Imperfect Reworking Process Consideration in Integrated Inventory Model under Permissible Delay in Payments

Algorithm 1

Step 1: set 𝑛=1.

Step 2: determine the 𝑝₀ by solving (4.16).

Step 3: if there exists a 𝑝₁ where 𝑝₁≤𝑝₀, and satisfies both the first-order condition as in (4.20) and

     the second-order condition as in (4.21), then we compute 𝐿₁(𝑝₁) by (4.6) and

     EJTP₁(𝑛,𝑝₁,𝐿₁(𝑝₁)) by (4.8). If not, we set EJTP₁(𝑛,𝑝₁,𝐿₁(𝑝₁))=0.

Step 4: if there exists a 𝑝₂ where 𝑝₂≥𝑝₀, and satisfies both the first-order condition as in (4.22) and

     the second-order condition as in (4.23), then we compute 𝐿₂(𝑝₂) by (4.11) and

     EJTP₂(𝑛,𝑝₂,𝐿₂(𝑝₂)) by (4.13). If not, we set EJTP₂(𝑛,𝑝₂,𝐿₂(𝑝₂))=0.

Step 5: if EJTP₁(𝑛,𝑝₁,𝐿₁(𝑝₁))≥EJTP₂(𝑛,𝑝₂,𝐿₂(𝑝₂)). Set EJTP(𝑛,𝑝[𝑛],𝐿[𝑛])=EJTP₁(𝑛,𝑝₁,𝐿₁(𝑝₁)),

     then (𝑝[𝑛],𝐿[𝑛]) is an optimal solution for a given 𝑛. If not, EJTP(𝑛,𝑝[𝑛],𝐿[𝑛])=

     EJTP₂(𝑛,𝑝₂,𝐿₂(𝑝₂)).

Step 6: set 𝑛=𝑛+1, repeat steps 2–5 to obtain EJTP(𝑛,𝑝[𝑛],𝐿[𝑛]).

Step 7: if EJTP(𝑛,𝑝[𝑛],𝐿[𝑛])≥EJTP(𝑛−1,𝑝[𝑛−1],𝐿[𝑛−1]), go to step 6. If not, go to step 8

     and stop.

Step 8: set EJTP(𝑛^∗,𝑝^∗,𝐿^∗)=EJTP(𝑛−1,𝑝[𝑛−1],𝐿[𝑛−1]), so (𝑛^∗,𝑝^∗,𝐿^∗) is an optimal solution.

     Consequently, the buyer’s optimal order quantity per order is 𝑄^∗=𝐷(𝑝^∗)𝐿^∗.