Inventory rationing policy and backup supply with two classes of customers

they get better results calculation method

we get<disp formula id="pone.0214386.e009">removedGRAPHIC P 1 j=Z lambda 1 (1 + Z lambda 1) j -1 P 1 (0), j=1, 2, ⋯, T,</disp formula><disp formula id="pone.0214386.e010">removedGRAPHIC P 1 j=Z lambda 1 + lambda 2 (1 + Z lambda 1) T (1 + Z lambda 1 + lambda 2) j - T -1 P 1 (0), j=T + 1, ⋯, r,</disp formula><disp formula id="pone.0214386.e011">removedGRAPHIC P 1 j=Z lambda 1 + lambda 2 (1 + Z lambda 1) T (1 + Z lambda 1 + lambda 2) r - T P 1 (0), j=r + 1, ⋯, Q,</disp formula><disp formula id="pone.0214386.e012">removedGRAPHIC P 1 j=Z lambda 1 + lambda 2 [(1 + Z lambda 1) T (1 + Z lambda 1 + lambda 2) r - T - (1 + Z lambda 1) j - Q -1 ] P 1 (0), j=Q + 1, ⋯, Q + T,</disp formula><disp formula id="pone.0214386.e013">removedGRAPHIC P 1 j=Z lambda 1 + lambda 2 (1 + Z lambda 1) T [(1 + Z lambda 1 + lambda 2) r - T - (1 + Z lambda 1 + lambda 2) j - Q - T -1 ] P 1 (0), j=Q + T + 1, ⋯, Q + r,</disp formula><disp formula id="pone.0214386.e014">removedGRAPHIC P 1 (0)=(lambda 1 + lambda 2) Q Z (1 + Z lambda 1) T (1 + Z lambda 1 + lambda 2) r - T + lambda 2 (1 + Z lambda 1) T + lambda 1 ] -1
distributions P 3 j, j=1,⋯, r + Q are given by<disp formula id="pone.0214386.e020">removedGRAPHIC P 3 j=Z lambda 1 (1 + Z lambda 1) j -1 P 3 (0), j=1, 2, ⋯, r</disp formula><disp formula id="pone.0214386.e021">removedGRAPHIC P 3 j=Z lambda 1 + lambda 2 (1 + Z lambda 1) r P 3 (0), j=r + 1, ⋯, Q,</disp formula><disp formula id="pone.0214386.e022">removedGRAPHIC P 3 j=Z lambda 1 + lambda 2 [(1 + Z lambda 1) r - (1 + Z lambda 1) j - Q -1 ] P 3 (0), j=Q + 1, ⋯, Q + r,</disp formula><disp formula id="pone.0214386.e023">removedGRAPHIC P 3 (0)=(lambda 1 + lambda 2) Q Z (1 + Z lambda 1) r + lambda 2 (1 + Z lambda 1) r + lambda 1 ] -1
We have<disp formula id="pone.0214386.e028">removedGRAPHIC I ¯ n=delta 1 ∑ j=1
lambda 2 ∑ j=0 T P 1 j + delta 2 lambda 2 ∑ j=0 T P 2 j + delta 3 lambda 2 ∑ j=0 T P 3
pone.0214386.e043">removedGRAPHIC r - Q<=-1,</disp formula><disp formula id="pone.0214386.e044">removedGRAPHIC 0<T<=Q,</disp formula><disp formula id="pone.0214386.e045">removedGRAPHIC 0