3. Desirable constraints

Sun Woong Kim, Steven G. Heeringa and Peter W. Solenberger

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Each controlled selection problem of the form illustrated in Table 2.1 has many possible integer solutions. Let B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OqamaaBaaaleaacaWGRbaabeaaaaa@39E5@ denote one such solution, whose internal entries b i j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OyamaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaaaa@3BE2@ are the replacement of the real numbers a i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yyamaaBaaaleaacaWGPbGaamOAaaqabaaaaa@3AF1@ in the controlled selection problem A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yqaaaa@38C8@ by the adjacent nonnegative integers. The entry, b i j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaia ilvababKaGSSqaiailcGaAenyAaiacOr0GQbGaiGgrdUgaaeqcaYsc Knay=hacaYIaiaildkgaaaaaaa@4412@ , equals either [ a i j ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaam WabeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaay5waiaa w2faaaaa@3CEE@ or [ a i j ] + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaam WabeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaay5waiaa w2faaiabgUcaRiaaigdaaaa@3E8B@ , where [ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai 4waiaaykW7caaMc8UaaGPaVlaayIW7caaMc8Uaaiyxaaaa@417F@ is the greatest integer function. If a i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yyamaaBaaaleaacaWGPbGaamOAaaqabaaaaa@3AF1@ is a nonnegative integer, b i j k = a i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaia ilvababKaGSSqaiailcGaAenyAaiacOr0GQbGaiGgrdUgaaeqcaYsc Knay=hacaYIaiaildkgaaaGccqGH9aqpcaWGHbWaaSbaaSqaaiaadM gacaWGQbaabeaaaaa@4811@ for all k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam 4Aaaaa@38F2@ . The same rule is applied to the marginal expectations. As noted by Jessen (1970) and Causey et al. (1985), we primarily pay attention to B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OqamaaBaaaleaacaWGRbaabeaaaaa@39E5@ that simultaneously satisfy the following constraints for all i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yAaaaa@38F0@ and j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OAaaaa@38F1@ :

b i j k 0                   ( 3.1 ) | b i j k a i j | < 1                  ( 3.2 ) | b i . k a i . | < 1   and                ( 3.3 ) | b . j k a . j | < 1 ,                 ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbae GabqqbaaaaaeaacaWGIbWaaSbaaSqaaiaadMgacaWGQbGaam4Aaaqa baaakeaacqGHLjYSaeaacaaIWaGaaeiiaaqaaaqaaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaacIcacaaIZaGaaiOlai aaigdacaGGPaaabaWaaqWaaeaacaWGIbWaaSbaaSqaaiaadMgacaWG QbGaam4AaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaadMgacaWGQb aabeaaaOGaay5bSlaawIa7aaqaaiabgYda8aqaaiaaigdacaqGGaaa baaabaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaGGOaGa aG4maiaac6cacaaIYaGaaiykaaqaamaaemaabaGaamOyamaaBaaale aacaWGPbGaaiOlaiaadUgaaeqaaOGaeyOeI0IaamyyamaaBaaaleaa caWGPbGaaiOlaaqabaaakiaawEa7caGLiWoaaeaacqGH8aapaeaaca aIXaGaaeiiaaqaaiaabggacaqGUbGaaeizaaqaaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaiikaiaaiodacaGGUaGaaG4maiaa cMcaaeaadaabdaqaaiaadkgadaWgaaWcbaGaaiOlaiaadQgacaWGRb aabeaakiabgkHiTiaadggadaWgaaWcbaGaaiOlaiaadQgaaeqaaaGc caGLhWUaayjcSdaabaGaeyipaWdabaGaaGymaiaacYcaaeaaaeaaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaGGOaGaaG 4maiaac6cacaaI0aGaaiykaaaaaaa@9C07@

where b i . k = j = 1 C b i j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OyamaaBaaaleaacaWGPbGaaiOlaiaadUgaaeqaaOGaeyypa0Zaaabm aeaacaWGIbWaaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaaabaGaam OAaiabg2da9iaaigdaaeaacaWGdbaaniabggHiLdaaaa@4604@ equals either [ a i . ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaam WabeaacaWGHbWaaSbaaSqaaiaadMgacaGGUaaabeaaaOGaay5waiaa w2faaaaa@3CB1@ or [ a i . ] + 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaam WabeaacaWGHbWaaSbaaSqaaiaadMgacaGGUaaabeaaaOGaay5waiaa w2faaiabgUcaRiaaigdaaaa@3E4D@ , b . j k = i = 1 R b i j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OyamaaBaaaleaacaGGUaGaamOAaiaadUgaaeqaaOGaeyypa0Zaaabm aeaacaWGIbWaaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaaabaGaam yAaiabg2da9iaaigdaaeaacaWGsbaaniabggHiLdaaaa@4613@ equals either [ a . j ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaam WabeaacaWGHbWaaSbaaSqaaiaac6cacaWGQbaabeaaaOGaay5waiaa w2faaaaa@3CB2@ or [ a . j ] + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaam WabeaacaWGHbWaaSbaaSqaaiaac6cacaWGQbaabeaaaOGaay5waiaa w2faaiabgUcaRiaaigdaaaa@3E4F@ , i = 1 R b i . k = a ..  and  j = 1 C b . j k = a . . . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaa bmaeaacaWGIbWaaSbaaSqaaiaadMgacaGGUaGaam4AaaqabaaabaGa amyAaiabg2da9iaaigdaaeaacaWGsbaaniabggHiLdGccqGH9aqpca WGHbWaaSbaaSqaaiaac6cacaGGUaaabeaakiaabccacaqGHbGaaeOB aiaabsgacaqGGaWaaabmaeaacaWGIbWaaSbaaSqaaiaac6cacaWGQb Gaam4AaaqabaaabaGaamOAaiabg2da9iaaigdaaeaacaWGdbaaniab ggHiLdGccqGH9aqpcaWGHbWaaSbaaSqaaiaac6caaeqaaOWaaSbaaS qaaiaac6caaeqaaOGaaiOlaaaa@563F@

Consider the set of all possible arrays, B = { B k , k = 1 , , L } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv 3ySLgzgjxyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbbiab=fa8cjab g2da9maacmaabaGaamOqamaaBaaaleaacaWGRbaabeaakiaacYcaca aMc8UaaGjcVlaadUgacqGH9aqpcaaIXaGaaiilaiablAciljaacYca caWGmbaacaGL7bGaayzFaaGaaGjcVlaayIW7aaa@5692@ , satisfying (3.1) - (3.4). Since a i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yyamaaBaaaleaacaWGPbGaamOAaaqabaaaaa@3AF1@ is the expectation of the sample allocation to each cell in A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yqaaaa@38C8@ , the following constraints (3.5) and (3.6) on b i j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaia ilvababKaGSSqaiailcGaAenyAaiacOr0GQbGaiGgrdUgaaeqcaYsc Knay=hacaYIaiaildkgaaaaaaa@4412@ in B k ( B ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OqamaaBaaaleaacaWGRbaabeaakiaacIcacqGHiiIZtuuDJXwAKzKC HTgD1jharyqr1ngBPrgigjxyRrxDYbacfeGae8xaWlKaaiykaaaa@4946@ are especially important.

E ( b i j k | i , j ) = B k B b i j k p ( B k ) = a i j ,    i = 1 , , R ,  and    j = 1 , , C ( 3.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yramaabmqabaWaaqGaaeaacaWGIbWaaSbaaSqaaiaadMgacaWGQbGa am4AaaqabaaakiaawIa7aiaadMgacaGGSaGaamOAaaGaayjkaiaawM caaiabg2da9maaqafabaGaamOyamaaBaaaleaacaWGPbGaamOAaiaa dUgaaeqaaOGaaGjcVlaadchadaqadeqaaiaadkeadaWgaaWcbaGaam 4AaaqabaaakiaawIcacaGLPaaacqGH9aqpcaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaacYcacaqGGaGaaeiiaiaaykW7caWGPbGaey ypa0JaaGymaiaacYcacqWIMaYscaGGSaGaamOuaiaacYcacaqGGaGa aeyyaiaab6gacaqGKbGaaeiiaiaabccacaqGGaGaamOAaiabg2da9i aaigdacaGGSaGaeSOjGSKaaiilaiaadoeaaSqaaiaadkeadaWgaaad baGaam4AaaqabaWccqGHiiIZtuuDJXwAKzKCHTgD1jharyqr1ngBPr gigjxyRrxDYbacfeGae8xaWleabeqdcqGHris5aOGaaGzbVlaaywW7 caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaI1aGaaiykaa aa@8430@

and

B k B p ( B k ) = 1 , ( 3.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaa buaeaacaWGWbWaaeWabeaacaWGcbWaaSbaaSqaaiaadUgaaeqaaaGc caGLOaGaayzkaaGaeyypa0JaaGymaaWcbaGaamOqamaaBaaameaaca WGRbaabeaaliabgIGioprr1ngBPrMrYf2A0vNCaeHbfv3ySLgzGyKC HTgD1jhaiuqacqWFbaVqaeqaniabggHiLdGccaGGSaGaaGzbVlaayw W7caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOl aiaaiAdacaGGPaaaaa@5F63@

where p ( B k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iCamaabmqabaGaamOqamaaBaaaleaacaWGRbaabeaaaOGaayjkaiaa wMcaaaaa@3C6E@ , which depends on a specified algorithm for solving the controlled selection problem, is the selection probability of the array B k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam OqamaaBaaaleaacaWGRbaabeaaaaa@39E5@ and p ( B k ) 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam iCamaabmqabaWaiaiovababKaG4SqaiaiocGaG4m4Aaaqajaiojq2a G9FaiaiocGaGGmOqaaaaaOGaayjkaiaawMcaaiabgwMiZkaaicdaaa a@45F2@ .

Note that (3.5) and (3.6) will define a rigorous probability sampling method when randomly selecting any array in B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv 3ySLgzgjxyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbbiab=fa8cbaa @447C@ . Also, note that since i = 1 R j = 1 C E ( b i j k | i , j ) = a .. B k B p ( B k ) = a .. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaa bmaeaadaaeWaqaaiaadweadaqadeqaamaaeiaabaGaamOyamaaBaaa leaacaWGPbGaamOAaiaadUgaaeqaaaGccaGLiWoacaWGPbGaaiilai aadQgaaiaawIcacaGLPaaaaSqaaiaadQgacqGH9aqpcaaIXaaabaGa am4qaaqdcqGHris5aaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGsb aaniabggHiLdGccqGH9aqpcaWGHbWaaSbaaSqaaiaac6cacaGGUaaa beaakmaaqababaGaamiCamaabmqabaGaamOqamaaBaaaleaacaWGRb aabeaaaOGaayjkaiaawMcaaaWcbaGaamOqamaaBaaameaacaWGRbaa beaaliabgIGioprr1ngBPrMrYf2A0vNCaeHbfv3ySLgzGyKCHTgD1j haiuqacqWFbaVqaeqaniabggHiLdGccqGH9aqpcaWGHbWaaSbaaSqa aiaac6cacaGGUaaabeaaaaa@6AB7@ , (3.5) implies (3.6) for any controlled selection problem such as those described in Problems 2.1 through 2.4. In addition, as an illustration, when applied to Problem 2.3, where a i j = n X i j q / X q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yyamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGUbWaaSGb aeaacaWGybWaaSbaaSqaaiaadMgacaWGQbGaamyCaaqabaaakeaaca WGybWaaSbaaSqaaiaadghaaeqaaaaaaaa@42EF@ , (3.5) yields

E ( b i j k / X i j q | i , j ) = a i j / X i j q = n / X q , ( 3.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam yramaabmaabaWaaqGaaeaadaWcgaqaaiaadkgadaWgaaWcbaGaamyA aiaadQgacaWGRbaabeaaaOqaaiaadIfadaWgaaWcbaGaamyAaiaadQ gacaWGXbaabeaaaaaakiaawIa7aiaadMgacaGGSaGaamOAaaGaayjk aiaawMcaaiabg2da9maalyaabaGaamyyamaaBaaaleaacaWGPbGaam OAaaqabaaakeaacaWGybWaaSbaaSqaaiaadMgacaWGQbGaamyCaaqa baaaaOGaeyypa0ZaaSGbaeaacaWGUbaabaGaamiwamaaBaaaleaaca WGXbaabeaaaaGccaGGSaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7 caaMf8UaaiikaiaaiodacaGGUaGaaG4naiaacMcaaaa@5FFE@

which indicates the equal allocation for each cell.

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