MATHEMATICAL APPENDIX to THREE REVENUE-SHARING VARIANTS: THEIR SIGNIFICANT PERFORMANCE DIFFERENCES UNDER SYSTEM-PARAMETER UNCERTAINTIES

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1 MATHEMATICAL APPENDIX to THEE EVENUE-SHAING VAIANTS: THEI SIGNIFICANT PEFOMANCE DIFFEENCES UNDE SYSTEM-PAAMETE UNCETAINTIES Yao-Yu WANG a,b, Hon-Shiang LAU a (corresponding author) and Zhong-Sheng HUA b NOTE: The paper can be understood ithout reading the folloing Appendix. Thus, if the paper is deemed publishable, this Appendix does not have to be published but can be supplied to interested readers by the authors separately. APPENDIX Assume: (i) demand curve D pl a-bp ; (ii) that the dominant manufacturer (Manu) perceives the retailer s (eta s) processing cost r as a stochastic variable r ~ U[0, μ r ], i.e., f(r) /(μ r ); and (iii) sub 0. We derive belo the optimal decisions under each [S]-variant i and each -restriction j. We then compare the optimal Mij s to identify the [S]-variant that is most profitable for Manu. For simplicity, μ r ill be ritten as μ from no on. A. Deriving the Optimal Decisions for Variant As shon in (a), the profit functions for Manu and eta are, respectively, П M (-φ)p(a-bp) + ( m)(a-bp), П φp(a-bp) (+r )(a-bp). (A.) For any set of {, φ}-values imposed by Manu, eta ill set her retail price p at p to maximize her on profit. Solving dп /dp0 gives the retailer s response retail price as p [φa + b(+r )]/(φb). (A.) Thus, for any {, φ}-values Manu sets, (A.) indicates that Manu perceives his profit as the stochastic variable M [( φ)p +( m)](a bp ). (A.) Hence Manu s problem can be stated as Max, ϕ μ 0 [( - ϕ ) p + ( -m )]( a bp ) df ( r), here p as stated in (A.). (A.4) A. estriction a ( can be any value) Solving the first-order conditions for (A.4).r.t. and φ gives Manu s optimal decisions as bμa B + a B + b μ A + bmab B + b μ A a ( ) ( ) bab( AB + B + b μ B + b μ A Under the above-stated {, φ}-values, Manu s profit is (A.5) ϕa (A.6) AB

2 here ABb ( μ AB + B + bμ Ma (A.7) bb ( + bμ ( AB+ B + bμ A a b m b ( ) [ ( μ + )] + μ, B b μ A ( ( A b μ ) A. estriction b ( must be non-negative) Substituting p in (A.) into (A.4) and simplifying gives E +. (A.8) ~ b ( + ϕ) + 6 b( ϕbm bμ+ aϕ ) + ϕb (4μ + 6 μm) +Τ M, (A.9) bϕ here Τ a ϕ ( ϕ) 4b μ 6abmϕ. With (A.9), it is easy to verify that E( M) is convex.r.t ; hence it attains its maximum at a stated in (A.5). That is, if a 0, restriction b does not bind, and Manu s optimal decisions are the same as those stated in (A.5) to (A.7) for restriction a. Hoever, if a 0, it is obvious that 0 maximizes E( M ) shon in (A.9) under restriction b. In summary, the optimal decisions under restriction b are: if a > 0, b a, ϕ b ϕ a Δ (A.0) if a 0, b 0, ϕ b aδ The corresponding Manu s profit is then: if if > 0, a Mb Ma 4 ( ( Δ + Δ Δ + Δ Δ + 8 μ Δ ) ΔΔ ( )) a ab a mb 0, a Mb bδδ ( ) / Δ ( ) b μ 8aμ+ μ(8b μ + 6b μ m+ 54b μm + 7b m + 6 μa ) here Δ Δ b μ b μm (A.) (A.) A. estriction c Using the same approach stated in the preceding subsections, Manu s optimal decisions under restriction c are: if a > m, c a, ϕ c ϕ a Δ4 bm 4bμ 6bμm (A.) if a m, c m, ϕ c aδ4 Manu s profit under restriction c can then be expressed as

3 if a > m, Mc M a a Δ Δ Δ Δ Δ Δ +Δ if a m, Mc 4 ( 4( 4 a) )( ) bδ4( Δ Δ4 ) here ( ( )) / b (4μ 6μm m ), 4 9a 8a (A.4) Δ + + Δ Δ + Δ +. (A.5) A. Deriving the Optimal Decisions for Variant As shon in (b), the profit functions for Manu and eta are, respectively, П M (-φ)(p )(a-bp) + ( m)(a-bp), П φ(p )(a-bp) r(a-bp). (A.) If Manu perceives eta s r as r, his problem becomes, similar to (A.4),, μ ϕ 0 Max [(-ϕ)( p ) + ( -m)]( a bp ) df( r), here p [ ϕa+ b( ϕ+ r)] / ( ϕb) (A.) Using the same approach stated in the preceding subsections, e can obtain Manu s optimal decisions and profit for Variant under each -restriction. For brevity, e skip the intermediate steps and state only the final solutions belo. A. estriction a The optimal solutions for problem (A.) are bμa B + a B + b μ A + bmab B + b μ A a ϕ ( ) ( ), (A.) bab( AB + B + b μ a B + b μ A. (A.4) AB Manu s profit is A B( b μ AB + B + b μ A ) Ma bb ( + bμ ( AB+ B + bμ, (A.5) here A a b m b ( ) [ ( μ + )] + μ, B b μ A ( ( A b μ ) A. estriction b The optimal solutions for problem (A.) under restriction b are if > 0,, ϕ ϕ Δ if a 0, b 0, ϕb aδ Manu s profit can then be stated as a b a b a +. (A.6) (A.7)

4 if if > 0, a Mb Ma 4 ( ( Δ + Δ Δ + Δ Δ + 8 μ Δ ) ΔΔ ( )) a ab a mb 0, a Mb bδδ ( ) here ( ) / Δ b μ 8aμ+ μ(8b μ + 6b μ m+ 54b μm + 7b m + 6 μa ) Δ Δ b μ b μm A. estriction c The optimal solutions for problem under restriction c are Manu s profit is if if if a > m, c a, ϕc φa Δ if a m, c m, ϕc Δ Δ > m, a Mc Ma (ΔΔ + Δ)( Δ b μ Δ + 6 b μ ΔΔ) a m, Mc bδδ 5 6 A. Deriving the Optimal Decisions for Variant As shon in (c), the profit functions for Manu and eta are, respectively, П M (-φ)(p -r)(a-bp) + ( m)(a-bp), П φ(p -r)(a-bp). If Manu perceives eta s r as r, his problem becomes, similar to (A.4), Max, ϕ μ 0 [(-ϕ)( p r) + ( -m)]( a bp (A.8) (A.9) (A.0) (A.) (A.) ) df ( r), here p [a+b(+r)]/(b). (A.) A. estriction a The first-order condition.r.t. for (A.) gives ( bϕμ+ ϕa+ bm) [ b( + ϕ)] (A.) Substituting (A.) into (A.) gives ~ ( ) E M ( ϕ ) b μ + [ a b( m+ μ)] [ b( + ϕ)]. (A.4) The first-order derivative.r.t. φ for (A.4) is ( ) ~ M ϕ [ ( μ) ] ( ϕ ) μ [ ( ϕ) ] 0 E a b m+ + + b b +. (A.5) It is obvious that (A.5) is less than zero, hich means E( M) is a decreasing function. Since the support interval of φ is [0, ], E( M) achieves its maximum at φ0. Substituting φ0 into (A.) gives m. Thus, Manu s optimal decisions are m, ϕ 0. (A.6) a a

5 Manu s profit is then [ a b( μ + m)] + b μ Ma. (A.7) b A. estriction b Noting that m>0, Manu s optimal decisions under restriction b are m, ϕ 0. (A.8) b b Manu s profit is then [ a b( μ + m)] + b μ Mb. (A.9) b A. estriction c Manu s optimal decisions under restriction c are m, ϕ 0. (A.0) c c Manu s profit is then [ a b( μ + m)] + b μ Mc. (A.) b A4. Comparing the Performance of the [S]-Variants We no compare the performance of each [S]-variant under each -restriction to see hich variant performs best. We denote the Manu-profit-difference beteen to variants as: UB ijk П * * Mik П Mjk (A4.) A4. estriction a Using the improvement measure defined in (A4.), e have UB a П Ma П Ma 0 (A4.) A( B + b μ AB + b μ A B + b μ A B + b μ A ) UB a П Ma П Ma bb ( + bμ ( AB+ B + bμ (A4.) A( B + b μ AB + b μ A B + b μ A B + b μ A ) UB a П Ma П Ma (A4.4) bb ( + bμ ( AB+ B + bμ From the definition in (A.8), it is easy to verify that A>0, B>0; hence UB a UB a <0. Thus, under restriction a, Variants and give the same П M, hile Variant performs better that both Variants and. A4. estriction b When a >0 (note that a a ). The situation here is the same as that in A4., hence the same conclusion. I.e., Variants and give the same П M, hile Variant performs better that both Variants and. 4

6 When a 0 Table 0 indicates the П Mb П Mb, hence UB b П Mb П Mb 0. Instead of obtaining the algebraic expressions of UB b and UB b and then proving their positivity (as done in A4. above), it is easier to deduce the relative magnitudes of П Mb, П Mb and П Mb as follos. Note that П Mb (or П Mb ) is the local optimal solution for problem (A.4) (or (A.)) under the condition a 0, hile П Ma (or П Ma ) is the global optimal solution for problem (A.4) (or (A.)). A global optimum is at least as good (if not better) than a local optimum. Therefore, e kno that П Mb П Ma (or П Mb П Ma ). A4. already shoed that П Ma П Ma <П Ma, hence П Mb П Mb <П Mb (note that П Ma П Mb П Mc ). Thus, e reach the same conclusion as that stated in A4.. A4. estriction c We can use the same approach stated in A4. to obtain the conclusion that UB c >0 and UB c >0. That is, Variant still performs better than both Variants and. Unfortunately, e are unable to obtain any analytical conclusion on the sign of UB c. To summarize, for the case of a stochastic r ~ U[0, μ r ] ith sub 0, e proved analytically that Variant alays performs better than both Variants and under all three -restrictions stated in (). We also proved analytically that Variants and perform identically under -restrictions a and b. The only relationship for hich e could not obtain an analytical proof is beteen Variants and under -restrictions c. More relevantly, the above derivations illustrate the impracticality of an analytical approach for the problem at hand. 5