our team is working toward creating a decision support software for a firm that manufactures different types of products, each of which has a monthly manufacturing limit$.$ Since the demand for some products changes from time to time, the firm produces a certain type of product, stocks it$,$ and then changes the setup of their machine to manufacture another type of product. A setup cost is incurred during a month in which any units of a product type is produced. In addition, there is a variable cost for every unit produced. When the quantity of a manufactured product is greater than the demand for a particular month, the excess is stored in inventories, which have associated inventory limits and inventory costs. The firm does not want to exceed this limit because they will then need to find a warehouse to store their products, which will be time and money consuming.

The firm therefore wants a module that can optimize their inventory and production levels for the next couple of months, such that the overall cost is minimized. For this, you need a core API that will solve the inventory and manufacturing planning problem and will optimize. Your team has concluded that this problem can be solved by dynamic programming.

The details of the solution steps as a dynamic programming algorithm is given as follows:

Define the problem in stages and states: In this problem, the stages are the months and the states are the starting inventory levels for each month. Thus the firm can enter any month with any inventory level ranging from zero to the inventory limit$.$ Each level is a separate state that affects decisions.

Find a recursive formula: Define a recursive formula that connects the decisions made in the previous stage $($for each state$)$ to the next stage.

Start from the last period and go back to the beginning: Since there is no later stage to effect decisions, it is easy to find optimal decisions at any given state for the last month. For example, at last stage $($month$),$ if your demand is $40$ units and you have $40$ products in your inventory at the start of the month, you manufacture none. However, if you have $35$ units in your inventory at the start of the last month, you need to manufacture $5$ units during that month. Once you calculate states for the last stage, you need to go backwards one month at a time to the beginning by using a recursive formula, which will connect last month$$s decisions to the previous month's. Once you reach the starting month, you will find the optimal solution easily since you would have connected all previous decisions to the states of the first month. This way, you will arrive at the minimum overall cost.

For product A$,$ the demand during the next four months has been forecasted as follows:

Month Units Required

$11$

$23$

$32$

$44$

the start of a month, the firm needs to decide the number of units to be produced in the current month. The following are the costs involved:

Fixed cost: Irrespective of the number of units produced, a fixed cost of $$3$ is incurred for setup $($as long as at least one unit is manufactured$).$

Variable cost: A variable cost of $$1$ is incurred for each unit manufactured.

Inventory cost: An inventory cost of $50$ per unit on hand is incurred at the end of each month.

Further, the manufacturing limit is $5$ units for a month and inventory limit is $4$ units.

To determine a production schedule that will meet all demands on time and will minimize the sum of production and inventory costs during the four months, the following steps are followed:

Stages and states: The stages are $4($since there are $4$ months to plan for$),$ and the states are the possible starting inventory levels, which can be $0,1,2,3,$ or $4($since the inventory limit is $4).$

The recursive formula: Except for the last month, a recursive formula can be defined as follows for month i:

Cost$($Starting inventory$)=$ Cost of Production $+$ Inventory cost$($Ending inventory$)+$ Costi$+1($Ending inventory$)$

Start from the last period and go back to the beginning: For the last month, depending on the starting inventory level, the following costs will be incurred:

Cost$4(0)=3+4$ x$1=7$

Cost$4(1)=3+3$ x$1=6$

Cost$4(2)=3+2$ x$1=5$

Cost$4(3)=3+1$ x$1=4$

Cost$4(4)=0=0$