Practical no.
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Aim: To determine the solution of a procurement/ production scheduling model.
Introduction:
The basic EOQ model and its variations assume that the demand rate is known and constant.
When this assumption is relaxed, i.e., when demand varies from period to period, the EOQ
formula no longer ensures a minimum-cost solution.
Let us consider a manufacturer who knows the demand for their product for each period over a
planning horizon. The manufacturer wants to determine how much (if any) to produce at the
beginning of each of the periods. In general, the manufacturer could have two options:
i. The manufacturer could produce the exact number of units required in each period.
However, this could lead to frequent production set-ups resulting in higher costs. Hence,
this type of production schedule might not be efficient.
ii. The manufacturer could produce all the demand in the first period itself and carry the
inventory for the remaining periods. However, this schedule could lead to high inventory
carrying costs.
The best approach would be an intermediate strategy where the demand for a few periods could
be combined for production, thereby decreasing the number of set-ups as well as reducing the
amount of inventory to be carried. One way to solve this problem is to enumerate all the
combinations of production decisions. However, the process would be lengthy and
cumbersome. The Wagner and Whitin algorithm helps us solve this problem by finding the
optimal trade-off between set-up and inventory carrying costs. The objective is to determine the
optimal production schedule over a planning horizon such that the demand is met (without any
shortages) at minimum cost.
Assumptions:
1. The demand is known but varying over time.
2. Production is done at the beginning of the period and demand is met at the end of the
period.
3. The lead time is constant.
4. Shortages are not allowed.
5. All costs are known.
6. Planning horizon is fixed and finite.
Notations used in the Model:
𝑟𝑖 : Requirement or demand in period 𝑖
𝑐: Unit cost
A: Set-up or ordering cost
𝐼𝑐 : Holding or carrying cost per unit per time
𝑐𝑛 : Total cost up to and including 𝑛𝑡ℎ period
Mathematical Model:
Let us assume that the last production is done at the beginning of the 𝑗 𝑡ℎ period (𝑗 = 1,2, … , 𝑛)
The total cost up to and including 𝑛𝑡ℎ period when the last production is done at the beginning of
the 𝑗 𝑡ℎ period is given by:
(𝑗)
𝑐𝑛 = 𝑐𝑗−1 + 𝐴 + 𝑐(𝑟𝑗 + 𝑟𝑗+1 + ⋯ + 𝑟𝑛 ) + 𝐼𝑐 (𝑟𝑗 + 𝑟𝑗+1 + ⋯ + 𝑟𝑛 ) + 𝐼𝑐 (𝑟𝑗+1 + 𝑟𝑗+2 + ⋯ + 𝑟𝑛 )
+ 𝐼𝑐 (𝑟𝑛 )
= 𝑐𝑗−1 + 𝐴 + 𝑐(𝑟𝑗 + 𝑟𝑗+1 + ⋯ + 𝑟𝑛 ) + 𝐼𝑐 [𝑟𝑗 + 2𝑟𝑗+1 + 3𝑟𝑗+2 … + (𝑛 − 𝑗 + 1)𝑟𝑛 ]
The aim is to minimize this cost:
(𝑗)
min 𝑐𝑛 = min{𝑐𝑗−1 + 𝐴 + 𝑐(𝑟𝑗 + 𝑟𝑗+1 + ⋯ + 𝑟𝑛 ) + 𝐼𝑐 [𝑟𝑗 + 2𝑟𝑗+1 + 3𝑟𝑗+2 … + (𝑛 − 𝑗 + 1)𝑟𝑛 ]}
𝑗 𝑗
Question:
A company manufactures a product such that it costs Rs. 50 for each production set-up. The
manufacturing or production cost per unit is Rs. 10 and the monthly holding cost is Rs. 2 per unit.
The monthly demand for the product is given below.
Time 1 2 3 4 5 6 7 8
Demand 10 15 20 22 8 20 28 5
Determine the optimal production schedule to ensure that the demand is met without any
shortages. Also determine the corresponding total cost.
Solution:
Screenshots of the solution from MS Excel have been attached.
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1. Screenshots of the solution from MS Excel.
2. Graph from MS Excel.
