Question

Bizim Mobilya has been in the furniture business since 1950s and has grown into a highly popular brand for the durability of their products. Bizim uses M manufacturing factories for their production and N warehouses to reach the customers in various markets. The vice president of the company responsible for its manufacturing operations wants to identify a production plan for the next T periods for a particular model. Factory i=1,…,M has a total manufacturing capacity of Ci

​ units and can store up to Ii
​ units of finished product in each period. The manager of warehouse j=1,…,N demands at least D jt units of this model in period t= 1,…,T and can store up to S j units of finished product in each period. Unit manufacturing cost of this product in factory i during period t is p it , inventory cost is h t for each item held in factory i at the end of period t and is kjt​ for each item held in warehouse j at the end of period t, and shipping cost is c ij
for each item transported from factory i to warehouse j during period t. Assume that transportation time is negligibly small when compared to the length of a planning period (e.g., periods can be defined as 3-month intervals, while transportation time is typically within a few days), so that items shipped arrive to their destination within the same period. The vice president also wants to avoid wild fluctuations in its production plan, as it results in serious long-term problems caused by hiring and firing workers. For this purpose, he wants to ensure that total production in any two successive periods should not be more than α% apart from each other (e.g., if production in t is 100 units and α=10%, then the production in t+1 should be between 90 and 110). Furthermore, to balance out the operations in each factory, it is required that the total production in any factory throughout the entire planning horizon should be within β% of the average production per factory. Your task is to build a linear optimization model that will identify a minimum-cost production, storage, shipment schedule. Assume fractional units of product are not acceptable. a. Clearly define your decision variables. b. Express the objective function using of the problem data and the decision variables you have listed in part (a).

Answer 1

a. Decision variables:

Let
`\(x_(it)\)` represent the number of units of the particular model produced in factory i during period t,
`\(y_(ijt)\)` represent the number of units shipped from factory i to warehouse j during period t, and
`\(z_(ijt)\)` represent the number of units stored in warehouse j at the end of period t.

b. Objective function:

The objective is to minimize the total cost, which includes manufacturing, inventory, and shipping costs. The objective function can be expressed as:

`\[ \text{Minimize} \sum_(i=1)^(M) \sum_(t=1)^(T) (p_(it)x_(it) + h_(t)\sum_(i=1)^(M) z_(ijt) + k_(jt)\sum_(j=1)^(N) z_(ijt) + c_(ij)y_(ijt)) \]`

Step-by-step explanation:

a. The decision variables are defined to capture the key elements of the problem:
`\(x_(it)\)` for production in factory i,
`\(y_(ijt)\)` for units shipped from factory i to warehouse j, and
`\(z_(ijt)\)` for units stored in warehouse j.

b. The objective function represents the total cost to be minimized. It comprises the manufacturing cost
`(\(p_(it)x_(it)\))`, inventory cost
`(\(h_(t)\sum_(i=1)^(M) z_(ijt) + k_(jt)\sum_(j=1)^(N) z_(ijt)\))`, and shipping cost
`(\(c_(ij)y_(ijt)\))`. The summations ensure the costs are aggregated over factories, warehouses, and periods. Minimizing this objective function will lead to an optimal production, storage, and shipment schedule for the given constraints. The linear optimization model aims to find the most cost-effective solution while satisfying the specified production constraints and avoiding fluctuations.