] and LoK sch [2] is explaining the location pattern of market and manufacturing centers selling goods to the local populations within given market areas. This theory is based on the idea of the urban area as a trading center. According to the theory, the smallest urban places, such as villages, sell commonplace goods like food, to the population inside it and to the population of its own small surroundings. Above the villages there is
0925-5273/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 1 2 5 - 0
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a smaller number of larger central places, these are small towns, which sell less commonly purchased goods, e.g. clothing, to the populations of its market areas. Larger towns sell even less frequently purchased goods, e.g. furniture, to still larger market areas. In the context of the central place theory, the whole land area is divided into a series of nested sets of hexagonal market areas. Market areas of all urban places are supposed to be hexagons since any area could be divided into a regular hexagon with no gaps or overlaps, while the partition to circles, though theoretically more understandable would lead to overlaps and gaps. The fact is, the hexagon is the nearest to a circle and it provides a pattern without gaps. While in non-industrial areas the predictions of the theory describing spatial system are in agreement with the empirical results, the explanation for industrialized economies is not good enough as industrial production in towns and cities is not only for distribution in the surrounding market areas but may also be for sale throughout the nation and for export all over the world. How to construct a theory, which would explain the urban system in an industrial and postindustrial economy? In the model suggested by Tinbergen [3] production approach was suggested as follows: The smallest towns and villages are producing some goods, e.g. the goods of type TY . The next largest towns are producing goods of type TY and TY . They meet the de mands of their own population and they are exporting goods of TY to the smaller central places and villages. Larger towns manufacture three kinds of goods, TY ,TY and TY ! for their own population, exporting TY and ! so on. The market area of goods is the result of the competition of spatial oligopoly. In case that some central places of a certain level are stronger than others, their market area starts growing and attracting customers from other central places which are on the same level. This process will be described by the models of spatial games. In real-world problems production and competition for customers is not so simple. The #ow of goods between central places can be described with
input}output matrices. The availability of certain goods in a certain central place depends on production } transportation and supply chains. The cumulative demand of a certain product or raw material can exceed the cumulative supply. In this case the shortage of goods can appear. If such a situation arises frequently, result in some market areas of the same level growing bigger and some of them becoming smaller. Various approaches describing production, inventory, transportation and consumption as the transformation of one set of resources to another set of resources and its distribution can be found in literature (for details see [4]). The study of spatial interactions and their in#uence on optimal inventory policy was presented in the paper `Inventories in Spatial Modelsa [5] and it was analyzed in detail in [6]. The framework of this problem as a market game was pointed out by Horvat and Bogataj [7}9] derived from [7,10], but there the net present value approach was not explicitly connected with spatial games though it appears in their approach. This has to be the main contribution of our paper. Many production}supply units, which form a spatial oligopoly, need shops, retails and warehouses to secure the "nal production stage and supply of "nal products. In our paper we are describing a two-level competition, used in the framework of Tinbergen consideration mentioned above. In central place B on a higher level, with more specialized and costly human resources, the manufacture of di!erent kinds of components takes place. TY denotes all these components. Final production and supply can take place in A with lower wages and land rent (warehousing), or in B. The games answer these questions: `What should the parameters in MRP for A and B that agree with "nal production in A be? What should the parameters in MRP for A and B that agree only marketing of "nal product in A be?a These conditions describe the hierarchical position of A in spatial hierarchy relative to B. The main answers we are looking for are: 1. When is A supposed to lose the position of a central place for a certain group of products?
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2. When is it supposed to be only a local distributor of a "nal product? 3. What are the conditions to maintain the role of the producer of "nal products? In case the "nal products are assembled at the location of supply, the components for the production of these items have to be available in time (JIT) with an acceptable shortage level. This production and supply policies consist of MRP decisions, determining the price and acceptable shortage of goods. Annuity stream approach is used here as it is often used in the papers of GrubbstroK m [4,11}15].
2. Customer:s traveling problem (CTP) In von Girlich's study [16] the in#uence of shortage of goods on the market area was presented as follows. We assume that customers locations are continuously distributed with the density of demand r(z)"r'0 on the area S , S "z3R: r(z)'0. Retailer or production unit of ith "rm, located at the point z of this area, is selling the "nal product G which is available at ith "rm at an arbitrary chosen moment with probability . This probability G depends on MRP policy. The set of all customers' locations supplied primarily from z is denoted by G S , de"ned as G S "z3S: EC (z))EC (z), jOi, (1) G G H where C (z) denotes the random cost for a unit of G the commodity for a customer with residence z patronizing warehouse i. EC (z) is its mathematical G expectation (see von Girlich [16]). We assume A"A , A A ", iOj, i"1, 2,2, n. (1a) G G H In this paper, the distance between the retailer i location (z "(x , y )) and consumer's location G G G z"(x, y) is assumed to be Euclidean. We are looking for a rational partition of total demand corresponding to market areas described
279
by (1). Every customer has to patronize one retail. His decision depends on E the mill price p for the commodity at i, G E the transportation cost, E the uncertainty at each i. G When a customer arrives at z , there are two G possible events: 1. supply unit i is empty .........................= "0 G 2. the commodity is available at i...........= "1. G The uncertainty of this oligopoly system can be described by random variables = , = , = ,2, = , with the state space 0, 1 of each L random variable. P(= "0)" 3[0, 1] and G G P(= "1)"1! 3[0, 1], i"1, 2,2, n. (2) G G Here is the probability of shortage of goods at ith G supply unit. In his work, the author simpli"es the situation so that there are only two production}supply enterprises with their supply in z "(1, 0) for the "rst unit and z "(0, 0) and he describes their market area if P(= "1= "0)"1 G H for iOj, i"1, 2, j"1, 2.
(3)
This assumption ensures that the commodity is available in the other supply unit in case there is a shortage of production in jth unit, j"1, 2. We call it a D model. In the games approach, which will describe a two-level model, we will assume that there are two centers, A on a lower level and B on a higher level and that B is a reliable supplier (has the goods with certainty). (i"A, j"B). Take C (z) as the cost of a unit of commodity for G a customer with residence z patronizing supply unit i. Because of uncertain supply in i, where he is going to buy this product at his "rst travel, C is a random G variable. If l is the Euclidean distance to the ith G supply unit and t the corresponding transportaG tion cost factor, we obtain for iOj:
p #2t l (z) for = "1, GG G C (z)" G G t l (z)#t l #p #t l (z) for = "0. GG GH GH H HH G (4)
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Here t is the transportation cost factor for travelGH ing from i to j which arises in case that the good are not available in i and l is the distance from ith to GH jth unit. Every customer from area S "rst travels to A and in case there is a shortage of this item there, he travels to B , where (according to our assumpH tion) the product is certainly available. From (4) follows the boundary between S and S : EC (z)"EC (z), (2! ! )(t l (z)!t l (z)) G H GG HH "t l ( ! )#(p !p )(1! ! ), GH GH H G H G G H i"A, j"B.
(5)
c
v
v
c
(5a)
In the paper [17] we have assumed "xed parameters for production and supply unit B. What is the result when B acts as a player competing for the same market area as A and is changing parameters? Which conditions have to be ful"lled for the market area for A, S '0, if B is able to change its para meters and if "; "0? As the transportation costs are known, boundaries S and S depend only on the shortage , and G "nal price p , i"A, B of duopolist. The boundary G divides S and S and therefore, the boundary determines the demand d , i"A, B at z . G G 3. MRP approach We use the following notation: A
c
production and supply unit (town) on lower level B production and supply unit (town) on higher level NPV annuity stream of production and supply unit A ("rst-order approximation) NPV annuity stream of production and supply unit B ("rst-order approximation) d external demand rate for "nal products in A d external demand rate for "nal products in B c the echelon stock value of a "nal product in A (value added in the process of manufacturing the item)
the echelon stock value of a "nal product in B (value added in the process of manufacturing the item), the echelon stock value of a component in A, in our case the following is valid: c "0, because components are only produced in B (value added in the process of manufacturing the item), the price of all components for one assembly into the "nal product (the purchase price of components, bought by A) the price of the "nal product (the purchase price of A, which can be reduced) the echelon stock value of a component in B (a value added in the process of manufacturing the item)
We can expect that the following two conditions are valid: v
'c , (6) c (c . (7) But, are they strong enough? In this study we show that those conditions are not strong enough. Some further notations:
K K K K ¹ ¹ ¹ ¹ K 20
continuous interest rate the set-up cost of "nal products in A the set-up cost of "nal products in B The set-up cost of components in A the set-up cost of components in B ordering interval for "nal products in A ordering interval for "nal products in B ordering interval for components in A ordering interval for components in B the cost of transportation of components from B to A (which is a part of setup costs for the production of a "nal product).
We have two cases. First, A is only ordering "nal products and secondly, A is only ordering components and assembles them into "nal products, therefore setup costs are K "0. m "¹ /¹ , (8) p selling price for "nal product in A, p selling price for "nal product in B, K "K #K !K . 20
(9)
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281
Here K is the di!erence of set-up costs between locations because of lower wages for workers performing setups (di!erent labor costs), and lower prices of land use at required warehouse facilities for A. From the theory of urban economics it follows that c 'c because of specialized human re sources on a higher level that is not available in smaller towns. Manufacturing the "nal product does not need specialized workers, so it can also be produced in A where wages are supposed to be lower. We assume that
More speci"cally: Duopolist i, i"A, B, determines the value of and p so that demand d , G G G
c "c #c , c '0. (10) Duopolist i (i"A, B) determines the value of shortage of goods so that demand d (i"A, B), G G d "r((1! )S # S ) (11) G G G H H will give the optimal annuity stream NPV . G Instead of minimizing average costs involved in production, inventory and distribution processes, a more correct procedure is to maximize the net present value (NPV) of all payments associated with the processes concerned (see [4,8,9,11}15]). Multiplying the NPV by an annuity factor , which in the in"nite horizon case is equal to the interest rate, we obtain the payment stream yielding the same NPV. This stream is called the annuity stream. When the payments are described in terms of their Laplace transforms, the NPV is directly obtained by exchanging the complex frequency s for the continuous interest rate in all relevant transform expressions [4].
will give the optimal annuity stream NPVH, G i"A, B. The annuity stream formulas (12) and (13) are derived from the general expression for the annuity stream using MRP theory, input output analysis and Laplace transform given in [7] which is a special two-level case of general expression of NPV, presented in (13). The annuity stream expressions here are "rst-order approximations in . Possible decisions of A are the following: (1) Decision A1: A is only buying the "nal products from B and can expect a shortage probability of this product according to uncertain demand and time intervals between orders. In this case his annuity stream is
NPV ( , , p , p ) "d ( , , p , p )(p !c !v ) 1 ¹ c , K !d ! K ! 2 2 ¹ NPV ( , , p , p , ¹ , m ) "d ( , , p , p )(p !c !c ) 1 K ! (K #K )! K # 2 ¹ m ¹ (c #m c ). !d 2
(12)
d "r(1!)S ,
(14)
d "r(S #S )"r(S #S!S )
(15)
and "nally d "r(S!(1!)S )
(16)
NPV (1, .)"d ( , , p , p )(p !v )
!
1 # K . 2 ¹
(17)
(2) Decision A2: A is only buying components and is involved in "nal production, selling the goods to its market area with a shortage probability of this product dependent on the uncertainty of demand. In this case, its annuity stream is NPV (2, .)"d ( , , p , p )(p !c !v )
(13)
1 ¹ c . ! K ! K !d 2 2 ¹ (18)
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Possible decisions of B are the following: (1) Decision B1: B is selling to A a quantity of "nal products at reduced price v : NPV (., 1)"d ( , , p , p )(p !c !c ) # d ( , , p , p )v ! (K #K ) 2
1 K ! K # ¹ m ¹ (c #m c ). !d 2
(19)
(2) Decision B2: B is selling to A a quantity of components at price v . NPV (., 2)"d ( , , p , p )(p !c !c ) ! (K #K ) 2 1 ! ¹
K K # m
¹ ! (d !d ) (c #m c ) 2 #d ( , , p , p )v . (20) For the sake of clarity of expressions, we choose the special case of equal mill prices and no shortage in B. The annuity streams are the following, concerning the optimal ordering policy (¹H, mH) for each reward matrix position separately: NPV (1, 1)"r(1!)S ((p !v )! K 2 1 r(1!)S c' ¹ , ! K ! ¹ 2 (21) where c' denotes storage costs of a unit of "nal product in A. NPV (1, 2)"r(1!)S (p !p )! K 2 1 ! K (0, ¹
(22)
NPV (2, 1)"r(1!)S (p !c !v ) 1 ! K ! K !r 2 ¹ ¹ c , (1!)S 2 NPV (2, 2)"r(1!)S (p !c !v ) 1 ! K ! K 2 ¹ ¹ c "0. !r(1!)S 2
(23)
(24)
In (23) and (24) the inventory costs are included in c . NPV (1, 1)"r(S!(1!)S )p !rS(c !c )#r(1!)S v ! (K #K ) 2
1 K ! K # ¹ m ¹ !rS (c #m c ), 2
(25)
NPV (2, 1)"r(S!(1!)S )(p !c ) !rSc #r(1!)S v ! (K #K ) 2 1 ! ¹
K K # m
¹ !r(S!(1!)S ) c 2 ¹ !rS m c . 2
(26)
Here v denotes the price by which B delivers components to A. When B covers the total market area S because B is not selling any goods to A, B acts as monopolist.
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annuity stream under constraints given by the second player. It is easy to see that the following conditions:
Because of equal mill prices we can write NPV (1, 2)"rS(p !c !c ) ! (K #K ) 2
1 ! ¹
NPVH (2, 1)' NPVH (1, 1), NPVH(2, 1)' NPVH(2, 2)
1 K ! K # ¹ m ¹ !rS (c #m c ), 2 NPV (2, 2)"rS(p !c !c ) ! (K #K ) 2 K K # m
(27)
(28)
4. The results of spatial games The natural question arises what kind of conditions should be ful"lled for A to patronize an area according to Christaler model on a lower level than B, so that it will manufacture the "nal product and sell it to his market area S . We have the following reward matrix of the game:
(30) (31)
should be ful"lled to achieve that neither player can gain advantage by unilateral change in strategy. Thus ( NPVH (2, 1), NPVH(2, 1)) is an equilibrium point. We get the optimal annuity stream NPVH (1, 1) considering the necessary condition of optimality NPVH (1, 1) "0 (32) ¹ from which follows that the optimal time between orders is
¹ !rS (c #m c ). 2
283
( NPVH (1, 1), NPVH(1, 1)) ( NPVH (1, 2), NPVH(1, 2)) ( NPVH (2, 1), NPVH(2, 1)) ( NPVH (2, 2), NPVH(2, 2))
.
(29) When (1, 1) is the solution of the game, A is selling only "nal products. When (2, 1) is the solution of the game, A is producing "nal products. When (1, 2) or (2, 2) is the solution of the game, A can only sell the goods in a small local area, but is not producing the "nal products. We have to answer the question, what conditions should be ful"lled so that the optimal strategy would be ( NPVH (2, 1), NPVH(2, 1)), which will make A the producer of a "nal product. Here the superscript H denotes the optimal value of the
2K ¹H " (33) r(1!)S c' and so the optimal value of the annuity stream NPVH (1, 1) can be expressed as NPVH (1, 1)"r(1!)S (p !v ) ! K !(2r(1!)S c' K . 2 (34) In a similar way, we also get the optimal annuity stream NPVH (2, 1) from the necessary condition NPVH (2, 1) "0, (35) ¹ from which we get
2K ¹H " r(1!)S c and
(36)
NPVH (2, 1)"r(1!)S (p !v ) ! K !(2r(1!)S c K . 2 (37) Considering (34) and (37) in (30) we "nd out that after some mathematical manipulations the follow-
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ing condition should be ful"lled:
We also get
r(1!)S (v !v !c )
1 mH " 0 ¹H 0 and
#(2r(1!)S K ((c !(c' )'0. (38) From (38) it follows: v
#c (v
2K Src
(48)
(S!(1!)S )c K . (49) K Sc The connection between optimal values for situations (2, 1) and (2, 2) is the following: mH " 0
(2r(1!)S K ((c !(c' ) . (39) # r(1!)S For player B, condition (31) should be ful"lled for the game. Let us denote situation (2, 1) by index ¸ and situation (2, 2) by index R. After stating the necessary conditions of optimality for situation (2, 1),
Let q denote the percentage of market area that belongs to B:
NPVH(2, 1) "0, m
(40)
S!(1!)S . q " S
(41)
It follows that the market share also strongly in#uences optimal parameters mH and mH , which * 0 can be seen from the relation
NPVH(2, 1) "0, ¹ from which follows 1 mH " * ¹H * and
2K Sr
(K #K /mH )2 * c SmH c ), ¹H " * * r((S!(1!)S we also get
(K #K /mH )2 0 . rS(c #mH c ) 0
(51)
(43)
(K #K /mH 1/q )2 0 . (53) ¹H " * r(S!(1!)S )c #mH c Sq ) 0 From (47) and (53) it is easy to see that the relationship among the times between two orders is the following:
(S!(1!)S )c K . (44) K SC To "nd the optimal value of NPV (2, 2), see Eq. (28), the necessary optimality conditions are
¹H " 0
(50)
mH "(q mH . * 0 In this case, we can also write
NPVH(2, 2) "0, ¹ from which follows
S!(1!)S mH . 0 S
(42)
mH " *
NPVH(2, 2) "0, m
mH " *
(45) (46)
¹H '¹H . (54) * 0 The condition that follows from (31) is the following: r(1!)S (v #c !p ) c K #(¹H !¹H ) Sr # '0, (55) 0 * 2 ¹H ¹H 0 * or, in a more explicit form,
v
(47)
(52)
#c 'p (¹H !¹H )(Src /2#K /¹H ¹H ) * 0 * . # 0 r(1!)S (56)
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Putting together both conditions (39) and (56) that should be ful"lled so that the optimal strategy for the game would be (2, 1), we get the main result: v
#c (v (2r(1!)S K ((c !(c' ) , (57) # r(1!)S v #c 'p (¹H !¹H )(Src /2#K /¹H ¹H ) * 0 * . # 0 r(1!)S
6. Conclusions and directions to di4erential games In this paper we have shown how to use MRP and input}output analysis (the NPV model) to investigate the results of customers' behavior when the supply units compete for customers and are exposed to an uncertain demand which results in a possible shortage of goods, assuming the total market area of both players to be constant. Player A will chose the optimal shortage of goods on that acceptable range. From the second term of right sides of conditions (52) we can see how important are the set-up costs which also include transportation expenses. This theory is only valid, if the total market area belonging to both players is constant. In case of the expansion of one or both players on the market area of a third central place, the theory of di!erential games has to be used. On the basis of these results, the optimal ordering policy which also includes the optimal shortage of goods can be worked out. These results can also be extended to the other customers' traveling behavior described by [16] and even to more general cases.
References [1] W. Chrtistaller, The Central Places of Southern Germany, Prentice-Hall, Englewood Cli!s, NJ, 1966. [2] A. LoK sch, The Economics of Location, New Haven, 1954.
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