TCp c p i ≥ + − () /( )1 , where [] $ () () u p Cp pp p c i11 111 =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − −− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ π . As a result, if there is an interior solution, we have: c pC p C p p i = − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ '( ) ( ) 1 2 . Note that, this requires as well that Cp Cp p'( ) ( )/> . For example, if Cp A p() /( )= − 1 , then, c A p pp i = − +− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 21 11 3 ()() with p > 1 2 . Alternatively, under the second equilibrium, we have: {} 01 21 <<p Max u $ = 01<<p Max [] π− −− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − − +− 11 1 1 2 pp p Cp c p p Tp i () () () () s.t. TCp c p i ≤ + − () /( )1 . Let () pp 12 ** , be the optimal solutions under both equilibria, then, obvi- ously, the supplier will adopt the solution leading to the largest expected payoff of the game. By changing the assumptions regarding the relative power each of the parties has, we will obtain, of course, other solutions. These are discussed below. Each of the solutions considered here can be altered by an appropriate selection of contract parameters which can lead to a pre-posterior game analysis evaluated in terms of ( ) pT, (see also Reyniers and Tapiero 1995a it is possible to create an incentive for the supplier to supply quality parts by the selection of contract parameters. A sensitivity analysis of some of these solutions follows. For conveni- ence, we consider only the case with interior solutions and study the effects of T on the propensity to sample. Obviously, the larger T, the less the manufacturer-customer will sample since ∂ ∂ y T < 0 . Further, () ∂ ∂ φ x T c pT b = −+ > 1 0 2 () . 460 8 QUALITY AND SUPPLY CHAIN MANAGEMENT for the analysis of contracts). If the supplier and the producer do not coop- erate (and thus the Nash equilibrium solutions defined here are appropriate), 8.3 YIELD AND CONTROL 461 Similar relationships can be found by treating other parameters. The implications are that increases in T provide an incentive for the supplier to sample while for the customer to sample less. When either the customer or the supplier is a leader and the other a follower, we define a Stackelberg game (Stackelberg 1952). For example, say that the supplier is a leader and the customer is a follower. Then, for a given () xp, , the customer problem is: ( )()()()() [] Max y Vyxp x p T x p c y b ,; =−−+−−−φ 11 11 and therefore the customer sampling policy is either to inspect all of the time (y=1) or none at all (y=0). Of course: () ( ) ( ) ( ) ()()()() y xpTxpc xpTxpc b b = −−+−−≥ −−+−−< ⎧ ⎨ ⎩ 111 11 011 11 if if φ φ The supplier’s problem consists then in selecting a strategy (x,p) based on the customer’s response y(x,p) given above. Namely, ( ) () () () ()() Max xp Uyxp Cp cx Cp c p p xy x T x py ii , ;; () (()) =− − − + − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ +−−− −π 1 111 2 s.t. yxp(, ) , = 10 , as sated above. In this case, we note that the sampling decision is always an all or nothing sampling policy. For example, say that y=1, then the supplier turns to full sampling if () Tpc i 1−≥, otherwise the supplier will not sample at all. How- ever, if the customer does not sample, then we note that the supplier does not sample either, since ( ) () () () Max xp Uyxp Cp xc Cp c p p ii , ;; () =− − + + − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ π 1 2 has always a solution x = 0. Of course, the supplier yield will then be minimal (and therefore the quality will be the worst possible). In this sense, when one of the parties has power over the other the quality will be low (as it is the case in Stackelberg games but which does not hold true in Nash conflict games). Stackelberg equilibrium In industrial situations, it is common that cooperative solutions are sought. In this case (if we do not consider for simplicity the distribution of spoils resulting from cooperation), the problem faced by the supplier and the manufacturer customer alike is given by: ()() () ()()()() () 2 ( 1 ) ;; ;; 1 1 1 ib i p V yxp Uyxp Cp cx cy x p y C p c p φφ ⎡⎤ − ⎢⎥ +=−−−−−−−−+ ⎢⎥ ⎢⎥ ⎣⎦ which we maximize with respect to (y,x,p). Of course, x=y=0 and therefore the optimum yield is found by a solution of: () ( ) ( ) VpUpCpp00 00;; ;;+=−+ φ and therefore, the optimal yield is: ( ) ∂ ∂ φ Cp p = which expresses the classical relationship between the marginal cost and the marginal revenue for the optimal yield. In this sense, cooperation will lead to the highest yield while an asymmetric power relationship as the one stated above will lead to the least yield. In conclusion, we note that producers’ and suppliers’ inspections are, as we discussed, function of the industrial contract in effect between a sup- plier and a customer. This provides a wide range of interpretations and poten- tial approaches for selecting a quality inspection policy. This section has shown that there is a clearly important relationship between the terms of a contract and the acceptance sampling policy. There are, of course, many facets to this problem, which could be considered and have not been consi- dered in sufficient depth. For example, risk aversion, more complex contracts and the design of yield delivery contracts have not been considered. Never- theless, these are topics for further research. The basic presumption of this section is that once supplier-customer contracts are negotiated and signed, there may be problems when enforcing these contracts. As a result, some controls are needed to ensure that contracts are carried out as agreed on. The approach is based on solving the post-contract game between the sup- plier and the producer where the resultant inspection and quality supplies equilibrium policies are given by the randomized strategies available to each of the parties. 462 8 QUALITY AND SUPPLY CHAIN MANAGEMENT Centralized problem (x + y(1−x)) 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 463 We consider next the supply chain organizational structures and their asso- ciated rules of leaderships. We also use the statistical Neyman-Pearson quantile risk framework for hypothesis testing (and quality control), as done earlier. Based on such risks we shall construct a variety of control programs that respond to the specific needs and the specific organizational structure of a supply chain. To demonstrate the usefulness of this app- roach, a number of problems are also solved. To keep matters tractable however, some simplifications are made. For simplicity and exposition purposes, assume that lots of size N are delivered by a supplier to a buyer (a producer of finished products), parts of which are sampled and tested. To assure contract compliance, both the supplier and the buyer can use a number of sampling programs, each with stringency tests of various degrees (spanning the no sampling case and thereby accepting the lot as is, to the full sampling case and thereby inspecting the whole lot) and assuming no risks. Let 1, j MN = ≤ be the M alternative sampling-control programs used by the client-buyer and 1, iN= be the alternative sampling-control programs used by the provider- supplier. Correspondingly, we denote by ( ) ( ) ,, , , ,; , pi pi S j S j αβ αβ , the producers and consumers risks for the producer and the supplier respectively. These are the probabilities of rejecting a good lot and accepting a bad one by a producer (indexed p) and a supplier (indexed S), under each specific and alternative statistical sample selected. These risks are summarized in the matrix below. () () () ( ) ( ) ( ) () () () () () () () () () () () () ,1 ,1 ,1 ,1 ,1 ,1 , , ,2 , 2 ,1 ,1 ,2 , 2 , , , , ,1 ,1 , , , , ,; , ,; , ,; , ,; , ,; , ,; , pp SS pp SmSM pp SS pp SmSM pN pN S S pN pN Sm SM αβ αβ αβ α β αβ αβ αβ αβ αβ αβ αβ αβ ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ (8.33) The selection of a control program can be unique and randomized, reflecting strategic considerations such as signals by a producer to indicate that they control their suppliers and vice versa for suppliers to indicate that they are careful to deliver acceptable quality items. These controls and their outcomes may also be negotiated and agreed on in contractual 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 8.4.1 A NEYMANN-PEARSON FRAMEWORK FOR RISK CONTROL agreements to include penalties and incentives based on the control-sample outcomes. In this sense, associated to the risk specifications of equation (8.33), there may be as well a bi-matrix of costs summarizing the expected and derived costs implied by the parties control strategies. For simplicity and brevity, this section will consider only a specification of type I risks and the collaborative minimization of type II risks, in the spirit of the traditional Neyman-Pearson theory. Explicitly, assume for simplicity binomial sampling distributions with parameters () ,, , pi pi nc for the producer and ( ) ,, , Sj Sj nc for the supplier where the indices i and j denote a set of finite and alternative sampling plans available to the producer and the supplier respectively. Let A QL be a contracted proportion of acceptable defectives (or the Acceptable Quality Limit) and LTFD be a contracted proportion of unacceptable defectives in a lot (or the Lowest Tolerance Fraction Defectives). Then the risks sustained by the producer (buyer) and by the supplier, when each selects sampling plans () ,, , pi pi nc and ( ) ,, , Sj Sj nc are respectively (see also Wetherhill, 1977, Tapiero, 1996): ()( ) ( )( ) ,, , , ,, ,, 00 11; 1. ki ki ki ki cc nn ki ki ki ki nn AQL AQL LTFD LTFD αβ −− == ⎛⎞ ⎛⎞ =− − = − ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑ ll l l ll ll (8.34) where ,kpS= . For example, if the supplier fully samples (i.e. j=N) and attends to all non conforming units, then ,, 1, 0 SN SN α β = = . If the buyer knew for sure that this were the case, he would use always a costless no- inspection alternative. Similarly, say that the supplier accepts a bad lot (the supplier’S consumer risk). The buyer-consumer risk will in this case be determined by the stringency of controls used by the supplier. If the buyer- producer also accepts this defective lot, the probability corresponding to the producer and the supplier sampling strategies defines bi-matrices with entries: ,, (1 ); pi S S j A αα ⎡⎤ − ⎣⎦ and ,, ; pi S S j B ββ ⎡ ⎤ ⎣ ⎦ for type I (producer) and type II (consumer) risks. In these entries, (,) SS A B denote the average supplier control risks, assumed known (or contracted) by the producer. These risks will be altered, of course, as a function of the mutual relationships established between the supplier and the producer. If the supplier assumes responsibility for a consumer’s risk only if it is detected by the producer, then the supplier and the producer consumer risks will rather be ,, ;(1 ) pi S S j p BB ββ ⎡ ⎤ − ⎣ ⎦ instead of ,Sj β , as stated in the type II 464 8 QUALITY AND SUPPLY CHAIN MANAGEMENT 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 465 risk bi-matrix above. Note that (, ) pp A B denote the average producer and consumers risks of the buyer-producer. Other cases may be considered as well, based on the exchange of information between the supplier and the producer. For example, if the supplier reports to the producer his choice of control techniques, then the risk bi-matrices for both, will be instead ,,, (1 ); pi S j S j ααα ⎡⎤ − ⎣⎦ and ,, , ;(1 ) pi S j S j p B ββ β ⎡ ⎤ − ⎣ ⎦ . In other words, the organization structure of the supply chain and the information-controls exchange combined with the “various degrees and forms” of collaboration (or none at all) will determine both the control programs applied and the risks sustained by the supply chain parties. Each of theses cases can be treated separately, although the approach we use here is essentially the same. Assume that average risks sustained by the supply chain parties are agreed on (or contracted) and let each of the parties selects a control program in randomized manner over the following risk bi-matrices ,, (1 ); pi S S j A αα ⎡⎤ − ⎣⎦ and ,, ; pi S S j B ββ ⎡⎤ ⎣⎦ . Let i x be the probability that the producer selects a control strategy i while j y is the probability that the supplier selects control strategy j. The average risks for the supplier are then: ,, 11 ; MM j Sj S j Sj S jj yy α αββ == == ∑∑ where ( ) , SS αβ is the average type I risk associated to a selection of sampling plans using the randomized sampling strategies , 1, 2, j yj M = used by the supplier. It is not, of course, the actual average risks sustained by the supplier, since such risks will depend on the action followed by producer as well. In this case, we use capital letters to denote the actual type I and II risks sustained. In this special case, ,(1) SSpp S A AA αα ==−and , SSppS B BB ββ == where () , pp A B are the corresponding average risks of the producer with sampling specific average risks ( ) , pp α β defined as randomized sampling strategies ,, 11 , NN ipi p ipi p ii xx α αββ == == ∑∑ . Note that in such notations, the sampling-control risk problems faced by both the producer and the supplier are then given by minimizing the consumers (type II) risks subject to some constraints on their producers (type I) risks, explicitly stated as follows. () () ,, 1 ,, 1 ,,01, 1, 1,2, ,,01, 1, 1,2, = = Subject to: = Subject to: N pi pi i i i M Sj Sj j j j ppSpS p PC nc x x i N SS S SC nc y y j M M in B B A A Min B A A βββ β = = ≤≤ = = ≤≤ = = ≤ ∑ ≤ ∑ (8.35) where risk minimization is reached with respect to the available alternative sampling plans and their randomization (namely, selecting a number of sampling plans through a randomization rule to be found by the solution of the game). Here, () , PC SC A A stands for specific parameters while ,(1) SSpp S AA α αα ==−. The solution of the constrained game (8.35) subject to risks (8.34) determines therefore an adaptation of the Neyman- Pearson lemma to a supplier-producer situation, which can be solved according to the available information we have regarding alternative sampling plans and assumptions on the behavioral relationships that exist between the supplier and the producer. For example, assuming power (leader-led) relationships and collaborative strategies that both the producer and the supplier will adopt, a number of games might be developed. Explicitly, if the producer is a leader in a Stackleberg game, fully informed of the supplier objectives, then the sampling-control selection problem is defined by: () () ,, 1 ,, 1 ,,01, 1, 1,2, ,,01, 1, 1,2, == Subject to: = N pi pi i i i M Sj Sj j j j ppSpS nc x x i N SS nc y y j M Min B B Min B β ββ β = = ≤≤ = = ≤≤ = = ∑ ∑ (8.36) where the type I risks, dropped out of equation (8.36) are implied as in equation (8.35). When it is the supplier who leads and the producer is led, then producer risk is minimized first and the supplier uses this information to minimize his risks. Further, if both the supplier and the producer collaborate in controlling risks, then the problem they face can be stated as a weighted (Pareto optimal) solution to the game (8.35). In this case, we presume that there is a parameter 01 λ ≤ ≤ expressing the negotiating of each of the parties, such that: 466 8 QUALITY AND SUPPLY CHAIN MANAGEMENT 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 467 () () { } ,, 1 ,, 1 , ,0 1, 1, 1,2, ; , ,0 1, 1, 1,2, (1 - ) N pi pi i i i M Sj Sj j j j pS nc x x i N nc y y j M M in B B λλ = = ≤≤ = = ≤≤ = = + ∑ ∑ (8.37) subject to both the producer and the supplier type I risks constraints is minimized. Alternatively, we may consider other objectives such as economic and sampling costs as well as the costs associated with the type I and type II risks of both the producer and the supplier. If we consider the ,, ;(1 ) pi S S j p BB ββ ⎡ ⎤ − ⎣ ⎦ and (1 ) , (1 ) 11 pS S SS p ppSpS p p SpS BB BB B β β β ββββ β βββ =−= = = −= ++ (8.38) and (1 ) (1 ), SS p S pS ppS BB B β βββ ββ =−=− = . (8.39) Evidently, other situations arise, a function of the information available to each of the parties and the exchange they engage in and the behavioral assumptions made regarding the potential collaboration and/or conflict that exists between the supplier and the producer. To obtain tractable results and for demonstration purposes we restrict ourselves to simple solutions for a supplier and a producer, each considering two alternative control pro- grams. Essential results are then summarized and discussed. Subsequently, special cases and numerical examples are treated to highlight both the implications and the applicability of the approach. Below, we begin with non-collaborating supplier and producer to subsequently compare to the effects of collaboration. Proposition 8.6. Let () ( ) ,, , , , , 1, 2 and , , 1, 2 pi pi S j S j ij αβ αβ == be type I and II and risks of a producer and a supplier engaged in mutual (and conflicting) binomial sampling-controls as in equation (8.34) with ,2 ,1 ,2 ,1 < and < pp SS β βββ . Then if type I risks are satisfied by both strategies, the optimal sampling-control is a pure strategy where both the supplier and the producer adopt intensive control strategies (with type II risks () ,2 ,2 , pS ββ ). If type I risks constraints are binding then the supplier type II risk bi-matrices ,, ,pi S j S j p ⎡⎤ ⎣⎦ ββ ;( β 1− B ) instead, then the samplin g-control problem’s formulations in (8.35)-(8.37) remain the same with average risks respectively defined instead by: and the producer can turn to randomized sampling strategies given by the solution of: = /(1 ) and pPC SC S SC A AA α α − = (8.40) While average type II risks minimized by each of the parties are explicitly given by: () () () ,, ,, ,1 ,2 0,1,2 ,1 ,2 ,1 ,2 0,1,2 (1 ) =(1) (1) Sj Sj pi pi SS S cnj pp p S S cni Min B y y Min B x x y y ββ ββββ << = << = =+− +− +− (8.41) where ,, (,),1,2 Sj Sj nc j= and ,, (,),1,2 pi pi nc i = are two known sampling plans available to the supplier and to the producer, while () ,, ,,1,2 Sk Sk k αβ = and ( ) ,, ,,1,2 pk pk k αβ = are the types I and II risks associated with each of these sampling plans by the supplier and the producer. Then the optimal randomized strategies for selecting one or the other sampling plans are: ,2 ,2 ** ,1 ,2 ,1 ,2 /(1 ) ; PC SC p SC S SS pp AA A yx α α αα αα − − − == −− . (8.42) Proof: The proof is straightforward since in the bi-matrix ,2 ,2 ,2 ,2 ,2 ;; pSS pS S B ββ βββ ⎡⎤⎡ ⎤ = ⎣⎦⎣ ⎦ , intensive sampling by both the supplier and the producer are dominating all other strategies. This observation might be practically misleading because it ignores the costs associated with sampling and of course all other risk costs. Of course, if the type I risks are set to their maximal values, then: (1 ) or = /(1 ) and p p S PC p PC SC S S SC A AAAAA αα α α =−= − == , which provides a system of equations in the randomizing parameters (x,y), 2 ,1xx x=− ): ()( ) , , 2 , 10 11, Sj Sj c n Sj j SC j n y AQL AQL A − == ⎛⎞ ⎛⎞ −−= ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑ ll l l and ()( ) ()( ) , , , , 22 , 10 1 0 11*111 pi Sj pi Sj cc nn pi j i j PC ij nm x AQL AQL y AQL AQL A −− == == ⎛⎞⎛ ⎞ ⎛⎞⎛⎞ ⎛⎞ ⎛⎞ −−−−−= ⎜⎟⎜ ⎟ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎜⎟⎜ ⎟ ⎝⎠ ⎝⎠ ⎝⎠⎝⎠ ⎝⎠⎝ ⎠ ∑∑ ∑∑ ll l l ll ll Given a solution for (x,y) in terms of the sampling-control parameters, the type II risks of the supplier, ,1 ,2 (1 ) SS S yy β ββ =+− is minimized 468 8 QUALITY AND SUPPLY CHAIN MANAGEMENT 12 1 0,1,0yyy y x≤ = =− ≤ = or using equation (8.34), we have (with 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 469 with respect to ,, (, ) Sj Sj cn while the risk of the producer 22 ,, 11 ijpiSj ji xy β β == ∑∑ is minimized with respect to ,, (, ) pi pi cn . This propo- sition remains valid when we use instead (8.38) and (8.39). Of course, the solution to this problem requires that we apply numerical techniques to select the appropriate sampling control parameters. Non-cooperation implies therefore that the firm uses as much as possible sampling-controls and do not randomize sampling strategies (unless type I risk constraints are violated, as stated in the proposition above). When type I risks constraints are binding, then ,2 ,2 ** ,1 ,2 ,1 ,2 /(1 ) ; PC SC p SC S SS pp AA A yx α α αα αα − − − == −− and therefore the sampling control problem is reduced to a nonlinear opti- mization problem stated in the proposition and explicitly given by: () () () ,, ,, ,2 ,2 ,1 ,2 0,1,2 ,1 ,2 ,2 ,2 ,2 ,1 ,2 ,2 ,1 ,2 0,1,2 ,1 , 2 ,1 ,2 /(1 ) = Sj Sj pi pi SC S SS S S cnj SS PC SC p SC S pp pp S SS cni pp SS A Min B AA A Min B α βββ αα α α βββ βββ αα αα << = << = ⎛⎞ − =+ − ⎜⎟ ⎜⎟ − ⎝⎠ ⎛⎞ ⎡ ⎤ ⎛⎞ ⎛⎞ −− − ⎜⎟ +− +− ⎜⎟⎢ ⎥ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ −− ⎢ ⎥ ⎝⎠ ⎝⎠ ⎣ ⎦ ⎝⎠ Producers and suppliers can reduce control costs if they collaborate. In this case, equation (8.37) is resolved subject to the type I risk constraints. Of course, if these risks are binding, then (8.37) is reduced to: () () () () ,, , , ,2 ,1 ,2 ,2 ,1 ,2 0,1,2;0 ,,1,2 1 + pi pi Sj Sj pppSSS cni cnij M i n x y λ λββββββ λ << = << = − ⎡⎤ +− +− ⎢⎥ ⎣⎦ (8.43) or () () ,, , , 0,1,2;0 ,,1,2 ,2 ,2 ,2 ,1 ,2 ,2 ,1 ,2 ,1 ,2 ,1 ,2 /(1 ) 1 +* pi pi S j S j cni c nij PC SC p SC S pppSSS pp SS Mi n AA A α α λ λβ βββ ββ λαα αα << = < < = Γ ⎡⎤ ⎛⎞ ⎛⎞ ⎡⎤ ⎡⎤ −− − − ⎢⎥ ⎜⎟ Γ= + − + − ⎜⎟ ⎢⎥ ⎢⎥ ⎜⎟ ⎜⎟ −− ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ ⎝⎠ ⎝⎠ ⎣⎦ (8.44) Examples will elaborate both the usefulness of this approach as well as deviations from a complete collaboration between the supplier and the producer. To keep our calculations simple, some simplifications are made. [...]... shared type 478 8 QUALITY AND SUPPLY CHAIN MANAGEMENT II risk is equal to 1134 compared to 424 as indicated in Table 8.3 The type I risks are equal to 240 and 401 however, compared to 15 and 20 which we used as type I constraints in Table 8.3 In this sense, the intricate relationship between a producer and a his supplier as well as the risk specifications for type I and II risks for each combined with... series assembly process supply chain, each producer with a risk attitude reflected by the consumer and producer risks assumed Thus, letting the first supplier be indexed “1”, we have: A(1) M1 j 1 y (1) j (1) j (1) ASC , B (1) M1 j 1 y (1) j (1) j , (8.56) 8 QUALITY AND SUPPLY CHAIN MANAGEMENT 474 while for subsequent producers-suppliers we have recursive equations for type I and II risks explicitly given... average type II risks for the supplier and the producer are calculated by: 0.85 S m 0 .10 1 1 p 0.85 n 0.95 1 n 0.85 m , 0.08 1 1 0.95 n 0 .10 1 0.95 m 1 0.85 n 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 477 Note again that these risks are also a function of the sample sizes only Using equation (8.54), or BS S / 1 p S , Bp p BS , we find that the average risk sustained by the supplier and the producer... Corbett C, Tang C (1999) Designing supply contracts: contract type and information asymmetry in Quantitative Models for Supply Chain Management Tayur S, Ganeshan R, Magazine M, eds., Kluwer Academic Publishers, Norwell, MA 480 8 QUALITY AND SUPPLY CHAIN MANAGEMENT Eppen GD, Hurst EG Jr (1974) Optimal location of inspection stations in a multistage production process Management Science 20: 1194-2000... producer and supplier may prefer not to sample), reducing thereby the amount of sampling The problem to be minimized is then given by simplifying equation (8.37) (1- ) (since B p which is reduced to: minimizing S p p S 8 QUALITY AND SUPPLY CHAIN MANAGEMENT 472 and BS S ) If both the supplier and the producer do not sample with proy (1 y ) S ,2 babilities y, x , then the resulting type II risks are S and. .. sizes (5 ,10, 15,20), selected by the supplier and the producer alike, we obtain the results in Table 8.3 Explicitly, if the producer and the supplier choose a sample size of 15 units each, then the probabilities of not sampling by the producer and the supplier are 56 and 72 respectively while the average type II risks for each are 308 and 514 The average type II risk, equally shared by the producer and the... the following risks: y, m 0 x, n 1 x, n 0 1 y, m 0 (0, 0);(1,1) 0 ( p , 0);( p ,1) ( (0, p 1 S S );( , S , S );( p S ) (8.46) S , S ) If type I risks are binding and the producer and the suppliers are not collaborating, we have the following randomizing parameters: y* 1 ASC ; x* 1 APC /(1 ASC ) S And the sampling-control problem is reduced to: p (8.47) 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN (a ) Min... (n,m) and randomization parameters (x,y) for the producer and the supplier that meet risk constraints on both type I and II risks may thus require extensive analysis and in some cases extensive sampling by both parties The sampling can, however, be significantly reduced if in fact, both the producer and the supplier turn to collaboration A more extensive analysis would, in this case, assess the risks... obtained for the supplier who p applies also a randomized curtailed sampling technique with a sample size m Let ( x, y ) be the probabilities that the producer and the suppliers do not sample Then, we have by equation (8.54), the following average type II risks: S y (1 y ) 1 LTFD For type I risks, we have: m and p x (1 x) 1 LTFD n 476 8 QUALITY AND SUPPLY CHAIN MANAGEMENT (1 y ) 1 1 AQL n (1 x) 1 1 AQL... 15051515 Tapiero CS (2004) Risk and Financial Management: Mathematical and Computational Concepts, Wiley, London Tapiero CS (2005a) Environmental Quality Control and Environmental Games, working paper Tapiero CS (2005b) Modeling Environmental Queue Control: A Game Model, Stochastic Environmental Research and Risk Assessment Tsay A, Nahmias S, Agrawal N (1998) Modeling supply chain contracts: A review, . 464 8 QUALITY AND SUPPLY CHAIN MANAGEMENT 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 465 risk bi-matrix above. Note that (, ) pp A B denote the average producer and consumers risks of the buyer-producer 474 8 QUALITY AND SUPPLY CHAIN MANAGEMENT ,A B=yB. (8.57) 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 475 while for ex-post assembly, we have the following consumers and producers risks: () (). problem is reduced to: 470 8 QUALITY AND SUPPLY CHAIN MANAGEMENT 8.4.2 SPECIAL CASES AND EXTENSIONS Example 8.1. Example 8.2. 8.4 RISK IN A COLLABORATIVE SUPPLY CHAIN 471 0 ** * 0 ( ) (1 ) 1 ;